ITP001 Axioms: ITP149_7.ax
%------------------------------------------------------------------------------
% File : ITP149_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : real_topology.ax [Gau19]
% : HL4149_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 2347 ( 540 unt; 302 typ; 0 def)
% Number of atoms : 7721 (2318 equ)
% Maximal formula atoms : 31 ( 3 avg)
% Number of connectives : 6072 ( 396 ~; 116 |;2209 &)
% ( 647 <=>;2704 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 7 avg)
% Maximal term depth : 16 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 227 ( 147 >; 80 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 298 ( 298 usr; 92 con; 0-6 aty)
% Number of variables : 10176 (9566 !; 459 ?;10176 :)
% ( 151 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(tyop_2Ereal__topology_2Enet,type,
tyop_2Ereal__topology_2Enet: $tType > $tType ).
tff(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
tff(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $tType > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2E_2A_2E0,type,
c_2Earithmetic_2E_2A_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2A_2E2,type,
c_2Earithmetic_2E_2A_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2E_2B_2E0,type,
c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2B_2E2,type,
c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Earithmetic_2E_2D_2E0,type,
c_2Earithmetic_2E_2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2D_2E2,type,
c_2Earithmetic_2E_2D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Ereal__topology_2E_2D_2D_3E_2E0,type,
c_2Ereal__topology_2E_2D_2D_3E_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool))) ).
tff(c_2Ereal__topology_2E_2D_2D_3E_2E3,type,
c_2Ereal__topology_2E_2D_2D_3E_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal * tyop_2Ereal__topology_2Enet(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Eiterate_2E_2E_2E_2E0,type,
c_2Eiterate_2E_2E_2E_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))) ).
tff(c_2Eiterate_2E_2E_2E_2E2,type,
c_2Eiterate_2E_2E_2E_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Ereal_2E_2F_2E0,type,
c_2Ereal_2E_2F_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2E_2F_2E2,type,
c_2Ereal_2E_2F_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Eprim__rec_2E_3C_2E0,type,
c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Eprim__rec_2E_3C_2E2,type,
c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3C_3D_2E0,type,
c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3C_3D_2E2,type,
c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3E_3D_2E0,type,
c_2Earithmetic_2E_3E_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3E_3D_2E2,type,
c_2Earithmetic_2E_3E_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_3F_21_2E0,type,
c_2Ebool_2E_3F_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_21_2E1,type,
c_2Ebool_2E_3F_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
tff(c_2Epred__set_2EBIGINTER_2E0,type,
c_2Epred__set_2EBIGINTER_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EBIGINTER_2E1,type,
c_2Epred__set_2EBIGINTER_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EBIGUNION_2E0,type,
c_2Epred__set_2EBIGUNION_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EBIGUNION_2E1,type,
c_2Epred__set_2EBIGUNION_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Earithmetic_2EBIT1_2E0,type,
c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT1_2E1,type,
c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EBIT2_2E0,type,
c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT2_2E1,type,
c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Epred__set_2ECARD_2E0,type,
c_2Epred__set_2ECARD_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).
tff(c_2Epred__set_2ECARD_2E1,type,
c_2Epred__set_2ECARD_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ) ).
tff(c_2Ereal__topology_2ECLOSED__interval_2E0,type,
c_2Ereal__topology_2ECLOSED__interval_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2ECLOSED__interval_2E1,type,
c_2Ereal__topology_2ECLOSED__interval_2E1: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Elist_2ECONS_2E0,type,
c_2Elist_2ECONS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).
tff(c_2Elist_2ECONS_2E2,type,
c_2Elist_2ECONS_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Epred__set_2ECROSS_2E0,type,
c_2Epred__set_2ECROSS_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2ECROSS_2E2,type,
c_2Epred__set_2ECROSS_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool) ) ).
tff(c_2Ereal__topology_2EClosed_2E0,type,
c_2Ereal__topology_2EClosed_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2EClosed_2E1,type,
c_2Ereal__topology_2EClosed_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2EDELETE_2E0,type,
c_2Epred__set_2EDELETE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EDELETE_2E2,type,
c_2Epred__set_2EDELETE_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * A_27a ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EDIFF_2E0,type,
c_2Epred__set_2EDIFF_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EDIFF_2E2,type,
c_2Epred__set_2EDIFF_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EDISJOINT_2E0,type,
c_2Epred__set_2EDISJOINT_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EDISJOINT_2E2,type,
c_2Epred__set_2EDISJOINT_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2EDist_2E0,type,
c_2Ereal__topology_2EDist_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2EDist_2E1,type,
c_2Ereal__topology_2EDist_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Epred__set_2EEMPTY_2E0,type,
c_2Epred__set_2EEMPTY_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2EFINITE_2E0,type,
c_2Epred__set_2EFINITE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2EFINITE_2E1,type,
c_2Epred__set_2EFINITE_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EFST_2E0,type,
c_2Epair_2EFST_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).
tff(c_2Epair_2EFST_2E1,type,
c_2Epair_2EFST_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27a ) ).
tff(c_2Epred__set_2EGSPEC_2E0,type,
c_2Epred__set_2EGSPEC_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EGSPEC_2E1,type,
c_2Epred__set_2EGSPEC_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Ecardinal_2EHAS__SIZE_2E0,type,
c_2Ecardinal_2EHAS__SIZE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Ecardinal_2EHAS__SIZE_2E2,type,
c_2Ecardinal_2EHAS__SIZE_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ) ).
tff(c_2Elist_2EHD_2E0,type,
c_2Elist_2EHD_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a) ).
tff(c_2Elist_2EHD_2E1,type,
c_2Elist_2EHD_2E1:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > A_27a ) ).
tff(c_2Ecombin_2EI_2E0,type,
c_2Ecombin_2EI_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(c_2Epred__set_2EIMAGE_2E0,type,
c_2Epred__set_2EIMAGE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EIMAGE_2E2,type,
c_2Epred__set_2EIMAGE_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) ).
tff(c_2Ebool_2EIN_2E0,type,
c_2Ebool_2EIN_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2EIN_2E2,type,
c_2Ebool_2EIN_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epred__set_2EINSERT_2E0,type,
c_2Epred__set_2EINSERT_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EINSERT_2E2,type,
c_2Epred__set_2EINSERT_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EINTER_2E0,type,
c_2Epred__set_2EINTER_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EINTER_2E2,type,
c_2Epred__set_2EINTER_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Earithmetic_2EMAX_2E0,type,
c_2Earithmetic_2EMAX_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EMAX_2E2,type,
c_2Earithmetic_2EMAX_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Elist_2ENIL_2E0,type,
c_2Elist_2ENIL_2E0:
!>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).
tff(c_2Earithmetic_2ENUMERAL_2E0,type,
c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2ENUMERAL_2E1,type,
c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ereal__topology_2EOPEN__interval_2E0,type,
c_2Ereal__topology_2EOPEN__interval_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2EOPEN__interval_2E1,type,
c_2Ereal__topology_2EOPEN__interval_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2EOpen_2E0,type,
c_2Ereal__topology_2EOpen_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2EOpen_2E1,type,
c_2Ereal__topology_2EOpen_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Epair_2ESND_2E0,type,
c_2Epair_2ESND_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b) ).
tff(c_2Epair_2ESND_2E1,type,
c_2Epair_2ESND_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27b ) ).
tff(c_2Epred__set_2ESUBSET_2E0,type,
c_2Epred__set_2ESUBSET_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2ESUBSET_2E2,type,
c_2Epred__set_2ESUBSET_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Eiterate_2ESum_2E0,type,
c_2Eiterate_2ESum_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Eiterate_2ESum_2E2,type,
c_2Eiterate_2ESum_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EUNCURRY_2E0,type,
c_2Epair_2EUNCURRY_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)) ).
tff(c_2Epair_2EUNCURRY_2E1,type,
c_2Epair_2EUNCURRY_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) ) ).
tff(c_2Epair_2EUNCURRY_2E2,type,
c_2Epair_2EUNCURRY_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) * tyop_2Epair_2Eprod(A_27a,A_27b) ) > A_27c ) ).
tff(c_2Epred__set_2EUNION_2E0,type,
c_2Epred__set_2EUNION_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EUNION_2E2,type,
c_2Epred__set_2EUNION_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EUNIV_2E0,type,
c_2Epred__set_2EUNIV_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EZERO_2E0,type,
c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Eabs_2E0,type,
c_2Ereal_2Eabs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Eabs_2E1,type,
c_2Ereal_2Eabs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Eat_2E0,type,
c_2Ereal__topology_2Eat_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal__topology_2Eat_2E1,type,
c_2Ereal__topology_2Eat_2E1: tyop_2Erealax_2Ereal > tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Eat__infinity_2E0,type,
c_2Ereal__topology_2Eat__infinity_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Eat__neginfinity_2E0,type,
c_2Ereal__topology_2Eat__neginfinity_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Eat__posinfinity_2E0,type,
c_2Ereal__topology_2Eat__posinfinity_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Eball_2E0,type,
c_2Ereal__topology_2Eball_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Eball_2E1,type,
c_2Ereal__topology_2Eball_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ebetween_2E0,type,
c_2Ereal__topology_2Ebetween_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Ebetween_2E2,type,
c_2Ereal__topology_2Ebetween_2E2: ( tyop_2Erealax_2Ereal * tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ebilinear_2E0,type,
c_2Ereal__topology_2Ebilinear_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ebilinear_2E1,type,
c_2Ereal__topology_2Ebilinear_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ebounded__def_2E0,type,
c_2Ereal__topology_2Ebounded__def_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ebounded__def_2E1,type,
c_2Ereal__topology_2Ebounded__def_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ecardinal_2Ecardeq_2E0,type,
c_2Ecardinal_2Ecardeq_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ecardinal_2Ecardeq_2E2,type,
c_2Ecardinal_2Ecardeq_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Ecauchy_2E0,type,
c_2Ereal__topology_2Ecauchy_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ecauchy_2E1,type,
c_2Ereal__topology_2Ecauchy_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ecball_2E0,type,
c_2Ereal__topology_2Ecball_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Ecball_2E1,type,
c_2Ereal__topology_2Ecball_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Etopology_2Eclosed__in_2E0,type,
c_2Etopology_2Eclosed__in_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Etopology_2Eclosed__in_2E2,type,
c_2Etopology_2Eclosed__in_2E2:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology(A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Eclosed__segment_2E0,type,
c_2Ereal__topology_2Eclosed__segment_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Eclosed__segment_2E1,type,
c_2Ereal__topology_2Eclosed__segment_2E1: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Eclosest__point_2E0,type,
c_2Ereal__topology_2Eclosest__point_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal__topology_2Eclosest__point_2E2,type,
c_2Ereal__topology_2Eclosest__point_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Eclosure_2E0,type,
c_2Ereal__topology_2Eclosure_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Eclosure_2E1,type,
c_2Ereal__topology_2Eclosure_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ecollinear_2E0,type,
c_2Ereal__topology_2Ecollinear_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ecollinear_2E1,type,
c_2Ereal__topology_2Ecollinear_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ecompact_2E0,type,
c_2Ereal__topology_2Ecompact_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ecompact_2E1,type,
c_2Ereal__topology_2Ecompact_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ecomplete_2E0,type,
c_2Ereal__topology_2Ecomplete_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Ecomplete_2E1,type,
c_2Ereal__topology_2Ecomplete_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ecomponents_2E0,type,
c_2Ereal__topology_2Ecomponents_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Ecomponents_2E1,type,
c_2Ereal__topology_2Ecomponents_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Econdensation__point__of_2E0,type,
c_2Ereal__topology_2Econdensation__point__of_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Econdensation__point__of_2E2,type,
c_2Ereal__topology_2Econdensation__point__of_2E2: ( tyop_2Erealax_2Ereal * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Econnected_2E0,type,
c_2Ereal__topology_2Econnected_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Econnected_2E1,type,
c_2Ereal__topology_2Econnected_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Econnected__component_2E0,type,
c_2Ereal__topology_2Econnected__component_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) ).
tff(c_2Ereal__topology_2Econnected__component_2E2,type,
c_2Ereal__topology_2Econnected__component_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Econnected__component_2E3,type,
c_2Ereal__topology_2Econnected__component_2E3: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) * tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Econtinuous_2E0,type,
c_2Ereal__topology_2Econtinuous_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Econtinuous_2E2,type,
c_2Ereal__topology_2Econtinuous_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) * tyop_2Ereal__topology_2Enet(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Econtinuous__on_2E0,type,
c_2Ereal__topology_2Econtinuous__on_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Econtinuous__on_2E2,type,
c_2Ereal__topology_2Econtinuous__on_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2Ecountable_2E0,type,
c_2Epred__set_2Ecountable_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2Ecountable_2E1,type,
c_2Epred__set_2Ecountable_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Edependent_2E0,type,
c_2Ereal__topology_2Edependent_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Edependent_2E1,type,
c_2Ereal__topology_2Edependent_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ediameter_2E0,type,
c_2Ereal__topology_2Ediameter_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Ediameter_2E1,type,
c_2Ereal__topology_2Ediameter_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Edim_2E0,type,
c_2Ereal__topology_2Edim_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).
tff(c_2Ereal__topology_2Edim_2E1,type,
c_2Ereal__topology_2Edim_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ).
tff(c_2Ereal__topology_2Eeuclidean_2E0,type,
c_2Ereal__topology_2Eeuclidean_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Eeventually_2E0,type,
c_2Ereal__topology_2Eeventually_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Eeventually_2E2,type,
c_2Ereal__topology_2Eeventually_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Ereal__topology_2Enet(A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Efrom_2E0,type,
c_2Ereal__topology_2Efrom_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Efrom_2E1,type,
c_2Ereal__topology_2Efrom_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Efrontier_2E0,type,
c_2Ereal__topology_2Efrontier_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Efrontier_2E1,type,
c_2Ereal__topology_2Efrontier_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Efsigma_2E0,type,
c_2Ereal__topology_2Efsigma_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Efsigma_2E1,type,
c_2Ereal__topology_2Efsigma_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Egdelta_2E0,type,
c_2Ereal__topology_2Egdelta_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Egdelta_2E1,type,
c_2Ereal__topology_2Egdelta_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ehausdist_2E0,type,
c_2Ereal__topology_2Ehausdist_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Ehausdist_2E1,type,
c_2Ereal__topology_2Ehausdist_2E1: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Ehomeomorphic_2E0,type,
c_2Ereal__topology_2Ehomeomorphic_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Ehomeomorphic_2E2,type,
c_2Ereal__topology_2Ehomeomorphic_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ehomeomorphism_2E0,type,
c_2Ereal__topology_2Ehomeomorphism_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Ehomeomorphism_2E2,type,
c_2Ereal__topology_2Ehomeomorphism_2E2: ( tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) * tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ) > tyop_2Emin_2Ebool ).
tff(c_2Etopology_2Ehull_2E0,type,
c_2Etopology_2Ehull_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Etopology_2Ehull_2E2,type,
c_2Etopology_2Ehull_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Ereal__topology_2Ein__direction_2E0,type,
c_2Ereal__topology_2Ein__direction_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal))) ).
tff(c_2Ereal__topology_2Ein__direction_2E2,type,
c_2Ereal__topology_2Ein__direction_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Eindependent_2E0,type,
c_2Ereal__topology_2Eindependent_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Eindependent_2E1,type,
c_2Ereal__topology_2Eindependent_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Eiterate_2Einf_2E0,type,
c_2Eiterate_2Einf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Eiterate_2Einf_2E1,type,
c_2Eiterate_2Einf_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Einfsum_2E0,type,
c_2Ereal__topology_2Einfsum_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal__topology_2Einfsum_2E2,type,
c_2Ereal__topology_2Einfsum_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Einterior_2E0,type,
c_2Ereal__topology_2Einterior_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Einterior_2E1,type,
c_2Ereal__topology_2Einterior_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Erealax_2Einv_2E0,type,
c_2Erealax_2Einv_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Einv_2E1,type,
c_2Erealax_2Einv_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Eis__interval_2E0,type,
c_2Ereal__topology_2Eis__interval_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Eis__interval_2E1,type,
c_2Ereal__topology_2Eis__interval_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Eisnet_2E0,type,
c_2Ereal__topology_2Eisnet_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Eisnet_2E1,type,
c_2Ereal__topology_2Eisnet_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Etopology_2Eistopology_2E0,type,
c_2Etopology_2Eistopology_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Etopology_2Eistopology_2E1,type,
c_2Etopology_2Eistopology_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Elim__def_2E0,type,
c_2Ereal__topology_2Elim__def_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal__topology_2Elim__def_2E2,type,
c_2Ereal__topology_2Elim__def_2E2:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet(A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ) ).
tff(c_2Ereal__topology_2Elimit__point__of_2E0,type,
c_2Ereal__topology_2Elimit__point__of_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Elimit__point__of_2E2,type,
c_2Ereal__topology_2Elimit__point__of_2E2: ( tyop_2Erealax_2Ereal * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Elinear_2E0,type,
c_2Ereal__topology_2Elinear_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Elinear_2E1,type,
c_2Ereal__topology_2Elinear_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Elocally_2E0,type,
c_2Ereal__topology_2Elocally_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Elocally_2E2,type,
c_2Ereal__topology_2Elocally_2E2: ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Emax_2E0,type,
c_2Ereal_2Emax_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Emax_2E2,type,
c_2Ereal_2Emax_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Emidpoint_2E0,type,
c_2Ereal__topology_2Emidpoint_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Emidpoint_2E1,type,
c_2Ereal__topology_2Emidpoint_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Emin_2E0,type,
c_2Ereal_2Emin_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Emin_2E2,type,
c_2Ereal_2Emin_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Emk__net_2E0,type,
c_2Ereal__topology_2Emk__net_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(A_27a)) ).
tff(c_2Ereal__topology_2Emk__net_2E1,type,
c_2Ereal__topology_2Emk__net_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Ereal__topology_2Enet(A_27a) ) ).
tff(c_2Ereal__topology_2Enetlimit_2E0,type,
c_2Ereal__topology_2Enetlimit_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),A_27a) ).
tff(c_2Ereal__topology_2Enetlimit_2E1,type,
c_2Ereal__topology_2Enetlimit_2E1:
!>[A_27a: $tType] : ( tyop_2Ereal__topology_2Enet(A_27a) > A_27a ) ).
tff(c_2Ereal__topology_2Enetord_2E0,type,
c_2Ereal__topology_2Enetord_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Ereal__topology_2Enetord_2E1,type,
c_2Ereal__topology_2Enetord_2E1:
!>[A_27a: $tType] : ( tyop_2Ereal__topology_2Enet(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ) ).
tff(c_2Ereal__topology_2Enetord_2E3,type,
c_2Ereal__topology_2Enetord_2E3:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet(A_27a) * A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ecombin_2Eo_2E0,type,
c_2Ecombin_2Eo_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Ecombin_2Eo_2E2,type,
c_2Ecombin_2Eo_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
tff(c_2Ecombin_2Eo_2E3,type,
c_2Ecombin_2Eo_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) * A_27a ) > A_27b ) ).
tff(c_2Etopology_2Eopen__in_2E0,type,
c_2Etopology_2Eopen__in_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Etopology_2Eopen__in_2E2,type,
c_2Etopology_2Eopen__in_2E2:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology(A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Eopen__segment_2E0,type,
c_2Ereal__topology_2Eopen__segment_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Eopen__segment_2E1,type,
c_2Ereal__topology_2Eopen__segment_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Epairwise_2E0,type,
c_2Ereal__topology_2Epairwise_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Epairwise_2E2,type,
c_2Ereal__topology_2Epairwise_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Epermutes_2E0,type,
c_2Ereal__topology_2Epermutes_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Epermutes_2E2,type,
c_2Ereal__topology_2Epermutes_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal_2Epow_2E0,type,
c_2Ereal_2Epow_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Epow_2E2,type,
c_2Ereal_2Epow_2E2: ( tyop_2Erealax_2Ereal * tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).
tff(c_2Eproduct_2Eproduct_2E0,type,
c_2Eproduct_2Eproduct_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Eproduct_2Eproduct_2E2,type,
c_2Eproduct_2Eproduct_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ) ).
tff(c_2Erealax_2Ereal__add_2E0,type,
c_2Erealax_2Ereal__add_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__add_2E2,type,
c_2Erealax_2Ereal__add_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__ge_2E0,type,
c_2Ereal_2Ereal__ge_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__ge_2E2,type,
c_2Ereal_2Ereal__ge_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__gt_2E0,type,
c_2Ereal_2Ereal__gt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__gt_2E2,type,
c_2Ereal_2Ereal__gt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Ereal__lt_2E0,type,
c_2Erealax_2Ereal__lt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Ereal__lt_2E2,type,
c_2Erealax_2Ereal__lt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__lte_2E0,type,
c_2Ereal_2Ereal__lte_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__lte_2E2,type,
c_2Ereal_2Ereal__lte_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Ereal__mul_2E0,type,
c_2Erealax_2Ereal__mul_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__mul_2E2,type,
c_2Erealax_2Ereal__mul_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__neg_2E0,type,
c_2Erealax_2Ereal__neg_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__neg_2E1,type,
c_2Erealax_2Ereal__neg_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__of__num_2E0,type,
c_2Ereal_2Ereal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Ereal__of__num_2E1,type,
c_2Ereal_2Ereal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__sub_2E0,type,
c_2Ereal_2Ereal__sub_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Ereal__sub_2E2,type,
c_2Ereal_2Ereal__sub_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Esequentially_2E0,type,
c_2Ereal__topology_2Esequentially_2E0: tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum) ).
tff(c_2Ereal__topology_2Esetdist_2E0,type,
c_2Ereal__topology_2Esetdist_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal__topology_2Esetdist_2E1,type,
c_2Ereal__topology_2Esetdist_2E1: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal__topology_2Espan_2E0,type,
c_2Ereal__topology_2Espan_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Espan_2E1,type,
c_2Ereal__topology_2Espan_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Esphere_2E0,type,
c_2Ereal__topology_2Esphere_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Esphere_2E1,type,
c_2Ereal__topology_2Esphere_2E1: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Esubspace_2E0,type,
c_2Ereal__topology_2Esubspace_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Esubspace_2E1,type,
c_2Ereal__topology_2Esubspace_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Esubtopology_2E0,type,
c_2Ereal__topology_2Esubtopology_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a))) ).
tff(c_2Ereal__topology_2Esubtopology_2E2,type,
c_2Ereal__topology_2Esubtopology_2E2:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology(A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Etopology_2Etopology(A_27a) ) ).
tff(c_2Ereal__topology_2Esummable_2E0,type,
c_2Ereal__topology_2Esummable_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Esummable_2E2,type,
c_2Ereal__topology_2Esummable_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Esums_2E0,type,
c_2Ereal__topology_2Esums_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))) ).
tff(c_2Ereal__topology_2Esums_2E3,type,
c_2Ereal__topology_2Esums_2E3: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Esup_2E0,type,
c_2Ereal_2Esup_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Esup_2E1,type,
c_2Ereal_2Esup_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
tff(c_2Etopology_2Etopology_2E0,type,
c_2Etopology_2Etopology_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a)) ).
tff(c_2Etopology_2Etopology_2E1,type,
c_2Etopology_2Etopology_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Etopology_2Etopology(A_27a) ) ).
tff(c_2Etopology_2Etopspace_2E0,type,
c_2Etopology_2Etopspace_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Etopology_2Etopspace_2E1,type,
c_2Etopology_2Etopspace_2E1:
!>[A_27a: $tType] : ( tyop_2Etopology_2Etopology(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Ereal__topology_2Etrivial__limit_2E0,type,
c_2Ereal__topology_2Etrivial__limit_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool) ).
tff(c_2Ereal__topology_2Etrivial__limit_2E1,type,
c_2Ereal__topology_2Etrivial__limit_2E1:
!>[A_27a: $tType] : ( tyop_2Ereal__topology_2Enet(A_27a) > tyop_2Emin_2Ebool ) ).
tff(c_2Ereal__topology_2Euniformly__continuous__on_2E0,type,
c_2Ereal__topology_2Euniformly__continuous__on_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ereal__topology_2Euniformly__continuous__on_2E2,type,
c_2Ereal__topology_2Euniformly__continuous__on_2E2: ( tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) * tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal__topology_2Ewithin_2E0,type,
c_2Ereal__topology_2Ewithin_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Ereal__topology_2Enet(A_27a))) ).
tff(c_2Ereal__topology_2Ewithin_2E2,type,
c_2Ereal__topology_2Ewithin_2E2:
!>[A_27a: $tType] : ( ( tyop_2Ereal__topology_2Enet(A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Ereal__topology_2Enet(A_27a) ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2A_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2A_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(A_27a,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(A_27b,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27e_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27e: $tType,X0_2E0: A_27e,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(A_27e,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(A_27e,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27i_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27i: $tType,X0_2E0: A_27i,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(A_27i,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool),app_2E2(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(A_27i,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27e_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27i_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Epair_2E_2C_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2E_2D_2D_3E_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool)),c_2Ereal__topology_2E_2D_2D_3E_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2E_2D_2D_3E_2E3_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Ereal__topology_2Enet(A_27b)] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27b),tyop_2Emin_2Ebool)),c_2Ereal__topology_2E_2D_2D_3E_2E0(A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2E_2D_2D_3E_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum)] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),c_2Ereal__topology_2E_2D_2D_3E_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2E_2D_2D_3E_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),c_2Ereal__topology_2E_2D_2D_3E_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2E_2E_2E_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Eiterate_2E_2E_2E_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2E_2F_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2E_2F_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3E_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_3D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a,c_2Emin_2E_40_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Emin_2E_40_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Emin_2E_40_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGINTER_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGINTER_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGINTER_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGINTER_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGINTER_2E1_2Emono_2EA_27e,axiom,
! [A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(A_27e,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGINTER_2E0(A_27e),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGINTER_2E1_2Emono_2EA_27i,axiom,
! [A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(A_27i,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGINTER_2E0(A_27i),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGINTER_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGINTER_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGUNION_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGUNION_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGUNION_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGUNION_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGUNION_2E1_2Emono_2EA_27e,axiom,
! [A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27e,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGUNION_2E0(A_27e),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGUNION_2E1_2Emono_2EA_27i,axiom,
! [A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27i,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGUNION_2E0(A_27i),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EBIGUNION_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGUNION_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2ECLOSED__interval_2E1,axiom,
! [X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2ECLOSED__interval_2E1(X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2ECLOSED__interval_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool,X2_2E0: tyop_2Emin_2Ebool] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Ebool),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ebool_2ECOND_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),c_2Elist_2ECONS_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ECROSS_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epred__set_2ECROSS_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2EClosed_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EClosed_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2EClosed_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDELETE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EDISJOINT_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2EDist_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2EDist_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ereal__topology_2EDist_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EFST_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a,c_2Epair_2EFST_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Epair_2EFST_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Epair_2EFST_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27a_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27a,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Erealax_2Ereal,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27e_20mono_2EA_27e,axiom,
! [A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27e,A_27e,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27e,A_27e),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27i_20mono_2EA_27i,axiom,
! [A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27i,A_27i,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27i,A_27i),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27d_29,axiom,
! [A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27e_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27d_29,axiom,
! [A_27c: $tType,A_27d: $tType,A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28A_27e_2Ctyop_2Epair_2Eprod_28A_27f_2CA_27g_29_29,axiom,
! [A_27b: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g))),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27i_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Epair_2Eprod_28A_27f_2Ctyop_2Epair_2Eprod_28A_27g_2CA_27h_29_29,axiom,
! [A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h))),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Epair_2Eprod_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecardinal_2EHAS__SIZE_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Ecardinal_2EHAS__SIZE_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2EHD_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Elist_2EHD_2E1(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Elist_2EHD_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27a_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27a,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27b,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27e,axiom,
! [A_27e: $tType,X0_2E0: A_27e,X1_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27e,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27e),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27i,axiom,
! [A_27i: $tType,X0_2E0: A_27i,X1_2E0: tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27i,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27i),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Erealax_2Ereal_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2EMAX_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMAX_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMAX_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2EOPEN__interval_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2EOPEN__interval_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2EOPEN__interval_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2EOpen_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EOpen_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2EOpen_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2ESND_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,c_2Epair_2ESND_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Epair_2ESND_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Epair_2ESND_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29_29_20mono_2Etyop_2Ereal__topology_2Enet_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2EUNCURRY_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),X1_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27c_20mono_2EA_27d_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27c_20mono_2EA_27d_20mono_2Etyop_2Epair_2Eprod_28tyop_2Emin_2Efun_28A_27e_2Ctyop_2Emin_2Ebool_29_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27c: $tType,A_27d: $tType,A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27c,A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27e_20mono_2Etyop_2Epair_2Eprod_28A_27f_2CA_27g_29_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27b: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27f_20mono_2EA_27g_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27b: $tType,A_27f: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27f_20mono_2Etyop_2Epair_2Eprod_28A_27g_2CA_27h_29_20mono_2Etyop_2Epair_2Eprod_28tyop_2Emin_2Efun_28A_27i_2Ctyop_2Emin_2Ebool_29_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h)),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27g_20mono_2EA_27h_20mono_2Etyop_2Epair_2Eprod_28tyop_2Emin_2Efun_28A_27i_2Ctyop_2Emin_2Ebool_29_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27g: $tType,A_27h: $tType,A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27g,A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27g,A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Epair_2Eprod_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ereal_2Eabs_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eat_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eat_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Eat_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eball_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Eball_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eball_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Ebetween_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Ebetween_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Ereal__topology_2Ebetween_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ebilinear_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2Ebilinear_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,c_2Ereal__topology_2Ebilinear_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ebounded__def_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ebounded__def_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Ebounded__def_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecardinal_2Ecardeq_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ecardinal_2Ecardeq_2E0(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecardinal_2Ecardeq_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ecardinal_2Ecardeq_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ecauchy_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Ecauchy_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Ereal__topology_2Ecauchy_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ecball_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Ecball_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecball_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Eclosed__in_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eclosed__in_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Etopology_2Eclosed__in_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Eclosed__in_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Etopology_2Eclosed__in_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eclosed__segment_2E1,axiom,
! [X0_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2Eclosed__segment_2E1(X0_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eclosed__segment_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Eclosest__point_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosest__point_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal__topology_2Eclosest__point_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eclosure_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eclosure_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ecollinear_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecollinear_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Ecollinear_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ecompact_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecompact_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Ecompact_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ecomplete_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecomplete_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Ecomplete_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ecomponents_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecomponents_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecomponents_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Econdensation__point__of_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Econdensation__point__of_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Econdensation__point__of_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Econnected_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Econnected_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Econnected_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Econnected__component_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Econnected__component_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2Econnected__component_2E3,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Econnected__component_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Econtinuous_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Econtinuous_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool),c_2Ereal__topology_2Econtinuous_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Econtinuous_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Ereal__topology_2Econtinuous_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Econtinuous__on_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Econtinuous__on_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Econtinuous__on_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecountable_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2Ecountable_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Edependent_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Edependent_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Edependent_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ediameter_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ediameter_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ediameter_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Edim_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Edim_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Ereal__topology_2Edim_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Eeventually_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Eeventually_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool),c_2Ereal__topology_2Eeventually_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Eeventually_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum)] : ( c_2Ereal__topology_2Eeventually_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Ebool),c_2Ereal__topology_2Eeventually_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Eeventually_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Ereal__topology_2Eeventually_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Efrom_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Efrom_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Ereal__topology_2Efrom_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Efrontier_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efrontier_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Efrontier_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Efsigma_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efsigma_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Efsigma_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Egdelta_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Egdelta_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Egdelta_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Ehausdist_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Ehausdist_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ehausdist_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Ehomeomorphic_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphic_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Ehomeomorphic_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Ehomeomorphism_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2Ehomeomorphism_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),c_2Ereal__topology_2Ehomeomorphism_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Ehull_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Ehull_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Etopology_2Ehull_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Ein__direction_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ein__direction_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)),c_2Ereal__topology_2Ein__direction_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eindependent_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eindependent_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Eindependent_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eiterate_2Einf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Einf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Eiterate_2Einf_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Einfsum_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Einfsum_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Ereal__topology_2Einfsum_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Einterior_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Einterior_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Einv_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eis__interval_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eis__interval_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Eis__interval_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eisnet_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Eisnet_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Ereal__topology_2Eisnet_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Eistopology_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eistopology_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Etopology_2Eistopology_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Elim__def_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Elim__def_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Ereal__topology_2Elim__def_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Elim__def_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Elim__def_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Ereal__topology_2Elim__def_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Elimit__point__of_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Elimit__point__of_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Elimit__point__of_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Elinear_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Elinear_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Ereal__topology_2Elinear_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Elocally_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Elocally_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Elocally_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Emax_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Emax_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Emidpoint_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Emidpoint_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Emin_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Emin_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Emk__net_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Emk__net_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(A_27a),c_2Ereal__topology_2Emk__net_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Emk__net_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Emk__net_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(A_27b),c_2Ereal__topology_2Emk__net_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Emk__net_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Emk__net_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),c_2Ereal__topology_2Emk__net_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Emk__net_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Emk__net_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Emk__net_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Enetlimit_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Enetlimit_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27a),A_27a,c_2Ereal__topology_2Enetlimit_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Enetlimit_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Enetlimit_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Ereal__topology_2Enetlimit_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Enetord_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Enetord_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Enetord_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2Enetord_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27a),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Ereal__topology_2Enetord_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Enetord_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Enetord_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27b)] : ( c_2Ereal__topology_2Enetord_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Enetord_2E0(A_27b),X0_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2Enetord_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Enetord_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Enetord_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2Enetord_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Enetord_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Ereal__topology_2Enetord_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ecombin_2Eo_2E3_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Eopen__in_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eopen__in_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Etopology_2Eopen__in_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Eopen__in_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Etopology_2Eopen__in_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Eopen__segment_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Eopen__segment_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eopen__segment_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Epairwise_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Epairwise_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Epairwise_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Epairwise_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Epairwise_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Epairwise_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Epairwise_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Erealax_2Ereal_2Ctyop_2Emin_2Ebool_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Epairwise_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Epairwise_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Epermutes_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Epermutes_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Epermutes_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Epermutes_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Epermutes_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Epermutes_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Epow_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ereal_2Epow_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eproduct_2Eproduct_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Eproduct_2Eproduct_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eproduct_2Eproduct_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eproduct_2Eproduct_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)] : ( c_2Eproduct_2Eproduct_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eproduct_2Eproduct_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__ge_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__ge_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__ge_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__gt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__gt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__gt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__lte_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Ereal__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Ereal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Esetdist_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Esetdist_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esetdist_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Espan_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Espan_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Espan_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Esphere_2E1,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Esphere_2E1(X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Esphere_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Esubspace_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esubspace_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Esubspace_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Esubtopology_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esubtopology_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a),app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a)),c_2Ereal__topology_2Esubtopology_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Esubtopology_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),app_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal)),c_2Ereal__topology_2Esubtopology_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Esummable_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Esummable_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Ereal__topology_2Esummable_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ereal__topology_2Esums_2E3,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esums_2E3(X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ereal__topology_2Esums_2E0,X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Esup_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Esup_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Etopology_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Etopology_2Etopology_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a),c_2Etopology_2Etopology_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Etopology_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Etopology_2Etopology_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),c_2Etopology_2Etopology_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Etopspace_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a)] : ( c_2Etopology_2Etopspace_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Etopology_2Etopspace_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Etopspace_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal)] : ( c_2Etopology_2Etopspace_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Etopology_2Etopspace_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Etrivial__limit_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Ebool,c_2Ereal__topology_2Etrivial__limit_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Etrivial__limit_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum)] : ( c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Enum_2Enum),tyop_2Emin_2Ebool,c_2Ereal__topology_2Etrivial__limit_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal__topology_2Etrivial__limit_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Ereal__topology_2Etrivial__limit_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Euniformly__continuous__on_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Euniformly__continuous__on_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ereal__topology_2Euniformly__continuous__on_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Ewithin_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ereal__topology_2Enet(A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ewithin_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Ereal__topology_2Enet(A_27a),app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Ereal__topology_2Enet(A_27a)),c_2Ereal__topology_2Ewithin_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal__topology_2Ewithin_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),app_2E2(tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal)),c_2Ereal__topology_2Ewithin_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Ereal__topology_2Efsigma,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Efsigma_2E1(V0s_2E0))
<=> ? [V1g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0))
& ! [V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,V1g_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V2c_2E0)) )
& ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1g_2E0) = V0s_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2Egdelta,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Egdelta_2E1(V0s_2E0))
<=> ? [V1g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0))
& ! [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0,V1g_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0)) )
& ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V1g_2E0) = V0s_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2Elocally,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V1s_2E0))
<=> ! [V2w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2w_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2w_2E0)) )
=> ? [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V5v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V4u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4u_2E0,V5v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5v_2E0,V2w_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Ehausdist,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V1t_2E0),V0s_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V6x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V0s_2E0)) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V1t_2E0),V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V1t_2E0),V0s_2E0))
<=> ( ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V1t_2E0),V0s_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V0s_2E0),V1t_2E0))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ? [V4b_2E0: tyop_2Erealax_2Ereal] :
! [V5d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5d_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V1t_2E0),V0s_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V0s_2E0),V1t_2E0)))))
=> p(c_2Ereal_2Ereal__lte_2E2(V5d_2E0,V4b_2E0)) ) ) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1t_2E0),V0s_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V8x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V1t_2E0),V0s_2E0),c_2Ereal_2Esup_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1t_2E0),V0s_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0),V1t_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2Esetdist,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0s_2E0),V1t_2E0))
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0s_2E0),V1t_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eiterate_2Einf_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ) ).
tff(thm_2Ereal__topology_2Eclosest__point,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1a_2E0),V2x_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2x_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3y_2E0)))) ) ) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1a_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Esummable,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,V1f_2E0))
<=> ? [V2l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Esums_2E3(V1f_2E0,V2l_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2Einfsum,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2l_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V0s_2E0),V2l_2E0) = c_2Ereal__topology_2Esums_2E3(V1f_2E0,V2l_2E0,V0s_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Einfsum_2E2(V0s_2E0,V1f_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Esums,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V0f_2E0),V3n_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V2s_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V3n_2E0)),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,V2s_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V0f_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0) ) ) ).
tff(thm_2Ereal__topology_2Ehomeomorphic,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
<=> ? [V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0,V3g_2E0))) ) ).
tff(thm_2Ereal__topology_2Ehomeomorphism,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0,V3g_2E0)))
<=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0)) = V4x_2E0 ) )
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V0s_2E0) = V1t_2E0 )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2f_2E0,V0s_2E0))
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V1t_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V5y_2E0)) = V5y_2E0 ) )
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V1t_2E0) = V0s_2E0 )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V3g_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Eis__interval,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
<=> ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0)) )
=> ( ( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
| ( p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V1a_2E0)) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Emidpoint,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1b_2E0)) ) ).
tff(thm_2Ereal__topology_2Ebetween,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebetween_2E2(V0x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0)))
<=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0x_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2Ecollinear,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecollinear_2E1(V0s_2E0))
<=> ? [V1u_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0)) )
=> ? [V4c_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V3y_2E0) = c_2Erealax_2Ereal__mul_2E2(V4c_2E0,V1u_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2Econdensation__point__of,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V0x_2E0,V1s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2Enetlimit,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0net_2E0),V1a_2E0))
<=> ! [V2x_2E0: A_27a] : ~ p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0net_2E0,V2x_2E0,V1a_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Enetlimit_2E1(A_27a,V0net_2E0) = c_2Emin_2E_40_2E1(A_27a,app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Ebounded__def,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
<=> ? [V1a_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__interval,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0l_2E0),V1x_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Epair_2EFST_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Elist_2EHD_2E1(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0l_2E0)),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Epair_2ESND_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Elist_2EHD_2E1(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0l_2E0)))) ) )
=> ( ! [V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0l_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0l_2E0),V1x_2E0)) )
=> ! [V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2ECLOSED__interval_2E1(V0l_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0l_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__interval,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V2x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V1b_2E0)) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Ediameter,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V0s_2E0))
<=> ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V1x_2E0),V0s_2E0),V2y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0),V1x_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V1x_2E0),V0s_2E0),V2y_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ediameter_2E1(V0s_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V0s_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Esup_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0))))) ) ) ) ) ).
tff(thm_2Ereal__topology_2Ecomponents,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Econnected__component,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V2y_2E0))
<=> ? [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V3t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Eball,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1e_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2y_2E0,c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2y_2E0)),V1e_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Ecball,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1e_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2y_2E0,c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2y_2E0)),V1e_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Esphere,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0x_2E0),V2y_2E0),V1e_2E0))
<=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2y_2E0)) = V1e_2E0 ) )
=> ( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1e_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2y_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0x_2E0),V2y_2E0),V1e_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Eclosed__segment,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1u_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V1u_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1u_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
=> ( ! [V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1u_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0l_2E0),V1u_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1u_2E0),c_2Epair_2EFST_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Elist_2EHD_2E1(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0l_2E0))),c_2Erealax_2Ereal__mul_2E2(V1u_2E0,c_2Epair_2ESND_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Elist_2EHD_2E1(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0l_2E0)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V1u_2E0)) )
=> ! [V0l_2E0: tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2Eclosed__segment_2E1(V0l_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0l_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Eopen__segment,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2Econnected,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1e1_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2e2_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0)))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0),V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V0s_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2e2_2E0,V0s_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2Elimit__point__of,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> ? [V3y_2E0: tyop_2Erealax_2Ereal] :
( ( V3y_2E0 != V0x_2E0 )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V2t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Einterior,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F1_2E0,V1x_2E0),V0s_2E0))
<=> ? [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_2E0,V0s_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F1_2E0,V1x_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Eclosure,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2Efrontier,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efrontier_2E1(V0s_2E0) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2Eisnet,axiom,
! [A_27a: $tType,A_27b: $tType,V0g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(c_2Ereal__topology_2Eisnet_2E1(A_27a,A_27b,V0g_2E0))
<=> ! [V1x_2E0: A_27b,V2y_2E0: A_27b] :
( ! [V3z_2E0: A_27a] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0g_2E0,V3z_2E0),V1x_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0g_2E0,V3z_2E0),V2y_2E0)) )
| ! [V4z_2E0: A_27a] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0g_2E0,V4z_2E0),V2y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0g_2E0,V4z_2E0),V1x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Enet__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Ereal__topology_2Enet(A_27a),c_2Ereal__topology_2Eisnet_2E0(A_27a,A_27a),V0rep_2E0)) ).
tff(thm_2Ereal__topology_2Eat,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0),V2y_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V0a_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2y_2E0,V0a_2E0)))) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eat_2E1(V0a_2E0) = c_2Ereal__topology_2Emk__net_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Eat__infinity,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0) = c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) )
=> ( c_2Ereal__topology_2Eat__infinity_2E0 = c_2Ereal__topology_2Emk__net_2E1(tyop_2Erealax_2Ereal,F0_2E0) ) ) ).
tff(thm_2Ereal__topology_2Eat__posinfinity,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0) = c_2Ereal_2Ereal__ge_2E2(V0x_2E0,V1y_2E0) )
=> ( c_2Ereal__topology_2Eat__posinfinity_2E0 = c_2Ereal__topology_2Emk__net_2E1(tyop_2Erealax_2Ereal,F0_2E0) ) ) ).
tff(thm_2Ereal__topology_2Eat__neginfinity,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) )
=> ( c_2Ereal__topology_2Eat__neginfinity_2E0 = c_2Ereal__topology_2Emk__net_2E1(tyop_2Erealax_2Ereal,F0_2E0) ) ) ).
tff(thm_2Ereal__topology_2Esequentially,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1n_2E0) = c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,V1n_2E0) )
=> ( c_2Ereal__topology_2Esequentially_2E0 = c_2Ereal__topology_2Emk__net_2E1(tyop_2Enum_2Enum,F0_2E0) ) ) ).
tff(thm_2Ereal__topology_2Ewithin,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
( ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a,V3y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0net_2E0),V1s_2E0),V2x_2E0),V3y_2E0))
<=> ( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0net_2E0,V2x_2E0,V3y_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ewithin_2E2(A_27a,V0net_2E0,V1s_2E0) = c_2Ereal__topology_2Emk__net_2E1(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0net_2E0),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Ein__direction,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V2b_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal,V1v_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2b_2E0),V0a_2E0),V1v_2E0))
<=> ? [V3c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3c_2E0))
& ( c_2Ereal_2Ereal__sub_2E2(V2b_2E0,V0a_2E0) = c_2Erealax_2Ereal__mul_2E2(V3c_2E0,V1v_2E0) ) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1v_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1v_2E0),V2b_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2b_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2b_2E0),V0a_2E0),V1v_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1v_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ein__direction_2E2(V0a_2E0,V1v_2E0) = c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1v_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2Etrivial__limit,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
<=> ( ! [V1a_2E0: A_27a,V2b_2E0: A_27a] : ( V1a_2E0 = V2b_2E0 )
| ? [V3a_2E0: A_27a,V4b_2E0: A_27a] :
( ( V3a_2E0 != V4b_2E0 )
& ! [V5x_2E0: A_27a] :
( ~ p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0net_2E0,V5x_2E0,V3a_2E0))
& ~ p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0net_2E0,V5x_2E0,V4b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2Eeventually,axiom,
! [A_27a: $tType,V0p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V0p_2E0,V1net_2E0))
<=> ( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V1net_2E0))
| ? [V2y_2E0: A_27a] :
( ? [V3x_2E0: A_27a] : p(c_2Ereal__topology_2Enetord_2E3(A_27a,V1net_2E0,V3x_2E0,V2y_2E0))
& ! [V4x_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V1net_2E0,V4x_2E0,V2y_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0p_2E0,V4x_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2Etendsto,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1l_2E0),V3e_2E0),V4x_2E0) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V1l_2E0)),V3e_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V0f_2E0,V1l_2E0,V2net_2E0))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1l_2E0),V3e_2E0),V2net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Elim__def,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V2l_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V0net_2E0),V2l_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Elim__def_2E2(A_27a,V0net_2E0,V1f_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Ereal__topology_2Enet(A_27a),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Efrom,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
( ! [V0n_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0n_2E0),V1m_2E0) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1m_2E0,c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,V1m_2E0)) )
=> ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Efrom_2E1(V0n_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0n_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Eeuclidean,axiom,
c_2Ereal__topology_2Eeuclidean_2E0 = c_2Etopology_2Etopology_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOpen_2E0) ).
tff(thm_2Ereal__topology_2Eclosed__def,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EClosed_2E1(V0s_2E0) = c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2Eopen__def,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& ! [V3x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_27_2E0,V1x_2E0)),V2e_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_27_2E0,V0s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2Esubtopology,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1u_2E0),V0top_2E0),V2s_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EINTER_2E2(A_27a,V2s_2E0,V1u_2E0),c_2Etopology_2Eopen__in_2E2(A_27a,V0top_2E0,V2s_2E0)) )
=> ! [V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1u_2E0) = c_2Etopology_2Etopology_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1u_2E0),V0top_2E0))) ) ) ).
tff(thm_2Ereal__topology_2Edim,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0v_2E0),V1n_2E0))
<=> ? [V2b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V2b_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0v_2E0,c_2Ereal__topology_2Espan_2E1(V2b_2E0)))
& p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V1n_2E0)) ) )
=> ! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Edim_2E1(V0v_2E0) = c_2Emin_2E_40_2E1(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0v_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Eindependent,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0))
<=> ~ p(c_2Ereal__topology_2Edependent_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2Edependent,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Edependent_2E1(V0s_2E0))
<=> ? [V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1a_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2Espan,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Espan_2E1(V0s_2E0) = c_2Etopology_2Ehull_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubspace_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2Esubspace,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0s_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0),V0s_2E0)) )
& ! [V3c_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V3c_2E0,V4x_2E0),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Ebilinear,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1x_2E0),V2y_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1x_2E0),V2y_2E0) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3y_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V3y_2E0),V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V4x_2E0),V3y_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0f_2E0))
<=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1x_2E0)))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V3y_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2Elinear,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
<=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0)) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0)) )
& ! [V3c_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Erealax_2Ereal__mul_2E2(V3c_2E0,V4x_2E0)) = c_2Erealax_2Ereal__mul_2E2(V3c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Edist,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal__topology_2Epermutes,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V0p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3y_2E0: A_27a,V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V0p_2E0),V3y_2E0),V4x_2E0))
<=> ( app_2E2(A_27a,A_27a,V0p_2E0,V4x_2E0) = V3y_2E0 ) )
=> ! [V0p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Epermutes_2E2(A_27a,V0p_2E0,V1s_2E0))
<=> ( ! [V2x_2E0: A_27a] :
( ~ p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> ( app_2E2(A_27a,A_27a,V0p_2E0,V2x_2E0) = V2x_2E0 ) )
& ! [V3y_2E0: A_27a] : p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V0p_2E0),V3y_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2Epairwise,axiom,
! [A_27a: $tType,V0r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,V1s_2E0))
<=> ! [V2x_2E0: A_27a,V3y_2E0: A_27a] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V3y_2E0,V1s_2E0))
& ( V2x_2E0 != V3y_2E0 ) )
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0r_2E0,V2x_2E0),V3y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2Euniformly__continuous__on,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_27_2E0,V4x_2E0)),V3d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0))),V2e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2Econtinuous__on,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_27_2E0,V2x_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))),V3e_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2Econtinuous,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V1net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V0f_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Enetlimit_2E1(A_27a,V1net_2E0)),V1net_2E0) ) ).
tff(thm_2Ereal__topology_2Ecomplete,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecomplete_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V2n_2E0),V0s_2E0))
& p(c_2Ereal__topology_2Ecauchy_2E1(V1f_2E0)) )
=> ? [V3l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3l_2E0,V0s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1f_2E0,V3l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Ecauchy,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Ecauchy_2E1(V0s_2E0))
<=> ! [V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3E_3D_2E2(V3m_2E0,V2N_2E0))
& p(c_2Earithmetic_2E_3E_3D_2E2(V4n_2E0,V2N_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V4n_2E0))),V1e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Ecompact,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V2n_2E0),V0s_2E0))
=> ? [V3l_2E0: tyop_2Erealax_2Ereal,V4r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3l_2E0,V0s_2E0))
& ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V5m_2E0,V6n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V5m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V6n_2E0))) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V1f_2E0,V4r_2E0),V3l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EFSIGMA__COMPLEMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efsigma_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) = c_2Ereal__topology_2Egdelta_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EGDELTA__COMPLEMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Egdelta_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) = c_2Ereal__topology_2Efsigma_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__OPEN__MAP__IMAGE,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V2f_2E0,V3s_2E0))
& ! [V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3s_2E0),V4t_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V3s_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4t_2E0))) )
& ! [V5t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5t_2E0,V3s_2E0))
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V5t_2E0)) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1Q_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V5t_2E0))) )
& p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V3s_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(V1Q_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V3s_2E0))) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V2f_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V5y_2E0) )
=> ( V4x_2E0 = V5y_2E0 ) ) )
=> ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V3s_2E0)) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1Q_2E0,V3s_2E0) ) )
=> ! [V6f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V6f_2E0))
& ! [V8x_2E0: tyop_2Erealax_2Ereal,V9y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6f_2E0,V8x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6f_2E0,V9y_2E0) )
=> ( V8x_2E0 = V9y_2E0 ) ) )
=> ( c_2Ereal__topology_2Elocally_2E2(V0P_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6f_2E0,V7s_2E0)) = c_2Ereal__topology_2Elocally_2E2(V1Q_2E0,V7s_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__TRANSLATION,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V4a_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V4a_2E0),V6x_2E0) = c_2Erealax_2Ereal__add_2E2(V4a_2E0,V6x_2E0) )
=> ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,V3x_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2s_2E0)) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V2s_2E0) )
=> ! [V4a_2E0: tyop_2Erealax_2Ereal,V5s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Elocally_2E2(V0P_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V4a_2E0),V5s_2E0)) = c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V5s_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__LOCALLY,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V2s_2E0,V3t_2E0))
=> ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V2s_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1Q_2E0,V3t_2E0) ) )
=> ! [V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V4s_2E0,V5t_2E0))
=> ( c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V4s_2E0) = c_2Ereal__topology_2Elocally_2E2(V1Q_2E0,V5t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__LOCALLY,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4s_2E0,V5t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0,V3g_2E0)))
=> ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V4s_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1Q_2E0,V5t_2E0) ) )
=> ! [V6s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6s_2E0,V7t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0,V3g_2E0)))
=> ( c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V6s_2E0) = c_2Ereal__topology_2Elocally_2E2(V1Q_2E0,V7t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__INTER,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V1s_2E0))
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V2t_2E0)) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))) )
=> ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V4t_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__SING,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elocally_2E2(V0P_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__EMPTY,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ELOCALLY__DIFF__CLOSED,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V1s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__OPEN__SUBSET,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__MONO,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V3t_2E0))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1Q_2E0,V3t_2E0)) )
& p(c_2Ereal__topology_2Elocally_2E2(V0P_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(V1Q_2E0,V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2EURYSOHN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V4f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V4f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V5x_2E0),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V6x_2E0) = V2a_2E0 ) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1t_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V7x_2E0) = V3b_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2EURYSOHN__STRONG,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2a_2E0 != V3b_2E0 ) )
=> ? [V4f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V4f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V5x_2E0),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V6x_2E0) = V2a_2E0 )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V0s_2E0)) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V7x_2E0) = V3b_2E0 )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EURYSOHN__LOCAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V5f_2E0,V2u_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2u_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V6x_2E0),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V7x_2E0) = V3a_2E0 ) )
& ! [V8x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1t_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V8x_2E0) = V4b_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2EURYSOHN__LOCAL__STRONG,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V3a_2E0 != V4b_2E0 ) )
=> ? [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V5f_2E0,V2u_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2u_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V6x_2E0),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2u_2E0))
=> ( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V7x_2E0) = V3a_2E0 )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V0s_2E0)) ) )
& ! [V8x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V2u_2E0))
=> ( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V8x_2E0) = V4b_2E0 )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1t_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EISOMETRY__IMP__HOMEOMORPHISM__COMPACT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V1s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0)) )
=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0))) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)) ) ) )
=> ? [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V1s_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V4g_2E0))) ) ).
tff(thm_2Ereal__topology_2EISOMETRY__IMP__EMBEDDING,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0)) )
=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0))) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0)) ) ) )
=> ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5g_2E0))) ) ).
tff(thm_2Ereal__topology_2EISOMETRY__IMP__OPEN__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V1s_2E0)) )
=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5y_2E0))) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V5y_2E0)) ) )
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3u_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DIAMETER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0)) )
=> ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V3t_2E0))
& ( V3t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3t_2E0)),V2d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2Ediameter_2E1(V0s_2E0),c_2Ereal__topology_2Ediameter_2E1(V3t_2E0))),V1e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__ALT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)))),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Esup_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1t_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__COMPACT__NONTRIVIAL,axiom,
! [F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6y_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V5x_2E0),V0s_2E0),V6y_2E0),V2e_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V6y_2E0),V2e_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F3_2E0,V0s_2E0),V2e_2E0),V5x_2E0),V6y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V5x_2E0,V6y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V5x_2E0),V0s_2E0),V6y_2E0),V2e_2E0)) )
=> ( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4y_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V3x_2E0),V1t_2E0),V4y_2E0),V2e_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1t_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V4y_2E0),V2e_2E0)) ) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1t_2E0),V2e_2E0),V3x_2E0),V4y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V3x_2E0,V4y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V3x_2E0),V1t_2E0),V4y_2E0),V2e_2E0)) )
=> ( ! [V2e_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2e_2E0),V0s_2E0),V1t_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1t_2E0),V2e_2E0)))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F3_2E0,V0s_2E0),V2e_2E0))))) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1t_2E0),V2e_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2e_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2e_2E0),V0s_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Eiterate_2Einf_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1t_2E0))) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__EQ__0,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> ( ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Ereal__topology_2Eclosure_2E1(V1t_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V2u_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2u_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V2u_2E0))))) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__COMPACT__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3z_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2y_2E0),V1t_2E0),V3z_2E0),V0s_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V1t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3z_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))))) ) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1t_2E0),V0s_2E0),V2y_2E0),V3z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2y_2E0,V3z_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2y_2E0),V1t_2E0),V3z_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1t_2E0),V0s_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__TRIANGLE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V2u_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2u_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V2u_2E0))))) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__COMPACT__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> ? [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__INSERT__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V0s_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V1t_2E0))),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__UNION__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V2u_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2u_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2u_2E0))),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V2u_2E0)))) ) ).
tff(thm_2Ereal__topology_2EREAL__HAUSDIST__LE__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),V2b_2E0))
<=> ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),V2b_2E0)) )
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V4y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),V2b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL__ALT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Emax_2E2(c_2Ereal_2Esup_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0))),c_2Ereal_2Esup_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__NONTRIVIAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Esup_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V7x_2E0),V6u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0)) )
=> ( ! [V5x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V3x_2E0),V1t_2E0)) )
& ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V4u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0)))) )
& ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V6u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0)))) ) )
=> ! [V8x_2E0: tyop_2Erealax_2Ereal,V9e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V8x_2E0))) )
=> ? [V10d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V10d_2E0))
& ! [V11x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11x_27_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8x_2E0,V11x_27_2E0)),V10d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V8x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V11x_27_2E0))),V9e_2E0)) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__HAUSDIST__POINT__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),V3d_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> ? [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V4y_2E0)),V3d_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__SETDIST__TRIANGLE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2u_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V2u_2E0))))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__HAUSDIST__TRIANGLE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V2u_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2u_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V2u_2E0))))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SING__LE__HAUSDIST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__LE__HAUSDIST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__HAUSDIST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4c_2E0: tyop_2Erealax_2Ereal,V5z_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V6x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),V3b_2E0)) )
& ! [V7y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7y_2E0,V1t_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V7y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),V4c_2E0)) )
& ( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5z_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V5z_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)))) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5z_2E0,V1t_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V5z_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)))) ) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2EREAL__HAUSDIST__LE__SUMS,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5y_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6z_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5y_2E0),V0s_2E0),V6z_2E0),V2b_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6z_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2b_2E0)))) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal,V6z_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V2b_2E0),V5y_2E0),V6z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V5y_2E0,V6z_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5y_2E0),V0s_2E0),V6z_2E0),V2b_2E0)) )
=> ( ! [V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4z_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V3y_2E0),V1t_2E0),V4z_2E0),V2b_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4z_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2b_2E0)))) ) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal,V4z_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1t_2E0),V2b_2E0),V3y_2E0),V4z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V3y_2E0,V4z_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V3y_2E0),V1t_2E0),V4z_2E0),V2b_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1t_2E0),V2b_2E0)))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V2b_2E0))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),V2b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__HAUSDIST__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),V2b_2E0)) )
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V4y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),V2b_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),V2b_2E0)) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__CLOSURE,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),V1t_2E0)) = c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) )
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,c_2Ereal__topology_2Eclosure_2E1(V3t_2E0))) = c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) = c_2Ereal_2Eabs_2E1(V3x_2E0) ) )
=> ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0))) = c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2t_2E0))) = c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V4b_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V5x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)),V4b_2E0)) )
& ! [V6y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6y_2E0,V1t_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V6y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),V4b_2E0)) ) )
<=> ( ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_27_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V7x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V3t_27_2E0)),V4b_2E0)) )
& ! [V8y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8y_2E0,V3t_27_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V8y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V2s_27_2E0)),V4b_2E0)) ) ) )
=> ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_27_2E0,V3t_27_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__SINGS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__EMPTY,axiom,
( ! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),V0t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__SYM,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V0s_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal__topology_2EHAUSDIST__POS__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Ehausdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ).
tff(thm_2Ereal__topology_2ECONNECTED__NEST__GEN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V5n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V5n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2EClosed_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0)))
& p(c_2Ereal__topology_2Econnected_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0))) )
& ? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2Ecompact_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2n_2E0)))
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3m_2E0))) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__NEST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V4n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V4n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Ecompact_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0)))
& p(c_2Ereal__topology_2Econnected_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0))) )
& ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2m_2E0))) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CHAIN__GEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> ( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0)) ) )
& ? [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V2s_2E0)) )
& ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3s_2E0,V0f_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0,V0f_2E0)) )
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))
| p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V3s_2E0)) ) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CHAIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> ( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0)) ) )
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V0f_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0,V0f_2E0)) )
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V3t_2E0))
| p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,V2s_2E0)) ) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__T0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != V1y_2E0 )
<=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2u_2E0) != c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V2u_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__T1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != V1y_2E0 )
<=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V3v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2u_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V2u_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V3v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V3v_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__T2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != V1y_2E0 )
<=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V3v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2u_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V3v_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__HAUSDORFF,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != V1y_2E0 )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V3v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2u_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V3v_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__NORMAL__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Eclosure_2E1(V2u_2E0)))
& p(c_2Ereal__topology_2EOpen_2E1(V3v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V3v_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__NORMAL__LOCAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V3s_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V3s_27_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V4t_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V3s_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V4t_27_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3s_27_2E0,V4t_27_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__NORMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V3v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V3v_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATION__CLOSURES,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V3v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V3v_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__BALLS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2r_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V2r_2E0),V3s_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V2r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(V3s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
=> ( ! [V6r_2E0: tyop_2Erealax_2Ereal,V7s_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V6r_2E0),V7s_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V6r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Erealax_2Ereal__lt_2E2(V7s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
=> ( ! [V10r_2E0: tyop_2Erealax_2Ereal,V11s_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V10r_2E0),V11s_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V10r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(V11s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
=> ( ! [V14r_2E0: tyop_2Erealax_2Ereal,V15s_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V14r_2E0),V15s_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V14r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Erealax_2Ereal__lt_2E2(V15s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V3s_2E0)))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V2r_2E0),V3s_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Emax_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Erealax_2Ereal__add_2E2(V2r_2E0,V3s_2E0)))) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V6r_2E0: tyop_2Erealax_2Ereal,V7s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V6r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5b_2E0,V7s_2E0)))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V6r_2E0),V7s_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Emax_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0)),c_2Erealax_2Ereal__add_2E2(V6r_2E0,V7s_2E0)))) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal,V10r_2E0: tyop_2Erealax_2Ereal,V11s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V10r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9b_2E0,V11s_2E0)))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V10r_2E0),V11s_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Emax_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V9b_2E0)),c_2Erealax_2Ereal__add_2E2(V10r_2E0,V11s_2E0)))) )
& ! [V12a_2E0: tyop_2Erealax_2Ereal,V13b_2E0: tyop_2Erealax_2Ereal,V14r_2E0: tyop_2Erealax_2Ereal,V15s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V14r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V13b_2E0,V15s_2E0)))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V14r_2E0),V15s_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Emax_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V13b_2E0)),c_2Erealax_2Ereal__add_2E2(V14r_2E0,V15s_2E0)))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__LE__SING,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SING__TRIANGLE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2y_2E0)))) ).
tff(thm_2Ereal__topology_2ESETDIST__SING__FRONTIER__CASES,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SING__IN__SET,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0))
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1s_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__IN,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0u_2E0)) )
=> ( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1s_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> ( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__EQ__0__SING,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) ) )
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V2s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Eclosure_2E1(V2s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__CLOSEST__POINT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V1s_2E0)) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V1s_2E0,V0a_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SING__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__FRONTIERS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal__topology_2ESETDIST__ZERO__STRONG,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ~ p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)))
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__ZERO,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ~ p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__UNIV,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__UNIQUE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3b_2E0,V1t_2E0))
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = V4d_2E0 )
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6y_2E0,V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V6y_2E0)))) ) )
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = V4d_2E0 ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) = c_2Ereal_2Eabs_2E1(V3x_2E0) ) )
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0))) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2t_2E0))) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__EQ__0__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
| p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> ( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__EQ__0__CLOSED__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
=> ( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__EQ__0__COMPACT__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> ( ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__CLOSED__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0))
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__COMPACT__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0))
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__FRONTIER,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),V1t_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) ) )
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V3t_2E0))
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,c_2Ereal__topology_2Efrontier_2E1(V3t_2E0))) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__CLOSURE,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),V1t_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) )
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,c_2Ereal__topology_2Eclosure_2E1(V3t_2E0))) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SUBSET__LEFT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V2u_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2u_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SUBSET__RIGHT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2u_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__DIFFERENCES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SETDIST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V2y_2E0) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__SETDIST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V2y_2E0) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__SETDIST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V2y_2E0) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),c_2Ereal__topology_2Eat_2E1(V1x_2E0))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__LIPSCHITZ,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: A_27a,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V0s_2E0)))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)))) ).
tff(thm_2Ereal__topology_2ESETDIST__SINGS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Ereal__topology_2ESETDIST__TRIANGLE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2t_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),V2t_2E0))))) ).
tff(thm_2Ereal__topology_2ESETDIST__SYM,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESETDIST__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V0s_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal__topology_2EREAL__SETDIST__LT__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),V2b_2E0)) )
=> ? [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0)),V2b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__SETDIST__EQ,axiom,
! [V0d_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0))))
<=> ( ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V2t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0)))) )
& ( ( ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V2t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__LE__DIST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)))) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__SETDIST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2d_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2d_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0)))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2d_2E0,c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESETDIST__SUBSETS__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_27_2E0,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_27_2E0,V1t_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V1t_2E0)) )
=> ? [V6x_27_2E0: tyop_2Erealax_2Ereal,V7y_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_27_2E0,V2s_27_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7y_27_2E0,V3t_27_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_27_2E0,V7y_27_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V5y_2E0)))) ) ) )
=> ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_27_2E0,V3t_27_2E0)) = c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESETDIST__POS__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0)))) ).
tff(thm_2Ereal__topology_2ESETDIST__EMPTY,axiom,
( ! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),V0t_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esetdist_2E1(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__IN__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1x_2E0),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__IN__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1x_2E0),c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__TO__POINT__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0)),V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__TO__CLOSEST__POINT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1a_2E0))),V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__DIST__CLOSEST__POINT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V2x_2E0) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V2x_2E0))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__DIST__CLOSEST__POINT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V2x_2E0) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V2x_2E0))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__DIST__CLOSEST__POINT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),V2x_2E0) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V2x_2E0))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0s_2E0),c_2Ereal__topology_2Eat_2E1(V1x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EDIST__CLOSEST__POINT__LIPSCHITZ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1x_2E0))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2y_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V2y_2E0))))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2y_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ( c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1x_2E0) = V1x_2E0 )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__SELF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ( c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1x_2E0) = V1x_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1a_2E0))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2x_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__IN__SET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1a_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSEST__POINT__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1a_2E0),V0s_2E0))
& ! [V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,c_2Ereal__topology_2Eclosest__point_2E2(V0s_2E0,V1a_2E0))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2y_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EDINI,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8e_2E0: tyop_2Erealax_2Ereal,V9n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V1g_2E0),V8e_2E0),V9n_2E0))
<=> ! [V10x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,V2s_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V9n_2E0),V10x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V10x_2E0))),V8e_2E0)) ) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal,V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V4x_2E0),V5n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5n_2E0),V4x_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V2s_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V3n_2E0),V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V4x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0),c_2Ereal__topology_2Esequentially_2E0)) )
& ! [V6n_2E0: tyop_2Enum_2Enum,V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6n_2E0),V7x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,c_2Earithmetic_2E_2B_2E2(V6n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V7x_2E0))) ) )
=> ! [V8e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V8e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Enum_2Enum,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V1g_2E0),V8e_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBANACH__FIX,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1s_2E0),V0f_2E0),V5x_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V5x_2E0 ) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecomplete_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2c_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2c_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V1s_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0))),c_2Erealax_2Ereal__mul_2E2(V2c_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0))))) ) )
=> p(c_2Ebool_2E_3F_21_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1s_2E0),V0f_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__REARRANGE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
& p(c_2Ereal__topology_2Epermutes_2E2(tyop_2Enum_2Enum,V2p_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0x_2E0,V2p_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__REARRANGE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V4n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
& p(c_2Ereal__topology_2Epermutes_2E2(tyop_2Enum_2Enum,V2p_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V0x_2E0,V3l_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0x_2E0,V2p_2E0),V3l_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__REARRANGE__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V4n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
& p(c_2Ereal__topology_2Epermutes_2E2(tyop_2Enum_2Enum,V2p_2E0,V1s_2E0)) )
=> ( c_2Ereal__topology_2Esums_2E3(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0x_2E0,V2p_2E0),V3l_2E0,V1s_2E0) = c_2Ereal__topology_2Esums_2E3(V0x_2E0,V3l_2E0,V1s_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V4n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esummable_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V1s_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
& ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5m_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V6n_2E0,V1s_2E0))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V5m_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V6n_2E0) ) )
=> ( V5m_2E0 = V6n_2E0 ) ) )
=> ( c_2Ereal__topology_2Esums_2E3(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0x_2E0,V2f_2E0),V3l_2E0,V1s_2E0) = c_2Ereal__topology_2Esums_2E3(V0x_2E0,V3l_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__INJECTIVE__IMAGE__STRONG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0)) )
=> ( ! [V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2f_2E0),V1s_2E0),V0x_2E0),V6n_2E0) = c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V1s_2E0),c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V6n_2E0)),V0x_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V6n_2E0)),c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0x_2E0,V2f_2E0))) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V1s_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
& ! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4m_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5n_2E0,V1s_2E0))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V4m_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2f_2E0,V5n_2E0) ) )
=> ( V4m_2E0 = V5n_2E0 ) ) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2f_2E0),V1s_2E0),V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__DIRICHLET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V4m_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V4m_2E0),V0f_2E0),V5n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V4m_2E0,V5n_2E0),V0f_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ( ! [V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V1g_2E0),V0f_2E0),V7n_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V7n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V7n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2N_2E0: tyop_2Enum_2Enum,V3k_2E0: tyop_2Enum_2Enum,V4m_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V4m_2E0),V0f_2E0))))
& ! [V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V2N_2E0,V6n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V6n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V6n_2E0))) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1g_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V3k_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V1g_2E0),V0f_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__DIRICHLET__BILINEAR,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V4m_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V4m_2E0),V0f_2E0),V7n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V4m_2E0,V7n_2E0),V0f_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V7n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V9n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V2h_2E0),V1g_2E0),V0f_2E0),V9n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V9n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V9n_2E0),V0f_2E0)) )
=> ( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V10n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F3_2E0,V2h_2E0),V1g_2E0),V0f_2E0),V10n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V10n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V10n_2E0)) )
=> ( ! [V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V8n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V1g_2E0),V8n_2E0) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V8n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V8n_2E0))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3k_2E0: tyop_2Enum_2Enum,V4m_2E0: tyop_2Enum_2Enum,V5p_2E0: tyop_2Enum_2Enum,V6l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V4m_2E0),V0f_2E0))))
& p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V5p_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V1g_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V2h_2E0),V1g_2E0),V0f_2E0),V6l_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V3k_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F3_2E0,V2h_2E0),V1g_2E0),V0f_2E0))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__BILINEAR__PARTIAL__PRE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V6n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V6n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V6n_2E0))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V6n_2E0)) )
=> ( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V5n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V5n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5n_2E0)) )
=> ( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V7n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V7n_2E0)),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V7n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2D_2E2(V7n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3l_2E0: tyop_2Erealax_2Ereal,V4k_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V3l_2E0,c_2Ereal__topology_2Esequentially_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V4k_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2h_2E0),V0f_2E0),V1g_2E0))) )
=> p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V4k_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V2h_2E0),V0f_2E0),V1g_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__PARTIAL__SUMS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1k_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1k_2E0),V0f_2E0),V2n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1k_2E0,V2n_2E0),V0f_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ( ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F2_2E0,V3m_2E0),V4n_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3m_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum)))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) ) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0f_2E0),V3m_2E0),V4n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),V0f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F2_2E0,V3m_2E0),V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1k_2E0),V0f_2E0))))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Epair_2EUNCURRY_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0f_2E0))))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__RATIO,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3N_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0c_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V4n_2E0,V3N_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1a_2E0,c_2Enum_2ESUC_2E1(V4n_2E0))),c_2Erealax_2Ereal__mul_2E2(V0c_2E0,c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1a_2E0,V4n_2E0))))) ) )
=> ? [V5l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Esums_2E3(V1a_2E0,V5l_2E0,V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__COMPARISON__UNIFORM,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ? [V4l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Esums_2E3(V1g_2E0,V4l_2E0,V3s_2E0))
& ? [V5N_2E0: tyop_2Enum_2Enum] :
! [V6n_2E0: tyop_2Enum_2Enum,V7x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V5N_2E0,V6n_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V6n_2E0,V3s_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V7x_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0,V7x_2E0),V6n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V6n_2E0))) ) )
=> ? [V8l_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
! [V9e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
=> ? [V10N_2E0: tyop_2Enum_2Enum] :
! [V11n_2E0: tyop_2Enum_2Enum,V12x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V10N_2E0,V11n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V12x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V3s_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V11n_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0,V12x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,V8l_2E0,V12x_2E0))),V9e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__COMPARISON__BOUND,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esums_2E3(V1g_2E0,V3a_2E0,V2s_2E0))
& ! [V4i_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4i_2E0,V2s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4i_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4i_2E0))) ) )
=> ? [V5l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V5l_2E0,V2s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V5l_2E0),V3a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__SUBSET__ABSCONV,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V0x_2E0),V4n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0)) )
=> ( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,V2t_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2t_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V0x_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__ABSCONV__IMP__CONV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V2n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esummable_2E2(V1k_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)))
=> p(c_2Ereal__topology_2Esummable_2E2(V1k_2E0,V0x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__COMPARISON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V1g_2E0))
& ? [V3N_2E0: tyop_2Enum_2Enum] :
! [V4n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3E_3D_2E2(V4n_2E0,V3N_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4n_2E0,V2s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0))) ) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0f_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__COMPARISON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ? [V3l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Esums_2E3(V1g_2E0,V3l_2E0,V2s_2E0))
& ? [V4N_2E0: tyop_2Enum_2Enum] :
! [V5n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3E_3D_2E2(V5n_2E0,V4N_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5n_2E0,V2s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5n_2E0))) ) )
=> ? [V6l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V6l_2E0,V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__IMP__SUMS__BOUNDED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1k_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1k_2E0),V0f_2E0),V2n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1k_2E0,V2n_2E0),V0f_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V1k_2E0),V0f_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1k_2E0),V0f_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__IMP__BOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esummable_2E2(V1k_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1k_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__IMP__TOZERO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,V1k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esummable_2E2(V1k_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__GOESTOZERO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2e_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1x_2E0),V2e_2E0),V3n_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3n_2E0,V0s_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1x_2E0,V3n_2E0)),V2e_2E0)) ) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,V1x_2E0))
=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Enum_2Enum,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1x_2E0),V2e_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__CAUCHY__UNIFORM,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ? [V3l_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> ? [V5N_2E0: tyop_2Enum_2Enum] :
! [V6n_2E0: tyop_2Enum_2Enum,V7x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V5N_2E0,V6n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V7x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V2k_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V6n_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1f_2E0,V7x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,V3l_2E0,V7x_2E0))),V4e_2E0)) ) )
<=> ! [V8e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V8e_2E0))
=> ? [V9N_2E0: tyop_2Enum_2Enum] :
! [V10m_2E0: tyop_2Enum_2Enum,V11n_2E0: tyop_2Enum_2Enum,V12x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V9N_2E0,V10m_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V12x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V2k_2E0,c_2Eiterate_2E_2E_2E_2E2(V10m_2E0,V11n_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1f_2E0,V12x_2E0))),V8e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__FROM__ELSEWHERE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V1m_2E0),V0f_2E0))
=> p(c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V2n_2E0),V0f_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__EQ__COFINITE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,V1s_2E0,V2t_2E0),c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,V2t_2E0,V1s_2E0))))
& p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2t_2E0,V0f_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__IFF__COFINITE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,V1s_2E0,V2t_2E0),c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,V2t_2E0,V1s_2E0))))
=> ( c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0) = c_2Ereal__topology_2Esummable_2E2(V2t_2E0,V0f_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__EQ__EVENTUALLY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ? [V3N_2E0: tyop_2Enum_2Enum] :
! [V4n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V3N_2E0,V4n_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4n_2E0,V2k_2E0)) )
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0) ) )
& p(c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V0f_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V1g_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__IFF__EVENTUALLY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ? [V3N_2E0: tyop_2Enum_2Enum] :
! [V4n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V3N_2E0,V4n_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4n_2E0,V2k_2E0)) )
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0) ) )
=> ( c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V0f_2E0) = c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V1g_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__CAUCHY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3N_2E0: tyop_2Enum_2Enum] :
! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V4m_2E0,V3N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,c_2Eiterate_2E_2E_2E_2E2(V4m_2E0,V5n_2E0)),V0f_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__CAUCHY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ? [V2l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V2l_2E0,V1s_2E0))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4N_2E0: tyop_2Enum_2Enum] :
! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V5m_2E0,V4N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,c_2Eiterate_2E_2E_2E_2E2(V5m_2E0,V6n_2E0)),V0f_2E0)),V3e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EABS__SUM__TRIVIAL__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1n_2E0: tyop_2Enum_2Enum,V2m_2E0: tyop_2Enum_2Enum,V3f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4P_2E0: tyop_2Emin_2Ebool,V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> ( ( p(V4P_2E0)
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V0s_2E0,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V1n_2E0)),V3f_2E0)),V5e_2E0)) )
<=> ( p(V4P_2E0)
=> ( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V2m_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V0s_2E0,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V1n_2E0)),V3f_2E0)),V5e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESUM__DIFF__LEMMA,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0))
=> ( c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V3n_2E0)),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V2m_2E0)),V0f_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1k_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3n_2E0)),V0f_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESEQUENCE__CAUCHY__WLOG,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal,V1P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V3m_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V4n_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2s_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2s_2E0,V4n_2E0))),V0e_2E0)) )
<=> ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V5m_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1P_2E0,V6n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V5m_2E0,V6n_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2s_2E0,V5m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2s_2E0,V6n_2E0))),V0e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EPARTIAL__SUMS__DROP__LE__INFSUM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2n_2E0: tyop_2Enum_2Enum] :
( ( ! [V3i_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3i_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3i_2E0))) )
& p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V2n_2E0)),V0f_2E0),c_2Ereal__topology_2Einfsum_2E2(V1s_2E0,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2EPARTIAL__SUMS__COMPONENT__LE__INFSUM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2n_2E0: tyop_2Enum_2Enum] :
( ( ! [V3i_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3i_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3i_2E0))) )
& p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V2n_2E0)),V0f_2E0),c_2Ereal__topology_2Einfsum_2E2(V1s_2E0,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2EINFSUM__RESTRICT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0k_2E0),V1a_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,V0k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1a_2E0,V2n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Einfsum_2E2(c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0k_2E0),V1a_2E0)) = c_2Ereal__topology_2Einfsum_2E2(V0k_2E0,V1a_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V1g_2E0))
& ! [V3x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3x_2E0,V2k_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) ) )
=> ( c_2Ereal__topology_2Einfsum_2E2(V2k_2E0,V0f_2E0) = c_2Ereal__topology_2Einfsum_2E2(V2k_2E0,V1g_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V1x_2E0),V2n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1x_2E0,V2n_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,V1x_2E0))
=> ( c_2Ereal__topology_2Einfsum_2E2(V0s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V1x_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Ereal__topology_2Einfsum_2E2(V0s_2E0,V1x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1x_2E0),V3n_2E0) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1x_2E0,V3n_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,V1x_2E0))
=> ( c_2Ereal__topology_2Einfsum_2E2(V0s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1x_2E0)) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,c_2Ereal__topology_2Einfsum_2E2(V0s_2E0,V1x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0),V3i_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1y_2E0,V3i_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0x_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V1y_2E0)) )
=> ( c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,V0x_2E0),c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,V1y_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0),V3i_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1y_2E0,V3i_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0x_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V1y_2E0)) )
=> ( c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,V0x_2E0),c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,V1y_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V1i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einfsum_2E2(V0s_2E0,F0_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINFSUM__LINEAR,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V0f_2E0),V3n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1h_2E0)) )
=> ( c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V0f_2E0)) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,V0f_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__BOUND,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V3a_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V1g_2E0,V4b_2E0,V2s_2E0))
& ! [V5i_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5i_2E0,V2s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5i_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5i_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V3a_2E0),V4b_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__DROP__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V2a_2E0,V1s_2E0))
& ! [V3x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2a_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__DROP__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V3a_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V1g_2E0,V4b_2E0,V2s_2E0))
& ! [V5x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5x_2E0,V2s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V4b_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__REINDEX,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0k_2E0: tyop_2Enum_2Enum,V3x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V0k_2E0),V3x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1a_2E0,c_2Earithmetic_2E_2B_2E2(V3x_2E0,V0k_2E0)) )
=> ! [V0k_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V0k_2E0)) = c_2Ereal__topology_2Esummable_2E2(c_2Ereal__topology_2Efrom_2E1(c_2Earithmetic_2E_2B_2E2(V2n_2E0,V0k_2E0)),V1a_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESUMS__OFFSET__REV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,c_2Ereal__topology_2Efrom_2E1(V2m_2E0)))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2m_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V3n_2E0,V2m_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Erealax_2Ereal__add_2E2(V1l_2E0,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V3n_2E0,c_2Earithmetic_2E_2D_2E2(V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0)),c_2Ereal__topology_2Efrom_2E1(V3n_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUMS__OFFSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,c_2Ereal__topology_2Efrom_2E1(V2m_2E0)))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V3n_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Ereal_2Ereal__sub_2E2(V1l_2E0,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,c_2Earithmetic_2E_2D_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0)),c_2Ereal__topology_2Efrom_2E1(V3n_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUMS__FINITE__UNION,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V2t_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V3l_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Erealax_2Ereal__add_2E2(V3l_2E0,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,V2t_2E0,V1s_2E0),V0f_2E0)),c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,V1s_2E0,V2t_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUMS__FINITE__DIFF,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,V1t_2E0,V2s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1t_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V3l_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Ereal_2Ereal__sub_2E2(V3l_2E0,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,V1t_2E0,V0f_2E0)),c_2Epred__set_2EDIFF_2E2(tyop_2Enum_2Enum,V2s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__RESTRICT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),V2n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,V1k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esummable_2E2(c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0)) = c_2Ereal__topology_2Esummable_2E2(V1k_2E0,V0f_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__TRIVIAL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Esummable_2E2(c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum),V0f_2E0)) ).
tff(thm_2Ereal__topology_2ESUMMABLE__SUBSET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0x_2E0),V3i_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3i_2E0,V1s_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3i_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,V1s_2E0,V2t_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(V2t_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0x_2E0))) )
=> p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__SUBSET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0x_2E0),V4i_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4i_2E0,V1s_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4i_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,V1s_2E0,V2t_2E0))
& p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0x_2E0),V3l_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(V0x_2E0,V3l_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__COMPONENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2i_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ! [V3x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3x_2E0,V2k_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) )
& p(c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V0f_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V1g_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__IFF,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V3x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3x_2E0,V2k_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) )
=> ( c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V0f_2E0) = c_2Ereal__topology_2Esummable_2E2(V2k_2E0,V1g_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V2n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0x_2E0))
=> p(c_2Ereal__topology_2Esummable_2E2(V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1x_2E0),V3n_2E0) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1x_2E0,V3n_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,V1x_2E0))
=> p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0),V3n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1y_2E0,V3n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0x_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V1y_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0),V3n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1y_2E0,V3n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0x_2E0))
& p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V1y_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V1y_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V1n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Esummable_2E2(V0s_2E0,F0_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMMABLE__LINEAR,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V0f_2E0),V3n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1h_2E0)) )
=> p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V0f_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUMS__REINDEX__GEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0k_2E0: tyop_2Enum_2Enum,V5i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F1_2E0,V0k_2E0),V5i_2E0) = c_2Earithmetic_2E_2B_2E2(V5i_2E0,V0k_2E0) )
=> ( ! [V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0k_2E0: tyop_2Enum_2Enum,V4x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V0k_2E0),V4x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1a_2E0,c_2Earithmetic_2E_2B_2E2(V4x_2E0,V0k_2E0)) )
=> ! [V0k_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V0k_2E0),V2l_2E0,V3s_2E0) = c_2Ereal__topology_2Esums_2E3(V1a_2E0,V2l_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F1_2E0,V0k_2E0),V3s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESUMS__REINDEX,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0k_2E0: tyop_2Enum_2Enum,V4x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V0k_2E0),V4x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1a_2E0,c_2Earithmetic_2E_2B_2E2(V4x_2E0,V0k_2E0)) )
=> ! [V0k_2E0: tyop_2Enum_2Enum,V1a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V0k_2E0),V2l_2E0,c_2Ereal__topology_2Efrom_2E1(V3n_2E0)) = c_2Ereal__topology_2Esums_2E3(V1a_2E0,V2l_2E0,c_2Ereal__topology_2Efrom_2E1(c_2Earithmetic_2E_2B_2E2(V3n_2E0,V0k_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__SUM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V3s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V3s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,V3s_2E0,V2k_2E0))
& ! [V4x_2E0: tyop_2Enum_2Enum] :
( ~ p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4x_2E0,V3s_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
& ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,V3s_2E0,V0f_2E0) = V1l_2E0 ) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,V2k_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__RESTRICT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),V3n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3n_2E0,V1k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2l_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),V2l_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum)) = c_2Ereal__topology_2Esums_2E3(V0f_2E0,V2l_2E0,V1k_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__TRIVIAL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum))) ).
tff(thm_2Ereal__topology_2ESERIES__DIFFS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1k_2E0),c_2Ereal__topology_2Efrom_2E1(V1k_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__COMPONENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3i_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V2l_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2l_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__FINITE__SUPPORT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,V2k_2E0)))
& ! [V3x_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3x_2E0,V2k_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3x_2E0,V1s_2E0)) )
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,V2k_2E0),V0f_2E0),V2k_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMS__0,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2n_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMS__EQ,axiom,
! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( ! [V4x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4x_2E0,V3k_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0) ) )
& p(c_2Ereal__topology_2Esums_2E3(V1f_2E0,V0l_2E0,V3k_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(V2g_2E0,V0l_2E0,V3k_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUMS__IFF,axiom,
! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V4x_2E0: tyop_2Enum_2Enum] :
( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4x_2E0,V3k_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0) ) )
=> ( c_2Ereal__topology_2Esums_2E3(V1f_2E0,V0l_2E0,V3k_2E0) = c_2Ereal__topology_2Esums_2E3(V2g_2E0,V0l_2E0,V3k_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esums_2E3(V0x_2E0,V1x0_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1x0_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V0x_2E0),V4n_2E0) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esums_2E3(V0x_2E0,V1x0_2E0,V3s_2E0))
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V1x0_2E0),V3s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V5n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V5n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal,V4s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0x_2E0,V1x0_2E0,V4s_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V2y_2E0,V3y0_2E0,V4s_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Ereal_2Ereal__sub_2E2(V1x0_2E0,V3y0_2E0),V4s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V5n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V5n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal,V4s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0x_2E0,V1x0_2E0,V4s_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V2y_2E0,V3y0_2E0,V4s_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x0_2E0,V3y0_2E0),V4s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V1n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Esums_2E3(F0_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__LINEAR,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V0f_2E0),V4n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V2l_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1h_2E0)) )
=> p(c_2Ereal__topology_2Esums_2E3(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V0f_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,V2l_2E0),V3s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESERIES__FINITE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1s_2E0))
=> p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,V1s_2E0,V0f_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESERIES__TERMS__TOZERO,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,c_2Ereal__topology_2Efrom_2E1(V2n_2E0)))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ).
tff(thm_2Ereal__topology_2EINFSUM__UNIQUE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,V2s_2E0))
=> ( c_2Ereal__topology_2Einfsum_2E2(V2s_2E0,V0f_2E0) = V1l_2E0 ) ) ).
tff(thm_2Ereal__topology_2ESERIES__UNIQUE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2l_27_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V2l_27_2E0,V3s_2E0)) )
=> ( V1l_2E0 = V2l_27_2E0 ) ) ).
tff(thm_2Ereal__topology_2ESERIES__FROM,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2k_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2k_2E0),V0f_2E0),V3n_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2k_2E0,V3n_2E0),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2k_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,c_2Ereal__topology_2Efrom_2E1(V2k_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2k_2E0),V0f_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0) ) ) ).
tff(thm_2Ereal__topology_2EFINITE__INTER__NUMSEG,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V0s_2E0,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0)))) ).
tff(thm_2Ereal__topology_2ESUMS__LIM,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0f_2E0),V2n_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,V1s_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V2n_2E0)),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Ereal__topology_2Elim__def_2E2(tyop_2Enum_2Enum,c_2Ereal__topology_2Esequentially_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0f_2E0)),V1s_2E0) = c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESUMS__INFSUM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Esums_2E3(V0f_2E0,c_2Ereal__topology_2Einfsum_2E2(V1s_2E0,V0f_2E0),V1s_2E0) = c_2Ereal__topology_2Esummable_2E2(V1s_2E0,V0f_2E0) ) ).
tff(thm_2Ereal__topology_2ESUMS__SUMMABLE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esums_2E3(V0f_2E0,V1l_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Esummable_2E2(V2s_2E0,V0f_2E0)) ) ).
tff(thm_2Ereal__topology_2EIMAGE__AFFINITY__INTERVAL,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V2m_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2m_2E0),V3c_2E0),V4x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2m_2E0,V4x_2E0),V3c_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2m_2E0),V3c_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2m_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2m_2E0,V0a_2E0),V3c_2E0),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2m_2E0,V1b_2E0),V3c_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2m_2E0,V1b_2E0),V3c_2E0),c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2m_2E0,V0a_2E0),V3c_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__EQ__AFFINITY,axiom,
! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( V0m_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ( V3y_2E0 = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V2x_2E0),V1c_2E0) )
<=> ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V3y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1c_2E0,V0m_2E0))) = V2x_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__AFFINITY__EQ,axiom,
! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( V0m_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V2x_2E0),V1c_2E0) = V3y_2E0 )
<=> ( V2x_2E0 = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V3y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1c_2E0,V0m_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__AFFINITY,axiom,
! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0m_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(V3y_2E0,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V2x_2E0),V1c_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V3y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1c_2E0,V0m_2E0))),V2x_2E0) ) ) ).
tff(thm_2Ereal__topology_2EREAL__AFFINITY__LT,axiom,
! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0m_2E0))
=> ( c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V2x_2E0),V1c_2E0),V3y_2E0) = c_2Erealax_2Ereal__lt_2E2(V2x_2E0,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V3y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1c_2E0,V0m_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__AFFINITY,axiom,
! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0m_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(V3y_2E0,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V2x_2E0),V1c_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V3y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1c_2E0,V0m_2E0))),V2x_2E0) ) ) ).
tff(thm_2Ereal__topology_2EREAL__AFFINITY__LE,axiom,
! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0m_2E0))
=> ( c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V2x_2E0),V1c_2E0),V3y_2E0) = c_2Ereal_2Ereal__lte_2E2(V2x_2E0,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V3y_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2E_2F_2E2(V1c_2E0,V0m_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EAFFINITY__INVERSES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0m_2E0),V1c_2E0),V6x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V6x_2E0),c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V1c_2E0))) )
=> ( ! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0,V0m_2E0),V1c_2E0),V7x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V7x_2E0),V1c_2E0) )
=> ( ! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0m_2E0),V1c_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V0m_2E0,V3x_2E0),V1c_2E0) )
=> ! [V0m_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal] :
( ( V0m_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0m_2E0),V1c_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,c_2Erealax_2Einv_2E1(V0m_2E0)),c_2Erealax_2Ereal__neg_2E1(c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(V0m_2E0),V1c_2E0))),V2x_2E0) = V2x_2E0 )
& ! [V5x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0m_2E0),V1c_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0,V0m_2E0),V1c_2E0),V5x_2E0) = V5x_2E0 ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EDIM__SUBSTANDARD,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0))
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0)) )
=> ( c_2Ereal__topology_2Edim_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0)) = c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SUBSTANDARD,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0))
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__SUBSTANDARD,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0))
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0)) )
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESYMMETRIC__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V1x_2E0),V0s_2E0)) )
=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V2x_2E0),c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__NEGATIONS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F1_2E0,V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(V2x_2E0) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F1_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESYMMETRIC__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V1x_2E0),V0s_2E0)) )
=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V2x_2E0),c_2Ereal__topology_2Einterior_2E1(V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__NEGATIONS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F1_2E0,V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(V2x_2E0) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F1_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V1f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4y_2E0) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( c_2Ereal__topology_2Elimit__point__of_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0x_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2s_2E0)) = c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V2s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> ( c_2Ereal__topology_2Ecomplete_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Ereal__topology_2Ecomplete_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECOMPLETE__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0) )
=> ( V1x_2E0 = V2y_2E0 ) ) )
=> ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecomplete_2E1(V3s_2E0))
=> p(c_2Ereal__topology_2Ecomplete_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__OPEN__MAPPING,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0))
& ( c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1g_2E0) = c_2Ecombin_2EI_2E0(tyop_2Erealax_2Ereal) ) )
=> ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EIN__INTERIOR__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0))
& ( c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1g_2E0) = c_2Ecombin_2EI_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Einterior_2E1(V2s_2E0))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__BIJECTIVE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) )
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
? [V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V4y_2E0 ) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__IMAGE__SUBSET__INTERIOR,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2y_2E0: tyop_2Erealax_2Ereal] :
? [V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2y_2E0 ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)),c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INTERIOR__IMAGE__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__BOUNDED__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0)) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__LINEAR__IMAGE__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)),c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> ( c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Ereal__topology_2EClosed_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INJECTIVE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0) )
=> ( V1x_2E0 = V2y_2E0 ) ) )
=> ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V3s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__BIJECTIVE__LINEAR__IMAGE__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) )
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
? [V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V4y_2E0 ) )
=> ( c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Ereal__topology_2EOpen_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SURJECTIVE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V1y_2E0: tyop_2Erealax_2Ereal] :
? [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = V1y_2E0 ) )
=> ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V3s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INJECTIVE__IMAGE__SUBSPACE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( V2x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
& p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EINJECTIVE__IMP__ISOMETRIC,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( V2x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)),c_2Erealax_2Ereal__mul_2E2(V3e_2E0,c_2Ereal_2Eabs_2E1(V4x_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPLETE__ISOMETRIC__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)),c_2Erealax_2Ereal__mul_2E2(V2e_2E0,c_2Ereal_2Eabs_2E1(V3x_2E0)))) )
& p(c_2Ereal__topology_2Ecomplete_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecomplete_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECAUCHY__ISOMETRIC,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)),c_2Erealax_2Ereal__mul_2E2(V2e_2E0,c_2Ereal_2Eabs_2E1(V4x_2E0)))) )
& ! [V5n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3x_2E0,V5n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2Ecauchy_2E1(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0))) )
=> p(c_2Ereal__topology_2Ecauchy_2E1(V3x_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__STANDARD__HYPERPLANE__HYPERPLANE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V3x_2E0),V2c_2E0))
<=> ( V3x_2E0 = V2c_2E0 ) )
=> ( ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V2c_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V3x_2E0),V2c_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V4x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V4x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V4x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V2c_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANE__STANDARD__HYPERPLANE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V4x_2E0),V2c_2E0))
<=> ( V4x_2E0 = V2c_2E0 ) )
=> ( ! [V2c_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2c_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F3_2E0,V4x_2E0),V2c_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2c_2E0)))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__HYPERPLANES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V4x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V4x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V4x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V2c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V2c_2E0),V3d_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) )
=> ? [V2a_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2a_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) != app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3e_2E0,c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))))) ) ) ) )
=> ? [V5a_2E0: tyop_2Erealax_2Ereal] :
! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = V5a_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT,axiom,
! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2s_2E0),V0t_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0t_2E0))
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0t_2E0,V3y_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) )
=> ? [V4a_2E0: tyop_2Erealax_2Ereal] :
! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = V4a_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__FINITE__RANGE__CONSTANT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0s_2E0))) )
=> ? [V2a_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) = V2a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DISCRETE__RANGE__CONSTANT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V0s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4y_2E0) != app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3e_2E0,c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0))))) ) ) ) )
=> ? [V5a_2E0: tyop_2Erealax_2Ereal] :
! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_2E0) = V5a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__RANGE__CONSTANT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0s_2E0),V2t_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V2t_2E0))
=> ( c_2Ereal__topology_2Econnected__component_2E2(V2t_2E0,V3y_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) )
=> ? [V4a_2E0: tyop_2Erealax_2Ereal] :
! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = V4a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DISCONNECTED__DISCRETE__FINITE__RANGE__CONSTANT__EQ,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0s_2E0),V2t_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V2t_2E0))
=> ( c_2Ereal__topology_2Econnected__component_2E2(V2t_2E0,V3y_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) )
=> ? [V4a_2E0: tyop_2Erealax_2Ereal] :
! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = V4a_2E0 ) ) ) )
& ! [V6s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V6s_2E0))
<=> ! [V7f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V7f_2E0,V6s_2E0))
& ! [V8x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V6s_2E0))
=> ? [V9e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
& ! [V10y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10y_2E0,V6s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7f_2E0,V10y_2E0) != app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7f_2E0,V8x_2E0) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V9e_2E0,c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7f_2E0,V10y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7f_2E0,V8x_2E0))))) ) ) ) )
=> ? [V11a_2E0: tyop_2Erealax_2Ereal] :
! [V12x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V12x_2E0,V6s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7f_2E0,V12x_2E0) = V11a_2E0 ) ) ) )
& ! [V13s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V13s_2E0))
<=> ! [V14f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V14f_2E0,V13s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V14f_2E0,V13s_2E0))) )
=> ? [V15a_2E0: tyop_2Erealax_2Ereal] :
! [V16x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V16x_2E0,V13s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V14f_2E0,V16x_2E0) = V15a_2E0 ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFINITE__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal_2Ereal__lte_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal__topology_2EFINITE__CBALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal_2Ereal__lte_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal__topology_2EEMPTY__INTERIOR__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
=> ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IMP__INFINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EFINITE__IMP__NOT__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ~ p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECARD__EQ__CBALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1r_2E0))
=> p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ) ).
tff(thm_2Ereal__topology_2ECARD__EQ__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1r_2E0))
=> p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ) ).
tff(thm_2Ereal__topology_2EUNCOUNTABLE__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECARD__EQ__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ) ).
tff(thm_2Ereal__topology_2ECOUNTABLE__OPEN__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EUNCOUNTABLE__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECARD__EQ__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ) ) ).
tff(thm_2Ereal__topology_2EUNCOUNTABLE__EUCLIDEAN,axiom,
~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECARD__EQ__EUCLIDEAN,axiom,
p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECARD__EQ__REAL__IMP__UNCOUNTABLE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)))
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EUNCOUNTABLE__REAL,axiom,
~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECARD__EQ__REAL,axiom,
p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Epred__set_2EUNIV_2E0(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVAL__UNIV,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V1b_2E0))
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__OPEN__INTERVALS,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V1b_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2c_2E0,V3d_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0)))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__ONE__POINT__COMPACTIFICATIONS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3b_2E0,V1t_2E0))
& p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2a_2E0),c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V3b_2E0))) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2d_2E0)),c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V3e_2E0)))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__CBALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2d_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V3e_2E0)))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2d_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V3e_2E0)))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__BALLS__CBALL__SPHERE,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2d_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V3e_2E0)))) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V6d_2E0: tyop_2Erealax_2Ereal,V7e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V6d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7e_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V6d_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5b_2E0,V7e_2E0)))) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal,V10d_2E0: tyop_2Erealax_2Ereal,V11e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V10d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V11e_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V10d_2E0)),c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9b_2E0,V11e_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__AFFINITY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V2c_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V3x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( V2c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V2c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( ( V1c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__CONNECTEDNESS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
=> ( c_2Ereal__topology_2Econnected_2E1(V0s_2E0) = c_2Ereal__topology_2Econnected_2E1(V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACTNESS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
=> ( c_2Ereal__topology_2Ecompact_2E1(V0s_2E0) = c_2Ereal__topology_2Ecompact_2E1(V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6y_2E0,V3t_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6y_2E0) ) )
=> ( V5x_2E0 = V6y_2E0 ) )
& ? [V7h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V4u_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V7h_2E0))) )
=> ( ? [V8f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V8f_27_2E0)))
& ? [V9g_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0,V4u_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0,V9g_27_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ? [V5h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V4u_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V5h_2E0))) )
=> ( ? [V6f_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6f_27_2E0)))
& ? [V7g_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0,V4u_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0,V7g_27_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V0s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V0s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V5g_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V2s_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) = V3x_2E0 ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__IMP__QUOTIENT__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0)) )
=> ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__IMP__CLOSED__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0)) )
=> ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3u_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EINJECTIVE__MAP__OPEN__IFF__CLOSED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) )
<=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6u_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5g_2E0)))
<=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6u_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5g_2E0)))
<=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6u_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__CLOSED__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0)))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V4u_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4u_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__OPEN__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0)))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V4u_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4u_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__CLOSED__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) )
& ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) ) )
=> ? [V6g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6g_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__INJECTIVE__OPEN__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) ) )
=> ( V3x_2E0 = V4y_2E0 ) )
& ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) ) )
=> ? [V6g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V2t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6g_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__CLOSED__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)) = V4x_2E0 ) )
& ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V5u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__INVERSE__OPEN__MAP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)) = V4x_2E0 ) )
& ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V5u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE__UNIQUE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V8x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V2s_2E0),V1g_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V1g_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V2s_2E0),V1g_2E0),V7v_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V2s_2E0) = V4u_2E0 )
& ! [V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5v_2E0,V3t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V5v_2E0) ) )
& ! [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V7v_2E0,V4u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V1g_2E0),V7v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),V7v_2E0) ) )
& ! [V9x_2E0: tyop_2Erealax_2Ereal,V10y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10y_2E0,V2s_2E0)) )
=> ( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V9x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V10y_2E0) )
<=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V9x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V10y_2E0) ) ) ) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(V3t_2E0,V4u_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE__STRONG,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
| p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0)) )
=> ( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
<=> ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0))
& ( c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V0s_2E0) = c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1t_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0)) )
=> ( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
<=> ( c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V0s_2E0) = c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__RIGHT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V3x_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0c_2E0))
=> ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V2t_2E0)) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,V2t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__SCALING__LEFT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V3x_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0c_2E0))
=> ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V1s_2E0),V2t_2E0) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,V2t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__IMP__QUOTIENT__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0)))
=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4u_2E0,V3t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V4u_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__RIGHT__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2t_2E0)) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,V2t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__LEFT__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0),V2t_2E0) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,V2t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANSLATION__SELF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__RIGHT__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0)) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,V2t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__LEFT__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1s_2E0,V2t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__INJECTIVE__LINEAR__IMAGE__SELF,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__MINIMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
<=> ? [V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0),V1t_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0)) = V4x_2E0 ) ) )
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V1t_2E0))
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V5y_2E0),V0s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V5y_2E0)) = V5y_2E0 ) ) )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2f_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V3g_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__EMPTY,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))
<=> ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),V1s_2E0))
<=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__FINITENESS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
=> ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0) = c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__IMP__CARD__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
=> p(c_2Ecardinal_2Ecardeq_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2Ehomeomorphic_2E2(V1t_2E0,V2u_2E0)) )
=> p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V2u_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__COMPOSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4s_2E0,V5t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0)))
& p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5t_2E0,V6u_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,V3k_2E0))) )
=> p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4s_2E0,V6u_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V0f_2E0),c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3k_2E0)))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__SYM,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0) = c_2Ereal__topology_2Ehomeomorphic_2E2(V1t_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__SYM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0)) = c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0,V2s_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0,V0f_2E0)) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V0s_2E0)) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V0s_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,F0_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM__OF__SUBSETS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4s_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V3t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4s_27_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5t_27_2E0,V3t_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4s_27_2E0) = V5t_27_2E0 ) )
=> p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4s_27_2E0,V5t_27_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V1g_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHISM,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Ehomeomorphism_2E2(c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1t_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0,V3g_2E0)))
<=> ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V2f_2E0,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V0s_2E0),V1t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V3g_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V1t_2E0),V0s_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0)) = V4x_2E0 ) )
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V1t_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V5y_2E0)) = V5y_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__DECREASING__CONVERGENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V1n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2n_2E0))) )
=> ? [V3l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V3l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__INCREASING__CONVERGENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V1n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V1n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))) )
=> ? [V3l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V3l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EMAPPING__CONNECTED__ONTO__SEGMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ~ ? [V3a_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) )
=> ? [V4f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V4f_2E0,V0s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4f_2E0,V0s_2E0) = c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__IVT__COMPONENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V3a_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V2y_2E0)) )
=> ? [V4z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4z_2E0,V0s_2E0))
& ( V4z_2E0 = V3a_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__IVT__HYPERPLANE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V3a_2E0,V1x_2E0),V4b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4b_2E0,c_2Erealax_2Ereal__mul_2E2(V3a_2E0,V2y_2E0))) )
=> ? [V5z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5z_2E0,V0s_2E0))
& ( c_2Erealax_2Ereal__mul_2E2(V3a_2E0,V5z_2E0) = V4b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EEXTENSION__FROM__CLOPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0),V3u_2E0))
& ( ( V3u_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ? [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V4g_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V1s_2E0),V3u_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2t_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__1,axiom,
! [F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V3a_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6t_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F2_2E0,V3a_2E0),V0f_2E0),V1g_2E0),V6t_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(V6t_2E0,V3a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6t_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6t_2E0)) )
=> ( ! [V5t_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0,V5t_2E0),V2s_2E0),V3a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5t_2E0,V2s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V5t_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V5t_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V2s_2E0),V3a_2E0),V5t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5t_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0,V5t_2E0),V2s_2E0),V3a_2E0)) )
=> ( ! [V4t_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V4t_2E0),V2s_2E0),V3a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V2s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4t_2E0,V3a_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V4t_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V3a_2E0),V4t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4t_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V4t_2E0),V2s_2E0),V3a_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V3a_2E0))))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V2s_2E0),V3a_2E0))))
& ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3a_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3a_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3a_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F2_2E0,V3a_2E0),V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LE,axiom,
! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8t_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),F2_2E0,V2h_2E0),V4a_2E0),V0f_2E0),V1g_2E0),V8t_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V8t_2E0),V4a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8t_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8t_2E0)) )
=> ( ! [V6t_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4a_2E0: tyop_2Erealax_2Ereal,V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool))),F4_2E0,V6t_2E0),V3s_2E0),V4a_2E0),V2h_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6t_2E0,V3s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4a_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V6t_2E0))) ) )
=> ( ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4a_2E0: tyop_2Erealax_2Ereal,V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6t_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3s_2E0),V4a_2E0),V2h_2E0),V6t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6t_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool))),F4_2E0,V6t_2E0),V3s_2E0),V4a_2E0),V2h_2E0)) )
=> ( ! [V5t_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5t_2E0),V3s_2E0),V2h_2E0),V4a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5t_2E0,V3s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V5t_2E0),V4a_2E0)) ) )
=> ( ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal,V5t_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V3s_2E0),V2h_2E0),V4a_2E0),V5t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5t_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5t_2E0),V3s_2E0),V2h_2E0),V4a_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V3s_2E0),V2h_2E0),V4a_2E0))))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3s_2E0),V4a_2E0),V2h_2E0))))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2h_2E0,V3s_2E0))
& ! [V7t_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7t_2E0,V3s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V7t_2E0) = V4a_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7t_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V7t_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),F2_2E0,V2h_2E0),V4a_2E0),V0f_2E0),V1g_2E0),V3s_2E0)) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V6x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0)),V3s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0)),V4t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2g_2E0,V4t_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0))
& ~ p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL,axiom,
! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0t_2E0)),V2s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0t_2E0)),V0t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0t_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V6x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V3s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V4t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2g_2E0,V4t_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0))
& ~ p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__UNION,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3l_2E0: tyop_2Erealax_2Ereal,V4f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V4f_2E0,V3l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0x_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V1t_2E0))))
<=> ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V4f_2E0,V3l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0x_2E0),V2s_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V4f_2E0,V3l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0x_2E0),V1t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__MAP__CLOSURES,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)))
<=> ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)),c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__GE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0))) )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__COMPONENT__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0),V3b_2E0)) )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),V3b_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__ABS__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0)),V3b_2E0)) )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)),V3b_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__IMAGE__BOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0)) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__IMAGE__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) = c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EIMAGE__CLOSURE__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)),V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIM__DROP__LBOUND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V3b_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V3b_2E0),V1f_2E0),V4x_2E0) = c_2Ereal_2Ereal__lte_2E2(V3b_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V3b_2E0),V1f_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,V2l_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__DROP__UBOUND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3b_2E0: tyop_2Erealax_2Ereal,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3b_2E0),V4x_2E0) = c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),V3b_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3b_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2l_2E0,V3b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__DROP__LE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal,V4m_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V4m_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3l_2E0,V4m_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPONENT__LE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal,V4m_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V4m_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3l_2E0,V4m_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPONENT__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4b_2E0: tyop_2Erealax_2Ereal,V5x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V4b_2E0),V5x_2E0))
<=> ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = V4b_2E0 ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2i_2E0: A_27b,V3l_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V4b_2E0),V0net_2E0)) )
=> ( V3l_2E0 = V4b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPONENT__LBOUND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V3b_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V3b_2E0),V1f_2E0),V4x_2E0) = c_2Ereal_2Ereal__lte_2E2(V3b_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V3b_2E0),V1f_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,V2l_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPONENT__UBOUND,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3b_2E0: tyop_2Erealax_2Ereal,V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3b_2E0),V5x_2E0) = c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),V3b_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4k_2E0: A_27b] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3b_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2l_2E0,V3b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECARD__FRONTIER__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
=> ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)))
& p(c_2Earithmetic_2E_3C_3D_2E2(c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__SCALING__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eis__interval_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0)))
<=> ( ( V1c_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Eis__interval_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Eis__interval_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2EINTERVAL__CONTAINS__COMPACT__NEIGHBOURHOOD,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) )
=> ? [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal,V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V4d_2E0)),V0s_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__SUBSET__IS__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),V0s_2E0))
<=> ( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Eis__interval_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Eis__interval_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__CASES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F11_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F7_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F10_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F6_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F9_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F8_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V3a_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V3a_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V4x_2E0)) )
=> ( ! [V5b_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V5b_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V5b_2E0)) )
=> ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Erealax_2Ereal__lt_2E2(V1a_2E0,V2x_2E0)) )
=> ( ! [V7b_2E0: tyop_2Erealax_2Ereal,V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V7b_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,c_2Erealax_2Ereal__lt_2E2(V8x_2E0,V7b_2E0)) )
=> ( ! [V18a_2E0: tyop_2Erealax_2Ereal,V20x_2E0: tyop_2Erealax_2Ereal,V19b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F11_2E0,V18a_2E0),V20x_2E0),V19b_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V18a_2E0,V20x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V20x_2E0,V19b_2E0)) ) )
=> ( ! [V18a_2E0: tyop_2Erealax_2Ereal,V19b_2E0: tyop_2Erealax_2Ereal,V20x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F7_2E0,V18a_2E0),V19b_2E0),V20x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V20x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F11_2E0,V18a_2E0),V20x_2E0),V19b_2E0)) )
=> ( ! [V15a_2E0: tyop_2Erealax_2Ereal,V17x_2E0: tyop_2Erealax_2Ereal,V16b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F10_2E0,V15a_2E0),V17x_2E0),V16b_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V15a_2E0,V17x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V17x_2E0,V16b_2E0)) ) )
=> ( ! [V15a_2E0: tyop_2Erealax_2Ereal,V16b_2E0: tyop_2Erealax_2Ereal,V17x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F6_2E0,V15a_2E0),V16b_2E0),V17x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V17x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F10_2E0,V15a_2E0),V17x_2E0),V16b_2E0)) )
=> ( ! [V12a_2E0: tyop_2Erealax_2Ereal,V14x_2E0: tyop_2Erealax_2Ereal,V13b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F9_2E0,V12a_2E0),V14x_2E0),V13b_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V12a_2E0,V14x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V14x_2E0,V13b_2E0)) ) )
=> ( ! [V12a_2E0: tyop_2Erealax_2Ereal,V13b_2E0: tyop_2Erealax_2Ereal,V14x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V12a_2E0),V13b_2E0),V14x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V14x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F9_2E0,V12a_2E0),V14x_2E0),V13b_2E0)) )
=> ( ! [V9a_2E0: tyop_2Erealax_2Ereal,V11x_2E0: tyop_2Erealax_2Ereal,V10b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F8_2E0,V9a_2E0),V11x_2E0),V10b_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V9a_2E0,V11x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V11x_2E0,V10b_2E0)) ) )
=> ( ! [V9a_2E0: tyop_2Erealax_2Ereal,V10b_2E0: tyop_2Erealax_2Ereal,V11x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V9a_2E0),V10b_2E0),V11x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V11x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F8_2E0,V9a_2E0),V11x_2E0),V10b_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V0s_2E0 = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) )
| ? [V1a_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1a_2E0)) )
| ? [V3a_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V3a_2E0)) )
| ? [V5b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V5b_2E0)) )
| ? [V7b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V7b_2E0)) )
| ? [V9a_2E0: tyop_2Erealax_2Ereal,V10b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V9a_2E0),V10b_2E0)) )
| ? [V12a_2E0: tyop_2Erealax_2Ereal,V13b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V12a_2E0),V13b_2E0)) )
| ? [V15a_2E0: tyop_2Erealax_2Ereal,V16b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F6_2E0,V15a_2E0),V16b_2E0)) )
| ? [V18a_2E0: tyop_2Erealax_2Ereal,V19b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F7_2E0,V18a_2E0),V19b_2E0)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
<=> ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0)) )
<=> ? [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__POINTWISE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
=> ( ? [V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V0s_2E0))
& ( V2a_2E0 = V1x_2E0 ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__UNIV,axiom,
p(c_2Ereal__topology_2Eis__interval_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__EMPTY,axiom,
p(c_2Ereal__topology_2Eis__interval_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eis__interval_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Ereal__topology_2Eis__interval_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__SCALAR__MULTIPLE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( ( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V1b_2E0)) )
| ( p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V0a_2E0)) ) ) )
=> ( ! [V2v_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2v_2E0),V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__mul_2E2(V3x_2E0,V2v_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ( ! [V4a_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V4a_2E0),V7x_2E0),V5b_2E0))
<=> ( ( p(c_2Erealax_2Ereal__lt_2E2(V4a_2E0,V7x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V7x_2E0,V5b_2E0)) )
| ( p(c_2Erealax_2Ereal__lt_2E2(V5b_2E0,V7x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V7x_2E0,V4a_2E0)) ) ) )
=> ( ! [V6v_2E0: tyop_2Erealax_2Ereal,V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V6v_2E0),V4a_2E0),V5b_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__mul_2E2(V7x_2E0,V6v_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V4a_2E0),V7x_2E0),V5b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2v_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2v_2E0),c_2Erealax_2Ereal__mul_2E2(V1b_2E0,V2v_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2v_2E0),V0a_2E0),V1b_2E0)) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V6v_2E0: tyop_2Erealax_2Ereal] :
( ( V6v_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V4a_2E0,V6v_2E0),c_2Erealax_2Ereal__mul_2E2(V5b_2E0,V6v_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V6v_2E0),V4a_2E0),V5b_2E0)) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ).
tff(thm_2Ereal__topology_2ESEGMENT,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V0a_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3b_2E0),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3b_2E0,V2a_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal_2E_2F_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E2(V2a_2E0,V3b_2E0),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),c_2Ereal_2E_2F_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__DIFF__OPEN__INTERVAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V1b_2E0))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V1b_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__INTERVAL__0,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0e_2E0)) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V0e_2E0),V0e_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2EBALL__INTERVAL__0,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0e_2E0)) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V0e_2E0),V0e_2E0)) ) ).
tff(thm_2Ereal__topology_2ECBALL__INTERVAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1e_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1e_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2EBALL__INTERVAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1e_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1e_2E0))) ) ).
tff(thm_2Ereal__topology_2EFINITE__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V0a_2E0) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))) = c_2Ereal_2Ereal__lte_2E2(V3b_2E0,V2a_2E0) ) ) ).
tff(thm_2Ereal__topology_2EFINITE__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)))) ).
tff(thm_2Ereal__topology_2ESPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0a_2E0,V1r_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1r_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2EBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1r_2E0)) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1r_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1r_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) )
& ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1r_2E0)) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1r_2E0),c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__OPEN__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__CASES,axiom,
! [V0b_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0))))
| ( V2x_2E0 = V1a_2E0 )
| ( V2x_2E0 = V0b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EEQ__BALLS,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1a_27_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal,V3r_27_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2r_2E0)) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_27_2E0,V3r_27_2E0)) )
<=> ( ( ( V0a_2E0 = V1a_27_2E0 )
& ( V2r_2E0 = V3r_27_2E0 ) )
| ( p(c_2Ereal_2Ereal__lte_2E2(V2r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(V3r_27_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5a_27_2E0: tyop_2Erealax_2Ereal,V6r_2E0: tyop_2Erealax_2Ereal,V7r_27_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V6r_2E0)) = c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5a_27_2E0,V7r_27_2E0)) )
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V6r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V7r_27_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal,V9a_27_2E0: tyop_2Erealax_2Ereal,V10r_2E0: tyop_2Erealax_2Ereal,V11r_27_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V10r_2E0)) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9a_27_2E0,V11r_27_2E0)) )
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V10r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(V11r_27_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
& ! [V12a_2E0: tyop_2Erealax_2Ereal,V13a_27_2E0: tyop_2Erealax_2Ereal,V14r_2E0: tyop_2Erealax_2Ereal,V15r_27_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V14r_2E0)) = c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V13a_27_2E0,V15r_27_2E0)) )
<=> ( ( ( V12a_2E0 = V13a_27_2E0 )
& ( V14r_2E0 = V15r_27_2E0 ) )
| ( p(c_2Erealax_2Ereal__lt_2E2(V14r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(V15r_27_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__INTERVAL__EQ,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTERVAL__EQ,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EOpen_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTERVAL__EQ,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EEQ__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) )
<=> ( ( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
| ( ( V0a_2E0 = V2c_2E0 )
& ( V1b_2E0 = V3d_2E0 ) ) ) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V6c_2E0: tyop_2Erealax_2Ereal,V7d_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6c_2E0,V7d_2E0)) )
<=> ( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6c_2E0,V7d_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal,V10c_2E0: tyop_2Erealax_2Ereal,V11d_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V9b_2E0)) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V10c_2E0,V11d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) )
<=> ( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V9b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V10c_2E0,V11d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ! [V12a_2E0: tyop_2Erealax_2Ereal,V13b_2E0: tyop_2Erealax_2Ereal,V14c_2E0: tyop_2Erealax_2Ereal,V15d_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V13b_2E0)) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V14c_2E0,V15d_2E0)) )
<=> ( ( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V13b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V14c_2E0,V15d_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
| ( ( V12a_2E0 = V14c_2E0 )
& ( V13b_2E0 = V15d_2E0 ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EIMAGE__TWIZZLE__INTERVAL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V3x_2E0) = V3x_2E0 )
=> ! [V0p_2E0: A_27a,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__INTERVAL,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V2a_2E0),V3b_2E0))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V1b_2E0))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ediameter_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V1b_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0))) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ediameter_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V2a_2E0),V3b_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3b_2E0,V2a_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__OPEN__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__INTERVAL__OPEN__INTERVAL,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),V0s_2E0)) ) ) )
& ! [V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V4s_2E0))
<=> ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4s_2E0))
=> ? [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6a_2E0,V7b_2E0))))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6a_2E0,V7b_2E0)),V4s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESUMS__INTERVALS,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))))] :
( ! [V10x_2E0: tyop_2Erealax_2Ereal,V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal,V11y_2E0: tyop_2Erealax_2Ereal,V8c_2E0: tyop_2Erealax_2Ereal,V9d_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0,V10x_2E0),V6a_2E0),V7b_2E0),V11y_2E0),V8c_2E0),V9d_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6a_2E0,V7b_2E0))))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11y_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8c_2E0,V9d_2E0)))) ) )
=> ( ! [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal,V8c_2E0: tyop_2Erealax_2Ereal,V9d_2E0: tyop_2Erealax_2Ereal,V10x_2E0: tyop_2Erealax_2Ereal,V11y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F1_2E0,V6a_2E0),V7b_2E0),V8c_2E0),V9d_2E0),V10x_2E0),V11y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V10x_2E0,V11y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0,V10x_2E0),V6a_2E0),V7b_2E0),V11y_2E0),V8c_2E0),V9d_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0,V4x_2E0),V0a_2E0),V1b_2E0),V5y_2E0),V2c_2E0),V3d_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F0_2E0,V0a_2E0),V1b_2E0),V2c_2E0),V3d_2E0),V4x_2E0),V5y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V4x_2E0,V5y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0,V4x_2E0),V0a_2E0),V1b_2E0),V5y_2E0),V2c_2E0),V3d_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F0_2E0,V0a_2E0),V1b_2E0),V2c_2E0),V3d_2E0))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2c_2E0),c_2Erealax_2Ereal__add_2E2(V1b_2E0,V3d_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) )
& ! [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal,V8c_2E0: tyop_2Erealax_2Ereal,V9d_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6a_2E0,V7b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8c_2E0,V9d_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F1_2E0,V6a_2E0),V7b_2E0),V8c_2E0),V9d_2E0))) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V6a_2E0,V8c_2E0),c_2Erealax_2Ereal__add_2E2(V7b_2E0,V9d_2E0))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTERVAL__IMAGE__UNIT__INTERVAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ( ! [V1b_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V4f_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V1b_2E0),V0a_2E0),V3x_2E0),V4f_2E0))
<=> ( V4f_2E0 = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V0a_2E0),V3x_2E0) ) )
=> ( ! [V1b_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V1b_2E0),V0a_2E0),V3x_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V1b_2E0),V0a_2E0),V3x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V1b_2E0),V0a_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__IMAGE__STRETCH__INTERVAL,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal,V6f_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2m_2E0),V5x_2E0),V6f_2E0))
<=> ( V6f_2E0 = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V5x_2E0) ) )
=> ( ! [V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2m_2E0),V5x_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2m_2E0),V5x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
? [V3u_2E0: tyop_2Erealax_2Ereal,V4v_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2m_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3u_2E0,V4v_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ) ) ).
tff(thm_2Ereal__topology_2EIMAGE__STRETCH__INTERVAL,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal,V4f_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0,V2m_2E0),V3x_2E0),V4f_2E0))
<=> ( V4f_2E0 = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3x_2E0) ) )
=> ( ! [V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2m_2E0),V3x_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0,V2m_2E0),V3x_2E0)) )
=> ( ! [V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V5f_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V2m_2E0),V0a_2E0),V1b_2E0),V5f_2E0))
<=> ( V5f_2E0 = c_2Ereal_2Emin_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0a_2E0),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1b_2E0)) ) )
=> ( ! [V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V6f_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V2m_2E0),V0a_2E0),V1b_2E0),V6f_2E0))
<=> ( V6f_2E0 = c_2Ereal_2Emax_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0a_2E0),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1b_2E0)) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2m_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2m_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0a_2E0),V1b_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0,V2m_2E0),V0a_2E0),V1b_2E0)),c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V2m_2E0),V0a_2E0),V1b_2E0))),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIT__INTERVAL__NONEMPTY,axiom,
( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EEMPTY__AS__INTERVAL,axiom,
c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2EINTERVAL__TRANSLATION,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V4c_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V4c_2E0),V7x_2E0) = c_2Erealax_2Ereal__add_2E2(V4c_2E0,V7x_2E0) )
=> ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0c_2E0,V3x_2E0) )
=> ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0c_2E0,V1a_2E0),c_2Erealax_2Ereal__add_2E2(V0c_2E0,V2b_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) )
& ! [V4c_2E0: tyop_2Erealax_2Ereal,V5a_2E0: tyop_2Erealax_2Ereal,V6b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V4c_2E0,V5a_2E0),c_2Erealax_2Ereal__add_2E2(V4c_2E0,V6b_2E0))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V4c_2E0),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5a_2E0,V6b_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTER__INTERVAL__MIXED__EQ__EMPTY,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__OPEN__INTERVAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V1b_2E0))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0a_2E0),V1b_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__CLOSED__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET__CLOSED__INTERVAL__SYMMETRIC,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V1a_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V1a_2E0),V1a_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET__OPEN__INTERVAL__SYMMETRIC,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V1a_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V1a_2E0),V1a_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__INTERVAL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V2a_2E0),V3b_2E0))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V2a_2E0),V3b_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__OPEN__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ) ).
tff(thm_2Ereal__topology_2EREAL__INV__LE__1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Einv_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__INV2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Einv_2E1(V1y_2E0),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__INV2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1y_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Einv_2E1(V1y_2E0),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__POW__LE__1,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__POW__1__LE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Epow_2E2(V1x_2E0,V0n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__BIGINTER,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EDIFF_2E2(A_27a,c_2Epred__set_2EUNIV_2E0(A_27a),V1t_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27a,V0s_2E0) = c_2Epred__set_2EDIFF_2E2(A_27a,c_2Epred__set_2EUNIV_2E0(A_27a),c_2Epred__set_2EBIGINTER_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EBIGINTER__BIGUNION,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EDIFF_2E2(A_27a,c_2Epred__set_2EUNIV_2E0(A_27a),V1t_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(A_27a,V0s_2E0) = c_2Epred__set_2EDIFF_2E2(A_27a,c_2Epred__set_2EUNIV_2E0(A_27a),c_2Epred__set_2EBIGUNION_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EDIFF__BIGINTER,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V0u_2E0),V1s_2E0),V2t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EDIFF_2E2(A_27a,V0u_2E0,V2t_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0,V1s_2E0)) )
=> ! [V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27a,V0u_2E0,c_2Epred__set_2EBIGINTER_2E1(A_27a,V1s_2E0)) = c_2Epred__set_2EBIGUNION_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V0u_2E0),V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUBSET__ANTISYM__EQ,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V0s_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V1t_2E0,V0s_2E0)) )
<=> ( V0s_2E0 = V1t_2E0 ) ) ).
tff(thm_2Ereal__topology_2EFUN__IN__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
=> p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0),c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__POW2,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0n_2E0))) ).
tff(thm_2Ereal__topology_2ESUBSET__BIGUNION,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0,V1g_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V0f_2E0),c_2Epred__set_2EBIGUNION_2E1(A_27a,V1g_2E0))) ) ).
tff(thm_2Ereal__topology_2ESIMPLE__IMAGE__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),F1_2E0,V1P_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V3x_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1P_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0),app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1P_2E0)) = c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),F1_2E0,V1P_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__LCANCEL__IMP,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Ereal__topology_2EPOWERSET__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0s_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0s_2E0,c_2Epred__set_2ESUBSET_2E2(A_27b,V0s_2E0,c_2Epred__set_2EEMPTY_2E0(A_27b))) )
=> ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V2t_2E0),V4s_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V4s_2E0,c_2Epred__set_2ESUBSET_2E2(A_27a,V4s_2E0,V2t_2E0)) )
=> ( ! [V1a_2E0: A_27a,V5s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F3_2E0,V1a_2E0),V5s_2E0) = c_2Epred__set_2EINSERT_2E2(A_27a,V1a_2E0,V5s_2E0) )
=> ( ! [V1a_2E0: A_27a,V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V1a_2E0),V2t_2E0),V3s_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V3s_2E0,c_2Epred__set_2ESUBSET_2E2(A_27a,V3s_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1a_2E0,V2t_2E0))) )
=> ( ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F0_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EEMPTY_2E0(A_27b),c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) )
& ! [V1a_2E0: A_27a,V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V1a_2E0),V2t_2E0)) = c_2Epred__set_2EUNION_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V2t_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F3_2E0,V1a_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V2t_2E0)))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__OF__NUM__GE,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Ereal__of__num_2E1(V0m_2E0),c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)) = c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Ereal__topology_2EABS__LE__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ENOT__EQ,axiom,
! [A_27a: $tType,V0a_2E0: A_27a,V1b_2E0: A_27a] :
( ( V0a_2E0 != V1b_2E0 )
<=> ( V0a_2E0 != V1b_2E0 ) ) ).
tff(thm_2Ereal__topology_2ELAMBDA__PAIR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V2x_2E0: A_27a,V3y_2E0: A_27b] : ( app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F0_2E0,V0P_2E0),V2x_2E0),V3y_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0P_2E0,V2x_2E0),V3y_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F0_2E0,V0P_2E0),V1x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0P_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V1x_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V1x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE,axiom,
! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V1x_2E0),V1x_2E0))
& ! [V2x_2E0: tyop_2Enum_2Enum,V3y_2E0: tyop_2Enum_2Enum,V4z_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V2x_2E0),V3y_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V3y_2E0),V4z_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V2x_2E0),V4z_2E0)) )
& ! [V5n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V5n_2E0),c_2Enum_2ESUC_2E1(V5n_2E0))) )
=> ! [V6m_2E0: tyop_2Enum_2Enum,V7n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V6m_2E0,V7n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V6m_2E0),V7n_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LE__EQ,axiom,
! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V1x_2E0),V1x_2E0))
& ! [V2x_2E0: tyop_2Enum_2Enum,V3y_2E0: tyop_2Enum_2Enum,V4z_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V2x_2E0),V3y_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V3y_2E0),V4z_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V2x_2E0),V4z_2E0)) ) )
=> ( ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V5m_2E0,V6n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V5m_2E0),V6n_2E0)) )
<=> ! [V7n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V7n_2E0),c_2Enum_2ESUC_2E1(V7n_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EADD__SUBR,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) = c_2Enum_2E0_2E0 ) ).
tff(thm_2Ereal__topology_2EADD__SUBR2,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0)) = c_2Enum_2E0_2E0 ) ).
tff(thm_2Ereal__topology_2EADD__SUB2,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0),V0m_2E0) = V1n_2E0 ) ).
tff(thm_2Ereal__topology_2ELE__ADDR,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0))) ).
tff(thm_2Ereal__topology_2ELE__ADD,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1n_2E0))) ).
tff(thm_2Ereal__topology_2EFINITE__POWERSET,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1t_2E0,c_2Epred__set_2ESUBSET_2E2(A_27a,V1t_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
=> p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INJECTIVE__BOUNDED__BELOW__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0) )
=> ( V1x_2E0 = V2y_2E0 ) ) )
=> ? [V3B_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3B_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V4x_2E0),V3B_2E0),c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0))
& ( c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0) = c_2Ecombin_2EI_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2B_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V2B_2E0,c_2Ereal_2Eabs_2E1(V3x_2E0)),c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INVERTIBLE__BOUNDED__BELOW__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0))
& ( c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0) = c_2Ecombin_2EI_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2B_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2B_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V2B_2E0,c_2Ereal_2Eabs_2E1(V3x_2E0)),c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INJECTIVE__IMP__SURJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0) )
=> ( V1x_2E0 = V2y_2E0 ) ) )
=> ! [V3y_2E0: tyop_2Erealax_2Ereal] :
? [V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V3y_2E0 ) ) ).
tff(thm_2Ereal__topology_2ESPAN__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eindependent_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> p(c_2Ereal__topology_2Eindependent_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__INJECTIVE__IMAGE__GEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eindependent_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) ) )
=> ( V2x_2E0 = V3y_2E0 ) ) )
=> p(c_2Ereal__topology_2Eindependent_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EDEPENDENT__EXPLICIT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4v_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2u_2E0),V4v_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2u_2E0,V4v_2E0),V4v_2E0) )
=> ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Edependent_2E1(V0p_2E0))
<=> ? [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V0p_2E0))
& ? [V3v_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3v_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
& ( c_2Eiterate_2ESum_2E2(tyop_2Erealax_2Ereal,V1s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2u_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__EXPLICIT,axiom,
! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4v_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F2_2E0,V3u_2E0),V4v_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3u_2E0,V4v_2E0),V4v_2E0) )
=> ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0p_2E0),V1y_2E0))
<=> ? [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0p_2E0))
& ( c_2Eiterate_2ESum_2E2(tyop_2Erealax_2Ereal,V2s_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F2_2E0,V3u_2E0)) = V1y_2E0 ) ) )
=> ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0p_2E0),V1y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1y_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0p_2E0),V1y_2E0)) )
=> ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Espan_2E1(V0p_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0p_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECARD__GE__DIM__INDEPENDENT,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal__topology_2Edim_2E1(V0v_2E0),c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1b_2E0))) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0v_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0))) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__INSERT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V1s_2E0),V0a_2E0))
<=> ( p(c_2Ereal__topology_2Eindependent_2E1(V1s_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0))) ) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eindependent_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0),c_2Ereal__topology_2Eindependent_2E1(V1s_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V1s_2E0),V0a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EIN__SPAN__INSERT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V2s_2E0))))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V2s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESPAN__BREAKDOWN,axiom,
! [V0b_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0b_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0))) )
=> ? [V3k_2E0: tyop_2Erealax_2Ereal] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V2a_2E0,c_2Erealax_2Ereal__mul_2E2(V3k_2E0,V0b_2E0)),c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V0b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__MONO,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eindependent_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__NONZERO,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0))
=> ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EMIDPOINT__COLLINEAR,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != V2c_2E0 )
=> ( ( V1b_2E0 = c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2c_2E0)) )
<=> ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__MIDPOINT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ).
tff(thm_2Ereal__topology_2EMIDPOINT__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2b_2E0))) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__MIDPOINT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebetween_2E2(c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))
& p(c_2Ereal__topology_2Ebetween_2E2(c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V0a_2E0))) ) ).
tff(thm_2Ereal__topology_2EMIDPOINT__EQ__ENDPOINT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = V0a_2E0 )
<=> ( V0a_2E0 = V1b_2E0 ) )
& ( ( c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = V1b_2E0 )
<=> ( V0a_2E0 = V1b_2E0 ) )
& ( ( V0a_2E0 = c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) )
<=> ( V0a_2E0 = V1b_2E0 ) )
& ( ( V1b_2E0 = c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) )
<=> ( V0a_2E0 = V1b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EDIST__MIDPOINT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) = c_2Ereal_2E_2F_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) )
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) = c_2Ereal_2E_2F_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) )
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),V0a_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) )
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),V1b_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EMIDPOINT__SYM,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V0a_2E0)) ) ).
tff(thm_2Ereal__topology_2EMIDPOINT__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__1,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Ecollinear_2E1(V0s_2E0)) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__DIST__BETWEEN,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) )
=> p(c_2Ereal__topology_2Ebetween_2E2(V2x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__BETWEEN__CASES,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
<=> ( p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_2E0)))
| p(c_2Ereal__topology_2Ebetween_2E2(V1b_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V0a_2E0)))
| p(c_2Ereal__topology_2Ebetween_2E2(V2c_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__IMP__COLLINEAR,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebetween_2E2(V2x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))
=> p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__ABS,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebetween_2E2(V2x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))
<=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V0a_2E0)),c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V2x_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1b_2E0,V2x_2E0)),c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V0a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__TRANS__2,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_2E0)))
& p(c_2Ereal__topology_2Ebetween_2E2(V3d_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))) )
=> p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0))) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__TRANS,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_2E0)))
& p(c_2Ereal__topology_2Ebetween_2E2(V3d_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2c_2E0))) )
=> p(c_2Ereal__topology_2Ebetween_2E2(V3d_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_2E0))) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__ANTISYM,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V2c_2E0)))
& p(c_2Ereal__topology_2Ebetween_2E2(V1b_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2c_2E0))) )
=> ( V0a_2E0 = V1b_2E0 ) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__SYM,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebetween_2E2(V2x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Ereal__topology_2Ebetween_2E2(V2x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V0a_2E0)) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__REFL__EQ,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebetween_2E2(V1x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V0a_2E0)))
<=> ( V1x_2E0 = V0a_2E0 ) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__REFL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))
& p(c_2Ereal__topology_2Ebetween_2E2(V1b_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))
& p(c_2Ereal__topology_2Ebetween_2E2(V0a_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V0a_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__3__TRANS,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3d_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& ( V1b_2E0 != V2c_2E0 ) )
=> p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3d_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__4__3,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != V1b_2E0 )
=> ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3d_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))))
<=> ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3d_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__TRIPLES,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( V1a_2E0 != V2b_2E0 )
=> ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0))))
<=> ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__3__EXPAND,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2c_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
<=> ( ( V0a_2E0 = V2c_2E0 )
| ? [V3u_2E0: tyop_2Erealax_2Ereal] : ( V1b_2E0 = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V3u_2E0,V0a_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3u_2E0),V2c_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EMUL__CAUCHY__SCHWARZ__EQUAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V1y_2E0,V1y_2E0)) )
<=> p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQUAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) )
<=> p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__LEMMA__ALT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
<=> ( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ? [V2c_2E0: tyop_2Erealax_2Ereal] : ( V1y_2E0 = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V0x_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__LEMMA,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
<=> ( ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ( V1y_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| ? [V2c_2E0: tyop_2Erealax_2Ereal] : ( V1y_2E0 = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V0x_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__3,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2z_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V2z_2E0,V1y_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__SMALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V0s_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) )
=> p(c_2Ereal__topology_2Ecollinear_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__EMPTY,axiom,
p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecollinear_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecollinear_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2z_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1y_2E0,V2z_2E0))) )
<=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)),c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V2z_2E0)),c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EABS__TRIANGLE__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) )
<=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V1y_2E0),V0x_2E0) ) ) ).
tff(thm_2Ereal__topology_2EREAL__EQ__RINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 = c_2Erealax_2Ereal__neg_2E1(V0x_2E0) )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EREAL__EQ__LINV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__neg_2E1(V0x_2E0) = V0x_2E0 )
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__ABS__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) )
<=> ( ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V1y_2E0),V0x_2E0) )
| ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__neg_2E1(c_2Ereal_2Eabs_2E1(V1y_2E0)),V0x_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EABS__CAUCHY__SCHWARZ__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) )
<=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V1y_2E0),V0x_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__UNION,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Ereal__topology_2Espan_2E1(V1t_2E0))) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
=> ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V0s_2E0) = V0s_2E0 )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__TRANSLATION__SELF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0)) )
=> ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V0s_2E0) = V0s_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ESURJECTIVE__IMAGE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( ! [V3y_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3y_2E0,V2t_2E0))
=> ? [V4x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V4x_2E0) = V3y_2E0 ) )
& ! [V5x_2E0: A_27a] : ( c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V0f_2E0,V5x_2E0),V2t_2E0) = c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V1s_2E0) ) )
=> ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V1s_2E0) = V2t_2E0 ) ) ).
tff(thm_2Ereal__topology_2ESPAN__SUBSET__SUBSPACE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1t_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Espan_2E1(V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ESPAN__EQ__SELF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Espan_2E1(V0s_2E0) = V0s_2E0 )
<=> p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESPAN__ADD__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0)))
=> ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0),c_2Ereal__topology_2Espan_2E1(V0s_2E0)) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__SUM,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2t_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),c_2Ereal__topology_2Espan_2E1(V0s_2E0))) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Eiterate_2ESum_2E2(A_27a,V2t_2E0,V1f_2E0),c_2Ereal__topology_2Espan_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),c_2Ereal__topology_2Espan_2E1(V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__NEG__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Ereal__topology_2Espan_2E1(V1s_2E0)) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESPAN__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V1s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V0x_2E0),c_2Ereal__topology_2Espan_2E1(V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__MUL__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( V1c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V0x_2E0),c_2Ereal__topology_2Espan_2E1(V2s_2E0)) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V0x_2E0),c_2Ereal__topology_2Espan_2E1(V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Ereal__topology_2Espan_2E1(V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__0,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Espan_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2ESPAN__UNIV,axiom,
c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2ESPAN__UNION__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Espan_2E1(V0s_2E0),c_2Ereal__topology_2Espan_2E1(V1t_2E0)),c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))) ).
tff(thm_2Ereal__topology_2ESPAN__INC,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2ESPAN__SUPERSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__INDUCT__ALT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1h_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
& ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1h_2E0,V4y_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1h_2E0,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V3x_2E0),V4y_2E0))) ) )
=> ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0)))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1h_2E0,V5x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__SUBSPACE,axiom,
! [V0b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0b_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,c_2Ereal__topology_2Espan_2E1(V0b_2E0)))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0)) )
=> ( c_2Ereal__topology_2Espan_2E1(V0b_2E0) = V1s_2E0 ) ) ).
tff(thm_2Ereal__topology_2EDEPENDENT__MONO,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Edependent_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Edependent_2E1(V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__SQUARE__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V1y_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ).
tff(thm_2Ereal__topology_2EREAL__EQ__SQUARE__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Eabs_2E1(V0x_2E0) = c_2Ereal_2Eabs_2E1(V1y_2E0) )
<=> ( c_2Ereal_2Epow_2E2(V0x_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Ereal_2Epow_2E2(V1y_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EREAL__HALF,axiom,
( ! [V0e_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0e_2E0) )
& ! [V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Ereal_2E_2F_2E2(V1e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2E_2F_2E2(V1e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = V1e_2E0 )
& ! [V2e_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2E_2F_2E2(V2e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = V2e_2E0 ) ) ).
tff(thm_2Ereal__topology_2EFINITE__SUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27b,V2t_2E0,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V1s_2E0))) )
<=> ? [V3s_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3s_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V3s_27_2E0,V1s_2E0))
& ( V2t_2E0 = c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V3s_27_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EEXISTS__FINITE__SUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ? [V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V3t_2E0,c_2Epred__set_2EIMAGE_2E2(A_27b,A_27a,V1f_2E0,V2s_2E0)))
& p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V3t_2E0)) )
<=> ? [V4t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27b,V4t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27b,V4t_2E0,V2s_2E0))
& p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EIMAGE_2E2(A_27b,A_27a,V1f_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__FINITE__SUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ! [V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V3t_2E0,c_2Epred__set_2EIMAGE_2E2(A_27b,A_27a,V1f_2E0,V2s_2E0))) )
=> p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V3t_2E0)) )
<=> ! [V4t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V4t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27b,V4t_2E0,V2s_2E0)) )
=> p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EIMAGE_2E2(A_27b,A_27a,V1f_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__IN__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),V16w_2E0: A_27e,V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V17x_2E0: A_27f,V18y_2E0: A_27g] : ( app_2E2(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F3_2E0,V14f_2E0),V16w_2E0),V13P_2E0),V17x_2E0),V18y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27g,A_27b,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,A_27b),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)),V14f_2E0,V16w_2E0),V17x_2E0),V18y_2E0),app_2E2(A_27g,tyop_2Emin_2Ebool,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),V13P_2E0,V16w_2E0),V17x_2E0),V18y_2E0)) )
=> ( ! [V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V16w_2E0: A_27e] : ( app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F2_2E0,V14f_2E0),V13P_2E0),V16w_2E0) = c_2Epair_2EUNCURRY_2E1(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F3_2E0,V14f_2E0),V16w_2E0),V13P_2E0)) )
=> ( ! [V7f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),V6P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V9x_2E0: A_27c,V10y_2E0: A_27d] : ( app_2E2(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F1_2E0,V7f_2E0),V6P_2E0),V9x_2E0),V10y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27d,A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,A_27b),V7f_2E0,V9x_2E0),V10y_2E0),app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V6P_2E0,V9x_2E0),V10y_2E0)) )
=> ( ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1P_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,A_27b,V2f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V4x_2E0)) )
=> ! [V0Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V3z_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1P_2E0))))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V3z_2E0)) )
<=> ! [V5x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V5x_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27a,A_27b,V2f_2E0,V5x_2E0))) ) )
& ! [V6P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V7f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b))] :
( ! [V8z_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V8z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F1_2E0,V7f_2E0),V6P_2E0)))))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V8z_2E0)) )
<=> ! [V11x_2E0: A_27c,V12y_2E0: A_27d] :
( p(app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V6P_2E0,V11x_2E0),V12y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27d,A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,A_27b),V7f_2E0,V11x_2E0),V12y_2E0))) ) )
& ! [V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)))] :
( ! [V15z_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V15z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),c_2Epair_2EUNCURRY_2E1(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F2_2E0,V14f_2E0),V13P_2E0)))))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V15z_2E0)) )
<=> ! [V19w_2E0: A_27e,V20x_2E0: A_27f,V21y_2E0: A_27g] :
( p(app_2E2(A_27g,tyop_2Emin_2Ebool,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),V13P_2E0,V19w_2E0),V20x_2E0),V21y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27g,A_27b,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,A_27b),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)),V14f_2E0,V19w_2E0),V20x_2E0),V21y_2E0))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EEXISTS__IN__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),V16w_2E0: A_27e,V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V17x_2E0: A_27f,V18y_2E0: A_27g] : ( app_2E2(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F3_2E0,V14f_2E0),V16w_2E0),V13P_2E0),V17x_2E0),V18y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27g,A_27b,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,A_27b),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)),V14f_2E0,V16w_2E0),V17x_2E0),V18y_2E0),app_2E2(A_27g,tyop_2Emin_2Ebool,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),V13P_2E0,V16w_2E0),V17x_2E0),V18y_2E0)) )
=> ( ! [V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V16w_2E0: A_27e] : ( app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F2_2E0,V14f_2E0),V13P_2E0),V16w_2E0) = c_2Epair_2EUNCURRY_2E1(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F3_2E0,V14f_2E0),V16w_2E0),V13P_2E0)) )
=> ( ! [V7f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),V6P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V9x_2E0: A_27c,V10y_2E0: A_27d] : ( app_2E2(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F1_2E0,V7f_2E0),V6P_2E0),V9x_2E0),V10y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27d,A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,A_27b),V7f_2E0,V9x_2E0),V10y_2E0),app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V6P_2E0,V9x_2E0),V10y_2E0)) )
=> ( ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1P_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,A_27b,V2f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V4x_2E0)) )
=> ! [V0Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ? [V3z_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1P_2E0))))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V3z_2E0)) )
<=> ? [V5x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V5x_2E0))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27a,A_27b,V2f_2E0,V5x_2E0))) ) )
& ! [V6P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V7f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b))] :
( ? [V8z_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V8z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F1_2E0,V7f_2E0),V6P_2E0)))))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V8z_2E0)) )
<=> ? [V11x_2E0: A_27c,V12y_2E0: A_27d] :
( p(app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V6P_2E0,V11x_2E0),V12y_2E0))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27d,A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,A_27b),V7f_2E0,V11x_2E0),V12y_2E0))) ) )
& ! [V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)))] :
( ? [V15z_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V15z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),c_2Epair_2EUNCURRY_2E1(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F2_2E0,V14f_2E0),V13P_2E0)))))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V15z_2E0)) )
<=> ? [V19w_2E0: A_27e,V20x_2E0: A_27f,V21y_2E0: A_27g] :
( p(app_2E2(A_27g,tyop_2Emin_2Ebool,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),V13P_2E0,V19w_2E0),V20x_2E0),V21y_2E0))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27g,A_27b,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,A_27b),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)),V14f_2E0,V19w_2E0),V20x_2E0),V21y_2E0))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EEMPTY__BIGUNION,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( c_2Epred__set_2EBIGUNION_2E1(A_27a,V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(A_27a) )
<=> ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0))
=> ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(A_27a) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__RNEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Erealax_2Ereal__neg_2E1(V1y_2E0)) = c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ).
tff(thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
=> ? [V2a_2E0: tyop_2Enum_2Enum] :
! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),V2a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EREAL__BOUNDS__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1k_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__neg_2E1(V1k_2E0),V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1k_2E0)) )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1k_2E0)) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2y_2E0: A_27b,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F1_2E0,V1s_2E0),V2y_2E0),V0f_2E0))
<=> ? [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(A_27b,V2y_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0,V3x_2E0))) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V2y_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F1_2E0,V1s_2E0),V2y_2E0),V0f_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0,V1s_2E0)) = c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBIGINTER__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2y_2E0: A_27b,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F1_2E0,V1s_2E0),V2y_2E0),V0f_2E0))
<=> ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
=> p(c_2Ebool_2EIN_2E2(A_27b,V2y_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0,V3x_2E0))) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V2y_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F1_2E0,V1s_2E0),V2y_2E0),V0f_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGINTER_2E1(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0,V1s_2E0)) = c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__LMUL1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2z_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0),c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V2z_2E0))) ) ).
tff(thm_2Ereal__topology_2ELE__EXISTS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
<=> ? [V2p_2E0: tyop_2Enum_2Enum] : ( V1n_2E0 = c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELT__EXISTS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
<=> ? [V2d_2E0: tyop_2Enum_2Enum] : ( V1n_2E0 = c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V2d_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT__EQ,axiom,
! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V1x_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Enum_2Enum,V3z_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V1x_2E0),V2y_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V2y_2E0),V3z_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V1x_2E0),V3z_2E0)) )
=> ( ! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V5n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V4m_2E0),V5n_2E0)) )
<=> ! [V6n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V6n_2E0),c_2Enum_2ESUC_2E1(V6n_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ETRANSITIVE__STEPWISE__LT,axiom,
! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Enum_2Enum,V3z_2E0: tyop_2Enum_2Enum] :
( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V1x_2E0),V2y_2E0))
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V2y_2E0),V3z_2E0)) )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V1x_2E0),V3z_2E0)) )
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V4n_2E0),c_2Enum_2ESUC_2E1(V4n_2E0))) )
=> ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V5m_2E0,V6n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0R_2E0,V5m_2E0),V6n_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__SEQUENTIAL__INJ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0))
<=> ? [V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V3n_2E0),c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V0x_2E0)))
& ! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V4m_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V5n_2E0) )
<=> ( V4m_2E0 = V5n_2E0 ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2f_2E0,V0x_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__SEQUENTIAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0))
<=> ? [V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V3n_2E0),c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V0x_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2f_2E0,V0x_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINFINITE__SUPERSET,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V0s_2E0,V1t_2E0)) )
=> ~ p(c_2Epred__set_2EFINITE_2E1(A_27a,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN__BALL__CBALL,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2t_2E0))) ) )
& ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V4x_2E0,V3s_2E0))
<=> ! [V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3s_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V5e_2E0))))) ) )
& ! [V6s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V7x_2E0,V6s_2E0))
<=> ! [V8e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V8e_2E0))
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V6s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7x_2E0,V8e_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__INFINITE__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__INFINITE__BALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__INFINITE__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2EINFINITE__OPEN__IN,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V0u_2E0)) ) )
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONDENSATION__POINT__OF__SUBSET,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V0x_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Econdensation__point__of_2E2(V0x_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONDENSATION__POINT__IMP__LIMPT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V0x_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL__CBALL,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0))))) ) )
& ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V4x_2E0,V3s_2E0))
<=> ! [V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V5e_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__BALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2ECONDENSATION__POINT__INFINITE__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econdensation__point__of_2E2(V1x_2E0,V0s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ~ p(c_2Epred__set_2Ecountable_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2ELIM__LINEAR,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V2f_2E0),V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1h_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1h_2E0),V2f_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,V3l_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__CONST,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2x_2E0) = V1a_2E0 )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V1a_2E0,V0net_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIM__CMUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V3c_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V3c_2E0),V1f_2E0),V4x_2E0) = c_2Erealax_2Ereal__mul_2E2(V3c_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V3c_2E0),V1f_2E0),c_2Erealax_2Ereal__mul_2E2(V3c_2E0,V2l_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__CMUL__EQ,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V3c_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V3c_2E0),V1f_2E0),V4x_2E0) = c_2Erealax_2Ereal__mul_2E2(V3c_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal] :
( ( V3c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V3c_2E0),V1f_2E0),c_2Erealax_2Ereal__mul_2E2(V3c_2E0,V2l_2E0),V0net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NEG,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),c_2Erealax_2Ereal__neg_2E1(V2l_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NEG__EQ,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),c_2Erealax_2Ereal__neg_2E1(V2l_2E0),V0net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELIM__ADD,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal,V4m_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V4m_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Erealax_2Ereal__add_2E2(V3l_2E0,V4m_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__ABS,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),c_2Ereal_2Eabs_2E1(V2l_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__SUB,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal,V4m_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V4m_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Ereal__sub_2E2(V3l_2E0,V4m_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__MAX,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = c_2Ereal_2Emax_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal,V4m_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V4m_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Emax_2E2(V3l_2E0,V4m_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__MIN,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = c_2Ereal_2Emin_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal,V4m_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V4m_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Emin_2E2(V3l_2E0,V4m_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2l_2E0),V3x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),V2l_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2l_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__ABS,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V2x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__CMUL__EQ,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal] :
( ( V2c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__CMUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__ADD,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__SUB,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V3x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__COMPARISON,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V3x_2E0) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0)) )
=> ( ! [V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F1_2E0,V2g_2E0),V4x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F1_2E0,V2g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPONENT,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4a_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V4a_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4a_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2i_2E0: A_27b,V3l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3l_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__BOUND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V3n_2E0) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3n_2E0)),c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V3n_2E0))) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__CMUL__BOUNDED,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3B_2E0: tyop_2Erealax_2Ereal,V4a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2g_2E0),V1f_2E0),V3B_2E0),V4a_2E0))
<=> ( ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4a_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
| p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4a_2E0)),V3B_2E0)) ) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V2g_2E0),V5n_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5n_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V5n_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3B_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2g_2E0),V1f_2E0),V3B_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V5x_2E0: A_27b,V6i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V5x_2E0),V6i_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V6i_2E0),V5x_2E0) )
=> ( ! [V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V5x_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F0_2E0,V3s_2E0),V1f_2E0),V5x_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V3s_2E0,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27b),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V2l_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3s_2E0))
& ! [V4i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V4i_2E0,V3s_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V4i_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2l_2E0,V4i_2E0),V0net_2E0)) ) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F0_2E0,V3s_2E0),V1f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V3s_2E0,V2l_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__NULL__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2s_2E0),V1f_2E0),V5x_2E0) = c_2Eiterate_2ESum_2E2(A_27b,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V5x_2E0)) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V3a_2E0: A_27b,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V3a_2E0),V4x_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V4x_2E0),V3a_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
& ! [V3a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V3a_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2s_2E0),V1f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__IN__CLOSED__SET,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2s_2E0),V4x_2E0) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),V2s_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V2s_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2s_2E0),V0net_2E0))
& ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3l_2E0,V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__ABS__UBOUND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3b_2E0: tyop_2Erealax_2Ereal,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3b_2E0),V4x_2E0) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)),V3b_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3b_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2l_2E0),V3b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__ABS__LBOUND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V3b_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V3b_2E0),V1f_2E0),V4x_2E0) = c_2Ereal_2Ereal__lte_2E2(V3b_2E0,c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0))) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V3b_2E0),V1f_2E0),V0net_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,c_2Ereal_2Eabs_2E1(V2l_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELIM__UNIQUE,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3l_27_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_27_2E0,V0net_2E0)) )
=> ( V2l_2E0 = V3l_27_2E0 ) ) ).
tff(thm_2Ereal__topology_2ETENDSTO__LIM,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0)) )
=> ( c_2Ereal__topology_2Elim__def_2E2(A_27a,V0net_2E0,V1f_2E0) = V2l_2E0 ) ) ).
tff(thm_2Ereal__topology_2ELIM__CONST__EQ,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V3x_2E0) = V1c_2E0 )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1c_2E0: tyop_2Erealax_2Ereal,V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2d_2E0,V0net_2E0))
<=> ( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
| ( V1c_2E0 = V2d_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORM__LIM__ADD,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))))] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V5m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V9e_2E0: tyop_2Erealax_2Ereal,V10x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V1P_2E0),V3g_2E0),V5m_2E0),V9e_2E0),V10x_2E0))
<=> ! [V11n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V11n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0,V11n_2E0),V10x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V5m_2E0,V11n_2E0))),V9e_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V6e_2E0: tyop_2Erealax_2Ereal,V7x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1P_2E0),V2f_2E0),V4l_2E0),V6e_2E0),V7x_2E0))
<=> ! [V8n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V8n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2f_2E0,V8n_2E0),V7x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V4l_2E0,V8n_2E0))),V6e_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V5m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V12e_2E0: tyop_2Erealax_2Ereal,V13x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))),F2_2E0,V1P_2E0),V2f_2E0),V3g_2E0),V4l_2E0),V5m_2E0),V12e_2E0),V13x_2E0))
<=> ! [V14n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V14n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2f_2E0,V14n_2E0),V13x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0,V14n_2E0),V13x_2E0)),c_2Erealax_2Ereal__add_2E2(app_2E2(A_27b,tyop_2Erealax_2Ereal,V4l_2E0,V14n_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V5m_2E0,V14n_2E0)))),V12e_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V5m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)] :
( ( ! [V6e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V6e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1P_2E0),V2f_2E0),V4l_2E0),V6e_2E0),V0net_2E0)) )
& ! [V9e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V1P_2E0),V3g_2E0),V5m_2E0),V9e_2E0),V0net_2E0)) ) )
=> ! [V12e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V12e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))),F2_2E0,V1P_2E0),V2f_2E0),V3g_2E0),V4l_2E0),V5m_2E0),V12e_2E0),V0net_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORM__LIM__SUB,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))))] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V5m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V9e_2E0: tyop_2Erealax_2Ereal,V10x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V1P_2E0),V3g_2E0),V5m_2E0),V9e_2E0),V10x_2E0))
<=> ! [V11n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V11n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0,V11n_2E0),V10x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V5m_2E0,V11n_2E0))),V9e_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V6e_2E0: tyop_2Erealax_2Ereal,V7x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1P_2E0),V2f_2E0),V4l_2E0),V6e_2E0),V7x_2E0))
<=> ! [V8n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V8n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2f_2E0,V8n_2E0),V7x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V4l_2E0,V8n_2E0))),V6e_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V5m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V12e_2E0: tyop_2Erealax_2Ereal,V13x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))),F2_2E0,V1P_2E0),V2f_2E0),V3g_2E0),V4l_2E0),V5m_2E0),V12e_2E0),V13x_2E0))
<=> ! [V14n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V14n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2f_2E0,V14n_2E0),V13x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0,V14n_2E0),V13x_2E0)),c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27b,tyop_2Erealax_2Ereal,V4l_2E0,V14n_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V5m_2E0,V14n_2E0)))),V12e_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V5m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)] :
( ( ! [V6e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V6e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1P_2E0),V2f_2E0),V4l_2E0),V6e_2E0),V0net_2E0)) )
& ! [V9e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V1P_2E0),V3g_2E0),V5m_2E0),V9e_2E0),V0net_2E0)) ) )
=> ! [V12e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V12e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))),F2_2E0,V1P_2E0),V2f_2E0),V3g_2E0),V4l_2E0),V5m_2E0),V12e_2E0),V0net_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORM__LIM__BILINEAR,axiom,
! [A_27a: $tType,A_27b: $tType,F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))] :
( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V5l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V6m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V19e_2E0: tyop_2Erealax_2Ereal,V20x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))))),F4_2E0,V1P_2E0),V2h_2E0),V3f_2E0),V4g_2E0),V5l_2E0),V6m_2E0),V19e_2E0),V20x_2E0))
<=> ! [V21n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V21n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3f_2E0,V21n_2E0),V20x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4g_2E0,V21n_2E0),V20x_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(A_27b,tyop_2Erealax_2Ereal,V5l_2E0,V21n_2E0)),app_2E2(A_27b,tyop_2Erealax_2Ereal,V6m_2E0,V21n_2E0)))),V19e_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V6m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V8b2_2E0: tyop_2Erealax_2Ereal,V11x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V1P_2E0),V6m_2E0),V8b2_2E0),V11x_2E0))
<=> ! [V12n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V12n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27b,tyop_2Erealax_2Ereal,V6m_2E0,V12n_2E0)),V8b2_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V5l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V7b1_2E0: tyop_2Erealax_2Ereal,V9x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1P_2E0),V5l_2E0),V7b1_2E0),V9x_2E0))
<=> ! [V10n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V10n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27b,tyop_2Erealax_2Ereal,V5l_2E0,V10n_2E0)),V7b1_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V4g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V6m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V16e_2E0: tyop_2Erealax_2Ereal,V17x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F3_2E0,V1P_2E0),V4g_2E0),V6m_2E0),V16e_2E0),V17x_2E0))
<=> ! [V18n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V18n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4g_2E0,V18n_2E0),V17x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V6m_2E0,V18n_2E0))),V16e_2E0)) ) )
=> ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V5l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V13e_2E0: tyop_2Erealax_2Ereal,V14x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F2_2E0,V1P_2E0),V3f_2E0),V5l_2E0),V13e_2E0),V14x_2E0))
<=> ! [V15n_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V15n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3f_2E0,V15n_2E0),V14x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V5l_2E0,V15n_2E0))),V13e_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V5l_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V6m_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V7b1_2E0: tyop_2Erealax_2Ereal,V8b2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1P_2E0),V5l_2E0),V7b1_2E0),V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V1P_2E0),V6m_2E0),V8b2_2E0),V0net_2E0))
& ! [V13e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V13e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F2_2E0,V1P_2E0),V3f_2E0),V5l_2E0),V13e_2E0),V0net_2E0)) )
& ! [V16e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V16e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F3_2E0,V1P_2E0),V4g_2E0),V6m_2E0),V16e_2E0),V0net_2E0)) ) )
=> ! [V19e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V19e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))))),F4_2E0,V1P_2E0),V2h_2E0),V3f_2E0),V4g_2E0),V5l_2E0),V6m_2E0),V19e_2E0),V0net_2E0)) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__BILINEAR,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))))] :
( ! [V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V6x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V1h_2E0),V2f_2E0),V3g_2E0),V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1h_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V6x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,V3g_2E0,V6x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4l_2E0: tyop_2Erealax_2Ereal,V5m_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2f_2E0,V4l_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V3g_2E0,V5m_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Ebilinear_2E1(V1h_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V1h_2E0),V2f_2E0),V3g_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1h_2E0,V4l_2E0),V5m_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V2x_2E0) = V2x_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,F0_2E0,V0a_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,F0_2E0,V0a_2E0,c_2Ereal__topology_2Eat_2E1(V0a_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__ZERO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2a_2E0),V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Erealax_2Ereal__add_2E2(V2a_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat_2E1(V2a_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2a_2E0),V1l_2E0,c_2Ereal__topology_2Eat_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ENETLIMIT__WITHIN,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),V1s_2E0)))
=> ( c_2Ereal__topology_2Enetlimit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),V1s_2E0)) = V0a_2E0 ) ) ).
tff(thm_2Ereal__topology_2ENETLIMIT__AT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Enetlimit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0)) = V0a_2E0 ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V4x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V3l_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__EVENTUALLY,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V4x_2E0))
<=> ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2g_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V3l_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN,axiom,
! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal,V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5d_2E0))
& ! [V6x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_27_2E0,V4s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_27_2E0,V3x_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_27_2E0,V3x_2E0)),V5d_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_27_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_27_2E0) ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V0l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V4s_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V0l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V4s_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__AT,axiom,
! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal,V4d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_27_2E0,V3x_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_27_2E0,V3x_2E0)),V4d_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_27_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_27_2E0) ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V0l_2E0,c_2Ereal__topology_2Eat_2E1(V3x_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V0l_2E0,c_2Ereal__topology_2Eat_2E1(V3x_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__EQ,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V4x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0net_2E0))
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3l_2E0,V0net_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V3l_2E0,V0net_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V3s_2E0),V4t_2E0),V5x_2E0))
<=> ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4t_2E0) ) )
=> ! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V3s_2E0),V4t_2E0),c_2Ereal__topology_2Eat_2E1(V2a_2E0)))
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V0l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V3s_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V0l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V4t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__SET__IMP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V4t_2E0),V3s_2E0),V5x_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4t_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0)) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V4t_2E0),V3s_2E0),c_2Ereal__topology_2Eat_2E1(V2a_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V3s_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__WITHIN,axiom,
! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal,V5s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( V3a_2E0 != V4b_2E0 )
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V5s_2E0))
& ( V6x_2E0 != V3a_2E0 )
& ( V6x_2E0 != V4b_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_2E0) ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V0l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3a_2E0),V5s_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V0l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3a_2E0),V5s_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__AWAY__AT,axiom,
! [V0l_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] :
( ( ( V3a_2E0 != V4b_2E0 )
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( ( V5x_2E0 != V3a_2E0 )
& ( V5x_2E0 != V4b_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0) ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V0l_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V0l_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal,V4l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3a_2E0,V2s_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& ( V5x_2E0 != V3a_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V4l_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1g_2E0,V4l_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__TRANSFORM__WITHIN__OPEN__IN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4a_2E0: tyop_2Erealax_2Ereal,V5l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4a_2E0,V2s_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& ( V6x_2E0 != V4a_2E0 ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6x_2E0) ) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V5l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4a_2E0),V3t_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1g_2E0,V5l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4a_2E0),V3t_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__CASES__FINITE__SEQUENTIALLY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0),V4n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V4n_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4n_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V5n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2g_2E0,V5n_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0))))
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V3l_2E0,c_2Ereal__topology_2Esequentially_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2g_2E0,V3l_2E0,c_2Ereal__topology_2Esequentially_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__CASES__COFINITE__SEQUENTIALLY,axiom,
! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V4n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F2_2E0,V0P_2E0),V4n_2E0))
<=> ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4n_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0),V4n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V4n_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F2_2E0,V0P_2E0),V4n_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V5n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2g_2E0,V5n_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0))))
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V3l_2E0,c_2Ereal__topology_2Esequentially_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1f_2E0,V3l_2E0,c_2Ereal__topology_2Esequentially_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__CASES__SEQUENTIALLY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V3m_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V4n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0)) )
=> ( ! [V3m_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V6n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V6n_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V6n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V6n_2E0)) )
=> ( ! [V3m_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V5n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V3m_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5n_2E0)) )
=> ( ! [V3m_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F3_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V7n_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V7n_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V7n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V7n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Enum_2Enum] :
( ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) )
& ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) )
& ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1g_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) )
& ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F3_2E0,V3m_2E0),V0f_2E0),V1g_2E0),V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1g_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__CONG__WITHIN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V3f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V3f_2E0),V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3f_2E0,V6x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Erealax_2Ereal,V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4a_2E0: tyop_2Erealax_2Ereal] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( V5x_2E0 != V4a_2E0 )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V3f_2E0),V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4a_2E0),V0s_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4a_2E0),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__CONG__AT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2f_2E0),V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V5x_2E0) )
=> ! [V0l_2E0: tyop_2Erealax_2Ereal,V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3a_2E0: tyop_2Erealax_2Ereal] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( ( V4x_2E0 != V3a_2E0 )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0) ) )
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V2f_2E0),V0l_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1g_2E0,V0l_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__SEQUENTIAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1l_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
<=> ? [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V3n_2E0),V0s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2x_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__CONTAINS__SEQUENTIAL__LIMIT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1x_2E0,V3n_2E0),V0s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1x_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2l_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SEQUENTIAL__LIMITS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1x_2E0,V3n_2E0),V0s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V1x_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2l_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__APPROACHABLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
=> ( ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3y_2E0,V0x_2E0)),V2e_2E0)) ) )
<=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EIN__CLOSURE__DELETE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1x_2E0))) = c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EDENSE__IMP__PERFECT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) )
=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EDENSE__LIMIT__POINTS,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: A_27a] :
( ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) )
<=> ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ESEQ__OFFSET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V3i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3i_2E0,V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2k_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESEQ__OFFSET__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V3i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2D_2E2(V3i_2E0,V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2k_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESEQ__OFFSET__REV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V3i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3i_2E0,V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2k_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2k_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESEQ__HARMONIC__OFFSET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1n_2E0) = c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ).
tff(thm_2Ereal__topology_2ESEQ__HARMONIC,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V0n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V0n_2E0) = c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,F0_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2EIN__INTERIOR__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)))
<=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0)),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1y_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0))))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0))) = c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0))) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1e_2E0))) = c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__CBALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1e_2E0))) = c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1e_2E0)) ) ).
tff(thm_2Ereal__topology_2ECBALL__EQ__EMPTY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V1e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Ereal__topology_2ECBALL__EMPTY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__EQ__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) )
<=> ( V1e_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( ( V1e_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ).
tff(thm_2Ereal__topology_2ESPHERE__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( ( V1e_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ).
tff(thm_2Ereal__topology_2ESPHERE__EQ__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) )
<=> ( ( V2x_2E0 = V0a_2E0 )
& ( V1r_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__WITHIN__INTERIOR,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)))
=> ( c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V0p_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V1s_2E0)) = c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V0p_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__INTERIOR,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Einterior_2E1(V2s_2E0)))
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V2s_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat_2E1(V3x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ENETLIMIT__WITHIN__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0)))
=> ( c_2Ereal__topology_2Enetlimit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V0s_2E0)) = V1x_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ~ ? [V2a_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) )
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__IMP__PERFECT__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ~ ? [V2a_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) )
=> ( c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__EMPTY,axiom,
p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CLOSURE__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2EBOUNDED__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2EFINITE__IMP__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))
<=> ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__BIGUNION,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0))
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0)) ) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__POS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
<=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1b_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__POS__LT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
<=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1b_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
| p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__DIFF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__INSERT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) = c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2EBOUNDED__BIGINTER,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ? [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ENOT__BOUNDED__UNIV,axiom,
~ p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECOBOUNDED__IMP__UNBOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)))
=> ~ p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V2x_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__NEGATIONS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__TRANSLATION__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) = c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__DIFFS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGE,axiom,
! [A_27a: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V0f_2E0),V1g_2E0),V2t_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0)),c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2t_2E0)) )
=> ( ! [V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1g_2E0),V2t_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0),c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2t_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V2t_2E0))))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1g_2E0),V2t_2E0)))) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V0f_2E0),V1g_2E0),V2t_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUMS__IMAGES,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V1t_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Eiterate_2ESum_2E2(A_27b,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V0f_2E0,V5x_2E0)),c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V1t_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V3a_2E0: A_27b,V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0f_2E0),V3a_2E0),V1t_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V0f_2E0,V4x_2E0),V3a_2E0),c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V1t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
& ! [V3a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0f_2E0),V3a_2E0),V1t_2E0)))) ) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V1t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET__BALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V2r_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2r_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__SUBSET__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V2r_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2r_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EUNBOUNDED__INTER__COBOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V1t_2E0))) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECOBOUNDED__INTER__UNBOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)))
& ~ p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__BOUNDED__EQ__TRIVIAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
=> ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
<=> ( V0s_2E0 = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__COMPONENTWISE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0) = c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__HAS__SUP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Ereal_2Esup_2E1(V0s_2E0))) )
& ! [V2b_2E0: tyop_2Erealax_2Ereal] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V2b_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESUP__INSERT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0))
<=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0))
=> ( c_2Ereal_2Esup_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0),V0x_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,c_2Ereal_2Esup_2E1(V1s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__HAS__INF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1x_2E0)) )
& ! [V2b_2E0: tyop_2Erealax_2Ereal] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3x_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EINF__INSERT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0))
<=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0))
=> ( c_2Eiterate_2Einf_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0),V0x_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,c_2Eiterate_2Einf_2E1(V1s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSET__BALLS,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1a_27_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal,V3r_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_27_2E0,V3r_27_2E0))))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1a_27_2E0)),V2r_2E0),V3r_27_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V2r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5a_27_2E0: tyop_2Erealax_2Ereal,V6r_2E0: tyop_2Erealax_2Ereal,V7r_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V6r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5a_27_2E0,V7r_27_2E0))))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5a_27_2E0)),V6r_2E0),V7r_27_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V6r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal,V9a_27_2E0: tyop_2Erealax_2Ereal,V10r_2E0: tyop_2Erealax_2Ereal,V11r_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V10r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9a_27_2E0,V11r_27_2E0))))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V9a_27_2E0)),V10r_2E0),V11r_27_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V10r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) )
& ! [V12a_2E0: tyop_2Erealax_2Ereal,V13a_27_2E0: tyop_2Erealax_2Ereal,V14r_2E0: tyop_2Erealax_2Ereal,V15r_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V14r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V13a_27_2E0,V15r_27_2E0))))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V13a_27_2E0)),V14r_2E0),V15r_27_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V14r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EINTER__BALLS__EQ__EMPTY,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V3s_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V2r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(V3s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V2r_2E0,V3s_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal,V6r_2E0: tyop_2Erealax_2Ereal,V7s_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V6r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5b_2E0,V7s_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V6r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Erealax_2Ereal__lt_2E2(V7s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V6r_2E0,V7s_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0)))) ) )
& ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal,V10r_2E0: tyop_2Erealax_2Ereal,V11s_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V10r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9b_2E0,V11s_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V10r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(V11s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(V10r_2E0,V11s_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V9b_2E0)))) ) )
& ! [V12a_2E0: tyop_2Erealax_2Ereal,V13b_2E0: tyop_2Erealax_2Ereal,V14r_2E0: tyop_2Erealax_2Ereal,V15s_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V14r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V13b_2E0,V15s_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V14r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Erealax_2Ereal__lt_2E2(V15s_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
| p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(V14r_2E0,V15s_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12a_2E0,V13b_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CLOSED__INTERVAL__CONVEX,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal,V4e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V4e_2E0,V2x_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V4e_2E0),V3y_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTERVAL__MIDPOINT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1b_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ).
tff(thm_2Ereal__topology_2ENOT__INTERVAL__UNIV,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) != c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CLOSED__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ).
tff(thm_2Ereal__topology_2EINTERIOR__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CLOSED__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ).
tff(thm_2Ereal__topology_2EOPEN__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ).
tff(thm_2Ereal__topology_2EOPEN__INTERVAL__LEMMA,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V2x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V1b_2E0)) )
=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V4x_27_2E0,V2x_2E0)),V3d_2E0))
=> ( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V4x_27_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V4x_27_2E0,V1b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__MIN,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,V1y_2E0)))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2z_2E0,V1y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__OPEN__SUBSET__CLOSED,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ).
tff(thm_2Ereal__topology_2EINTER__INTERVAL,axiom,
! [V0d_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3a_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Emax_2E2(V3a_2E0,V1c_2E0),c_2Ereal_2Emin_2E2(V2b_2E0,V0d_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2EENDS__IN__UNIT__INTERVAL,axiom,
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))))) ) ).
tff(thm_2Ereal__topology_2EENDS__IN__INTERVAL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3b_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
& ! [V4a_2E0: tyop_2Erealax_2Ereal,V5b_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4a_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4a_2E0,V5b_2E0))))
& ! [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7b_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6a_2E0,V7b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EDISJOINT__INTERVAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V0a_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V3d_2E0,V2c_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V2c_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V3d_2E0,V0a_2E0)) ) )
& ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V0a_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V3d_2E0,V2c_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2c_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V3d_2E0,V0a_2E0)) ) )
& ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V0a_2E0))
| p(c_2Erealax_2Ereal__lt_2E2(V3d_2E0,V2c_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2c_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V3d_2E0,V0a_2E0)) ) )
& ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2c_2E0,V3d_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V0a_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V3d_2E0,V2c_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2c_2E0))
| p(c_2Ereal_2Ereal__lte_2E2(V3d_2E0,V0a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSET__INTERVAL,axiom,
! [V0d_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1c_2E0,V0d_2E0))
=> ( p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V1c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,V2b_2E0)) ) ) )
& ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0))))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1c_2E0,V0d_2E0))
=> ( p(c_2Erealax_2Ereal__lt_2E2(V3a_2E0,V1c_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0d_2E0,V2b_2E0)) ) ) )
& ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0)),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V1c_2E0,V0d_2E0))
=> ( p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V1c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,V2b_2E0)) ) ) )
& ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0))))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V1c_2E0,V0d_2E0))
=> ( p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V1c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,V2b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V0a_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) )
& ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V0a_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ESUBSET__INTERVAL__IMP,axiom,
! [V0d_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3a_2E0: tyop_2Erealax_2Ereal] :
( ( ( p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V1c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,V2b_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) )
& ( ( p(c_2Erealax_2Ereal__lt_2E2(V3a_2E0,V1c_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0d_2E0,V2b_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0)))) )
& ( ( p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V1c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,V2b_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0)),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) )
& ( ( p(c_2Ereal_2Ereal__lte_2E2(V3a_2E0,V1c_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0d_2E0,V2b_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V0d_2E0)),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V2b_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__NE__EMPTY,axiom,
! [V0b_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V0b_2E0)) )
& ( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,V0b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINTERVAL__EQ__EMPTY,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V0a_2E0))
<=> ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
& ( p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V0a_2E0))
<=> ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EREFLECT__INTERVAL,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F1_2E0,V5x_2E0) = c_2Erealax_2Ereal__neg_2E1(V5x_2E0) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(V2x_2E0) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V1b_2E0),c_2Erealax_2Ereal__neg_2E1(V0a_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) )
& ! [V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F1_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0))) = c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V4b_2E0),c_2Erealax_2Ereal__neg_2E1(V3a_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EIN__INTERVAL__REFLECT,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V2x_2E0),c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V1b_2E0),c_2Erealax_2Ereal__neg_2E1(V0a_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) )
& ! [V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V5x_2E0),c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V4b_2E0),c_2Erealax_2Ereal__neg_2E1(V3a_2E0)))) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EIN__INTERVAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V1b_2E0))))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V2a_2E0,V0x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V1b_2E0)) ) )
& ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1b_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2Einterval,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V0b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V1a_2E0),V3x_2E0),V0b_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V0b_2E0)) ) )
=> ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V0b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1a_2E0),V0b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V1a_2E0),V3x_2E0),V0b_2E0)) )
=> ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V0b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V1a_2E0),V2x_2E0),V0b_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,V2x_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V0b_2E0)) ) )
=> ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V0b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1a_2E0),V0b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V1a_2E0),V2x_2E0),V0b_2E0)) )
=> ! [V0b_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EOPEN__interval_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1a_2E0),V0b_2E0)) )
& ( c_2Ereal__topology_2ECLOSED__interval_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V1a_2E0),V0b_2E0)) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__UNION__COMPACT__SUBSETS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V5n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V5n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2Ecompact_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V2n_2E0)))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V3n_2E0),V0s_2E0))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V4n_2E0),c_2Ereal__topology_2Einterior_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,c_2Earithmetic_2E_2B_2E2(V4n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))
& ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0))) = V0s_2E0 )
& ! [V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V6k_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6k_2E0,V0s_2E0)) )
=> ? [V7N_2E0: tyop_2Enum_2Enum] :
! [V8n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V8n_2E0,V7N_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6k_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V8n_2E0))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__UNION__COMPACT__SUBSETS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V5n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V5n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> ? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2Ecompact_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V2n_2E0)))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V3n_2E0),V0s_2E0))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,c_2Earithmetic_2E_2B_2E2(V4n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))))
& ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0))) = V0s_2E0 )
& ! [V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V6k_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6k_2E0,V0s_2E0)) )
=> ? [V7N_2E0: tyop_2Enum_2Enum] :
! [V8n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V8n_2E0,V7N_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6k_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0,V8n_2E0))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATE__CLOSED__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2d_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATE__COMPACT__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2d_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V4y_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ESEPARATE__POINT__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0)) )
=> ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V2d_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3x_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ETRANSLATION__DIFF,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V0a_2E0),V4x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V4x_2E0) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V0a_2E0),V1s_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V0a_2E0),V2t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__COMPACT__DIFFERENCES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CLOSED__DIFFERENCES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__SUMS,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V4x_2E0),V0s_2E0),V5y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0),V4x_2E0),V5y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V4x_2E0,V5y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V4x_2E0),V0s_2E0),V5y_2E0),V1t_2E0)) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
| p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0)))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__COMPACT__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CLOSED__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__NEGATIONS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELEBESGUE__COVERING__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& ( V1c_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1c_2E0)))
& ! [V2b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0,V1c_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2b_2E0)) ) )
=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ediameter_2E1(V4t_2E0),V3d_2E0)) )
=> ? [V5b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5b_2E0,V1c_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V5b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ediameter_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0)))),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2Ediameter_2E1(V0s_2E0),c_2Ereal__topology_2Ediameter_2E1(V1t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ediameter_2E1(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__lt_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__CBALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ediameter_2E1(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__lt_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1d_2E0: tyop_2Erealax_2Ereal] :
( ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0)) )
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V3y_2E0)),V1d_2E0)) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ediameter_2E1(V0s_2E0),V1d_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__EQ__0,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ( ( c_2Ereal__topology_2Ediameter_2E1(V0s_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ? [V1a_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ? [V1z_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1z_2E0,c_2Ereal__topology_2Ediameter_2E1(V0s_2E0))))) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__SUBSET__CBALL__NONEMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V1z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1z_2E0,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1z_2E0,c_2Ereal__topology_2Ediameter_2E1(V0s_2E0))))) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ( c_2Ereal__topology_2Ediameter_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Ereal__topology_2Ediameter_2E1(V0s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1t_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2Ediameter_2E1(V0s_2E0),c_2Ereal__topology_2Ediameter_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__POS__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Ediameter_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ediameter_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__EMPTY,axiom,
c_2Ereal__topology_2Ediameter_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).
tff(thm_2Ereal__topology_2EDIAMETER__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) = c_2Ereal_2Eabs_2E1(V2x_2E0) ) )
=> ( c_2Ereal__topology_2Ediameter_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) = c_2Ereal__topology_2Ediameter_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__BOUNDED__BOUND,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0)),c_2Ereal__topology_2Ediameter_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EDIAMETER__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0)),c_2Ereal__topology_2Ediameter_2E1(V0s_2E0))) )
& ! [V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V3d_2E0,c_2Ereal__topology_2Ediameter_2E1(V0s_2E0))) )
=> ? [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__gt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V4x_2E0,V5y_2E0)),V3d_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__AFFINITY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V2c_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V3x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V2c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__NEGATIONS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__NEGATIONS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__INV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,V0f_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__INV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1a_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1a_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,V0f_2E0),c_2Ereal__topology_2Eat_2E1(V1a_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN__INV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V1s_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2a_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,V0f_2E0),c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__INV,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0))
& ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,c_2Ereal__topology_2Enetlimit_2E1(A_27a,V0net_2E0)) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,V1f_2E0),V0net_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIM__INV,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0))
& ( V2l_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,V1f_2E0),c_2Erealax_2Einv_2E1(V2l_2E0),V0net_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__PRODUCT,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V3i_2E0: A_27a,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V3i_2E0),V4x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V0f_2E0,V4x_2E0),V3i_2E0) )
=> ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V0f_2E0),V5x_2E0) = c_2Eproduct_2Eproduct_2E2(A_27a,V2t_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V0f_2E0,V5x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
& ! [V3i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3i_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V3i_2E0),V1s_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V0f_2E0),V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__PRODUCT,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V1f_2E0),V5x_2E0) = c_2Eproduct_2Eproduct_2E2(A_27b,V2t_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V5x_2E0)) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V3i_2E0: A_27b,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V3i_2E0),V4x_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V4x_2E0),V3i_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0))
& ! [V3i_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3i_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V3i_2E0),V0net_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V1f_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__POW,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V2n_2E0),V4x_2E0) = c_2Ereal_2Epow_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V2n_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V2n_2E0),V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__POW,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V2n_2E0),V4x_2E0) = c_2Ereal_2Epow_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),V2n_2E0) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V0net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V2n_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__MUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V3x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1c_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2f_2E0,V0s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2f_2E0),V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__VMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2v_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V3x_2E0),V2v_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1c_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2v_2E0),V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__MUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V2c_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V2c_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V1f_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__VMUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Erealax_2Ereal,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2v_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1c_2E0,V3x_2E0),V2v_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1c_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2v_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__VMUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Erealax_2Ereal,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V3v_2E0),V4x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1c_2E0,V4x_2E0),V3v_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2d_2E0: tyop_2Erealax_2Ereal,V3v_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1c_2E0,V2d_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V3v_2E0),c_2Erealax_2Ereal__mul_2E2(V2d_2E0,V3v_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__MUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V3c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V3c_2E0),V1f_2E0),V5x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V3c_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V3c_2E0,V4d_2E0,V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V3c_2E0),V1f_2E0),c_2Erealax_2Ereal__mul_2E2(V4d_2E0,V2l_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EDISTANCE__ATTAINS__INF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2x_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3y_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EDISTANCE__ATTAINS__SUP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3y_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2x_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__INF,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ATTAINS__SUP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__ATTAINS__INF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& ! [V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V2y_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__ATTAINS__SUP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& ! [V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V2y_2E0,V1x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__DIST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__DIST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal__topology_2Eat_2E1(V1x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__ABS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,c_2Ereal__topology_2Eat_2E1(V0x_2E0))) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__ABS__COMPOSE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ABS__COMPOSE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V2x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__RANGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V5x_27_2E0,V2x_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))),V3e_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__RANGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0)))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V4x_27_2E0,V1x_2E0)),V3d_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0))),V2e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_27_2E0,V0s_2E0))
& ( V3x_27_2E0 != V1x_2E0 )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3x_27_2E0,V1x_2E0)),V2e_2E0)) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EOPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& ! [V3x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3x_27_2E0,V1x_2E0)),V2e_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_27_2E0,V0s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__UNIFORM__LIMIT,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5e_2E0: tyop_2Erealax_2Ereal,V6n_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V3s_2E0),V1f_2E0),V2g_2E0),V5e_2E0),V6n_2E0))
<=> ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V3s_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V6n_2E0),V7x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V7x_2E0))),V5e_2E0)) ) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3s_2E0),V4n_2E0) = c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V4n_2E0),V3s_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V3s_2E0),V0net_2E0))
& ! [V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V3s_2E0),V1f_2E0),V2g_2E0),V5e_2E0),V0net_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V2g_2E0,V3s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EABS__TRIANGLE__LE,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Eabs_2E1(V1x_2E0),c_2Ereal_2Eabs_2E1(V2y_2E0)),V0e_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0)),V0e_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__CONTINUOUS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__UNIFORMLY__EQUICONTINUOUS,axiom,
! [V0fs_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0)) )
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5f_2E0,V0fs_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_27_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_27_2E0,V2x_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V6x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5f_2E0,V2x_2E0))),V3e_2E0)) ) ) )
& p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0)) )
=> ! [V7e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7e_2E0))
=> ? [V8d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V8d_2E0))
& ! [V9f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V10x_2E0: tyop_2Erealax_2Ereal,V11x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9f_2E0,V0fs_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11x_27_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V11x_27_2E0,V10x_2E0)),V8d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9f_2E0,V11x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9f_2E0,V10x_2E0))),V7e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOPEN__IN__COMPONENTS,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0u_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOPEN__BIGUNION__COMPONENTS,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0)) )
=> ? [V2k_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2k_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0u_2E0)))
& ( V1s_2E0 = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V2k_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__SUBSET__CLOPEN,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V2c_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2c_2E0,V0u_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2c_2E0,V1s_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2c_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS__BIGUNION__COMPLEMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
=> ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1c_2E0) = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EDELETE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0),V1c_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__INTERMEDIATE__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Ereal__topology_2Ecomponents_2E1(V2u_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Ereal__topology_2Ecomponents_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EEXISTS__COMPONENT__SUPERSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0)) )
=> ? [V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2c_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__UNIQUE__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1k_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = V1k_2E0 )
<=> ( ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1k_2E0) = V0s_2E0 )
& ! [V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,V1k_2E0))
=> ( p(c_2Ereal__topology_2Econnected_2E1(V2c_2E0))
& ( V2c_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V3c_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2c_2E0,V3c_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3c_27_2E0,V0s_2E0)) )
=> ( V3c_27_2E0 = V2c_2E0 ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__UNIQUE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1k_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1k_2E0) = V0s_2E0 )
& ! [V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,V1k_2E0))
=> ( p(c_2Ereal__topology_2Econnected_2E1(V2c_2E0))
& ( V2c_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V3c_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2c_2E0,V3c_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3c_27_2E0,V0s_2E0)) )
=> ( V3c_27_2E0 = V2c_2E0 ) ) ) ) )
=> ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = V1k_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__MAXIMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2c_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2c_2E0)) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS__SELF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
<=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ? [V1a_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EQ__COMPONENTS__SUBSET__SING,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Econnected_2E1(V0s_2E0) = c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__UNIV,axiom,
c_2Ereal__topology_2Ecomponents_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__EXISTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ? [V1a_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) )
<=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__EQ__SING,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) )
<=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__EQ__SING__N__EXISTS,axiom,
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) )
<=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ? [V2a_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecomponents_2E1(V1s_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))) )
<=> ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENTS__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_27_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0))) )
=> ( V1c_2E0 = V2c_27_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__EMPTY,axiom,
c_2Ereal__topology_2Ecomponents_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__EQ__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) )
<=> ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_27_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0))) )
=> ( ( V1c_2E0 = V2c_27_2E0 )
<=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1c_2E0,V2c_27_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPONENTS__NONOVERLAP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_27_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0))) )
=> ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1c_2E0,V2c_27_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( V1c_2E0 != V2c_27_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__FINITE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Ecomponents_2E1(V1s_2E0)))
& ! [V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V1s_2E0)))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2c_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENTS__GEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V1s_2E0)))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2c_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2c_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__COMPONENTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0))) )
=> p(c_2Ereal__topology_2Ecompact_2E1(V1c_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__COMPONENTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0))) )
=> p(c_2Ereal__topology_2EClosed_2E1(V1c_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__COMPONENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1c_2E0)) ) ).
tff(thm_2Ereal__topology_2EJOINABLE__COMPONENTS__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3c2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2c1_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3c2_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2c1_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3c2_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( V2c1_2E0 = V3c2_2E0 ) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS__MAXIMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
<=> ( ( V1c_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1c_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1c_2E0))
& ! [V2c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( V2c_27_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1c_2E0,V2c_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2c_27_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V2c_27_2E0)) )
=> ( V2c_27_2E0 = V1c_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS__CONNECTED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
=> p(c_2Ereal__topology_2Econnected_2E1(V1c_2E0)) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1c_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS__NONEMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0s_2E0)))
=> ( V1c_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EPAIRWISE__DISJOINT__COMPONENTS,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Epairwise_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Ecomponents_2E1(V0u_2E0))) ).
tff(thm_2Ereal__topology_2EBIGUNION__COMPONENTS,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( V0u_2E0 = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecomponents_2E1(V0u_2E0)) ) ).
tff(thm_2Ereal__topology_2EIN__COMPONENTS,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,c_2Ereal__topology_2Ecomponents_2E1(V0u_2E0)))
<=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0u_2E0))
& ( V1s_2E0 = c_2Ereal__topology_2Econnected__component_2E2(V0u_2E0,V2x_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__INTERMEDIATE__SUBSET,axiom,
! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V1u_2E0,V2a_2E0),V0t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0t_2E0,V1u_2E0)) )
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0t_2E0,V2a_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V1u_2E0,V2a_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQUIVALENCE__RELATION,axiom,
! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V2x_2E0),V3y_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V3y_2E0),V2x_2E0)) )
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal,V6z_2E0: tyop_2Erealax_2Ereal] :
( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V4x_2E0),V5y_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V5y_2E0),V6z_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V4x_2E0),V6z_2E0)) )
& ! [V7a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7a_2E0,V1s_2E0))
=> ? [V8t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V8t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7a_2E0,V8t_2E0))
& ! [V9x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V8t_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V7a_2E0),V9x_2E0)) ) ) ) )
=> ! [V10a_2E0: tyop_2Erealax_2Ereal,V11b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econnected__component_2E3(V1s_2E0,V10a_2E0,V11b_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V10a_2E0),V11b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__CONNECTED__COMPONENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2x_2E0),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__DIFF,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EDIFF_2E2(A_27a,V2x_2E0,V1t_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V0s_2E0),V1t_2E0) = c_2Epred__set_2EBIGUNION_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__CONNECTED__COMPONENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0))) ).
tff(thm_2Ereal__topology_2ECOMPLEMENT__CONNECTED__COMPONENT__BIGUNION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2y_2E0),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0)) = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EDELETE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)),c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__CONNECTED__COMPONENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0),c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))) = V0s_2E0 ) ) ).
tff(thm_2Ereal__topology_2EJOINABLE__CONNECTED__COMPONENT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2x_2E0),V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V3y_2E0),V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2x_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V3y_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIQUE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1c_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1c_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1c_2E0))
& ! [V3c_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V3c_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3c_27_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V3c_27_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3c_27_2E0,V1c_2E0)) ) )
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2x_2E0) = V1c_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__IDEMP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E2(c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0),V1x_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EQ__CONNECTED__COMPONENT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2y_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2y_2E0) )
<=> ( ( ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V2y_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V2y_2E0) = c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V2y_2E0,V1x_2E0) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__OVERLAP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1a_2E0),c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2b_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0))
& ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1a_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2b_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__NONOVERLAP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1a_2E0),c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2b_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
| ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0))
| ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1a_2E0) != c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2b_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__DISJOINT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1a_2E0),c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2b_2E0)))
<=> ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2b_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__CONNECTED__COMPONENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0)))
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2y_2E0) = c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EMPTY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E2(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal),V0x_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econnected_2E1(c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0))) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__MONO,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2x_2E0),c_2Ereal__topology_2Econnected__component_2E2(V1t_2E0,V2x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__MAXIMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1t_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V2x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTED__COMPONENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) )
=> p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V2y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__SELF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) )
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = V0s_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__EQ__UNIV,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) )
<=> ( V0s_2E0 = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__UNIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E2(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0x_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CONNECTED__COMPONENT__SET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = V0s_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0),V0s_2E0)) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__BIGUNION,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2t_2E0),V1x_2E0),V0s_2E0))
<=> ( p(c_2Ereal__topology_2Econnected_2E1(V2t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_2E0,V0s_2E0)) ) )
=> ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1x_2E0),V0s_2E0),V2t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V2t_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2t_2E0),V1x_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1x_2E0),V0s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SET,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0s_2E0),V1x_2E0),V2y_2E0))
<=> ? [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V3t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1x_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2y_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0s_2E0),V1x_2E0),V2y_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E2(V0s_2E0,V1x_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__OF__SUBSET,axiom,
! [V0y_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))
& p(c_2Ereal__topology_2Econnected__component_2E3(V1s_2E0,V3x_2E0,V0y_2E0)) )
=> p(c_2Ereal__topology_2Econnected__component_2E3(V2t_2E0,V3x_2E0,V0y_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__TRANS,axiom,
! [V0z_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected__component_2E3(V1s_2E0,V2x_2E0,V3y_2E0))
& p(c_2Ereal__topology_2Econnected__component_2E3(V1s_2E0,V3y_2E0,V0z_2E0)) )
=> p(c_2Ereal__topology_2Econnected__component_2E3(V1s_2E0,V2x_2E0,V0z_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__SYM,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V2y_2E0))
=> p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V2y_2E0,V1x_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V1x_2E0) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V1x_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__COMPONENT__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econnected__component_2E3(V0s_2E0,V1x_2E0,V2y_2E0))
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUPPER__LOWER__HEMICONTINUOUS__EXPLICIT,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V7x_2E0),V6u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0)) )
=> ( ! [V5x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V3x_2E0),V1t_2E0)) )
& ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V4u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0)))) )
& ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V6u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0)))) ) )
=> ! [V8x_2E0: tyop_2Erealax_2Ereal,V9e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V8x_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V8x_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V10d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V10d_2E0))
& ! [V11x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11x_27_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8x_2E0,V11x_27_2E0)),V10d_2E0)) )
=> ( ! [V12y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V12y_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V8x_2E0)))
=> ? [V13y_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V13y_27_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V11x_27_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V12y_2E0,V13y_27_2E0)),V9e_2E0)) ) )
& ! [V14y_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V14y_27_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V11x_27_2E0)))
=> ? [V15y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V15y_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V8x_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V14y_27_2E0,V15y_2E0)),V9e_2E0)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__IFF__UPPER__HEMICONTINUOUS__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = V5y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1s_2E0),V0f_2E0),V5y_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0)) )
=> ( ! [V5y_2E0: tyop_2Erealax_2Ereal,V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0,V5y_2E0),V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1s_2E0),V0f_2E0),V5y_2E0)),V4u_2E0)) ) )
=> ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0),V5y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5y_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0,V5y_2E0),V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3u_2E0))) )
<=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__IFF__LOWER__HEMICONTINUOUS__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = V5y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1s_2E0),V0f_2E0),V5y_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0)) )
=> ( ! [V5y_2E0: tyop_2Erealax_2Ereal,V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0,V5y_2E0),V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1s_2E0),V0f_2E0),V5y_2E0)),V4u_2E0)) ) )
=> ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0),V5y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5y_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0,V5y_2E0),V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3u_2E0))) )
<=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2t_2E0),V1s_2E0),V0f_2E0),V4u_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELOWER__HEMICONTINUOUS,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0)) )
=> ( ! [V7x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V7x_2E0),V6u_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V3x_2E0),V1t_2E0)) )
=> ( ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V4u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0)))) )
<=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V6u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EUPPER__HEMICONTINUOUS,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0)) )
=> ( ! [V7x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V7x_2E0),V6u_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V2s_2E0),V0f_2E0),V6u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0,V3x_2E0),V1t_2E0)) )
=> ( ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V4u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0)))) )
<=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V6u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V6u_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SUMS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
| p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__TRANSLATION,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V0a_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V0a_2E0),c_2Ereal__topology_2Einterior_2E1(V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__AFFINITY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V2c_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V3x_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& ( V2c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1a_2E0),V2c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__TRANSLATION__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) = c_2Ereal__topology_2EOpen_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__NEGATIONS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] :
( ( ( V1c_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2a_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0))))
& ? [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V2a_2E0 ) ) )
=> ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V2a_2E0 ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2a_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0))))
& ? [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V2a_2E0 ) ) )
=> ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V2a_2E0 ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__LEVELSET__OPEN__IN__CASES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2a_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0)))) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) != V2a_2E0 ) )
| ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V2a_2E0 ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__AVOID,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) != V3a_2E0 ) )
=> ? [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V5y_2E0)),V4e_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5y_2E0) != V3a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__AVOID,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) != V3a_2E0 ) )
=> ? [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V5y_2E0)),V4e_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5y_2E0) != V3a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__AVOID,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) != V2a_2E0 ) )
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V4y_2E0)),V3e_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) != V2a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__AVOID,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V2s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) != V3a_2E0 ) )
=> ? [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V5y_2E0)),V4e_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5y_2E0) != V3a_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__IMAGE__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2f_2E0),V0a_2E0),V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1z_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V2f_2E0,c_2Ereal__topology_2Eat_2E1(c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1z_2E0))) = c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2f_2E0),V0a_2E0),c_2Ereal__topology_2Eat_2E1(V1z_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V2g_2E0,c_2Ereal__topology_2Eat_2E1(V0x_2E0)))
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V3h_2E0,c_2Ereal__topology_2Eat_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V0x_2E0))))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3h_2E0,V4y_2E0)) = V4y_2E0 )
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3h_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V0x_2E0)) = V0x_2E0 ) )
=> ( c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1f_2E0,c_2Ereal__topology_2Eat_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V0x_2E0))) = c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),c_2Ereal__topology_2Eat_2E1(V0x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__UNIFORMLY__CONTINUOUS__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__IMP__CONTINUOUS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ecauchy_2E1(V2x_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V3n_2E0),V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecauchy_2E1(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__EXTENDS__TO__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
=> ? [V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V2g_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) ) )
& ! [V4h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V4h_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4h_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) ) ) )
=> ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4h_2E0,V6x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__EXTENDS__TO__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ecauchy_2E1(V2x_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V3n_2E0),V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecauchy_2E1(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) )
=> ? [V4g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V4g_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4g_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECAUCHY__CONTINUOUS__UNIQUENESS__LEMMA,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ecauchy_2E1(V2x_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V3n_2E0),V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecauchy_2E1(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) )
=> ! [V4a_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V6n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V5x_2E0,V6n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V5x_2E0,V4a_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> ? [V7l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V7l_2E0,c_2Ereal__topology_2Esequentially_2E0))
& ! [V8y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V9n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V8y_2E0,V9n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V8y_2E0,V4a_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8y_2E0),V7l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IMP__CAUCHY__CONTINUOUS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0)) )
=> ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ecauchy_2E1(V2x_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V3n_2E0),V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecauchy_2E1(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CAUCHY__CONTINUOUS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
=> ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Ecauchy_2E1(V2x_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V3n_2E0),V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecauchy_2E1(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE__SEQUENTIALLY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
<=> ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3a_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V4n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2x_2E0,V3a_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3a_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0)) )
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5y_2E0,V2x_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))),V3e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__CONSTANT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V2s_2E0),V1f_2E0),V0a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) = V0a_2E0 ) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V1f_2E0),V0a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V2s_2E0),V1f_2E0),V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2s_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V1f_2E0),V0a_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__CONSTANT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2a_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2a_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2a_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AGREE__ON__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2h_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V3x_2E0) ) ) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2h_2E0,V4x_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CONSTANT__ON__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = V2a_2E0 ) ) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V2a_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__GE__ON__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0))) ) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__LE__ON__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2a_2E0)) ) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V2a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__IN__CLOSURE__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0))) )
=> ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) )
<=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V2t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__IN__CLOSURE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V2t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__gt_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__GE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__ge_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EUNBOUNDED__HALFSPACE__COMPONENT__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__STANDARD__HYPERPLANE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V1x_2E0),V0a_2E0))
<=> ( V1x_2E0 = V0a_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V1x_2E0),V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__gt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ~ ( ( V0a_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1b_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ~ ( ( V0a_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1b_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__GE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__ge_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ~ ( ( V0a_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1b_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__HALFSPACE__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ~ ( ( V0a_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
& ( V1b_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__HYPERPLANE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__GT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__gt_2E2(V1x_2E0,V0a_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__ge_2E2(V2x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__COMPONENT__LT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V0a_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__GT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__gt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Ereal_2Ereal__ge_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__HALFSPACE__LT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__GE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__gt_2E2(V2x_2E0,V0a_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__ge_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__COMPONENT__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0a_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__GE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Ereal_2Ereal__gt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__ge_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__HALFSPACE__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__POSITIVE__ORTHANT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0x_2E0,c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTERVAL__RIGHT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V1x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTERVAL__LEFT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EOPEN__POSITIVE__MULTIPLES,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V1c_2E0),V2x_2E0),V0s_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1c_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1c_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V1c_2E0),V2x_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0))))) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__GT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__gt_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EOPEN__HALFSPACE__GT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__gt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EOPEN__HALFSPACE__COMPONENT__LT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Erealax_2Ereal__lt_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2EOPEN__HALFSPACE__LT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__GE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__ge_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__HALFSPACE__COMPONENT__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__STANDARD__HYPERPLANE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal,V0a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V1x_2E0),V0a_2E0))
<=> ( V1x_2E0 = V0a_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V1x_2E0),V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__HYPERPLANE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V0a_2E0),V3x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V3x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V0a_2E0),V3x_2E0),V1b_2E0)) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V2x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0a_2E0),V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__HYPERPLANE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0))
<=> ( c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0) = V1b_2E0 ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0a_2E0),V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__HALFSPACE__GE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__ge_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__HALFSPACE__LE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2x_2E0),V1b_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTERVAL__RIGHT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1x_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTERVAL__LEFT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal_2Ereal__lte_2E2(V1x_2E0,V0a_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__LIFT__DOT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2y_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__LIFT__DOT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0a_2E0,V2y_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal__topology_2Eat_2E1(V1x_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__LIFT__DOT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V3a_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V3a_2E0),V2f_2E0),V4y_2E0) = c_2Erealax_2Ereal__mul_2E2(V3a_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V4y_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2f_2E0,V1l_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V3a_2E0),V2f_2E0),c_2Erealax_2Ereal__mul_2E2(V3a_2E0,V1l_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS__CLOSED,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3k_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V3k_2E0),V4i_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V4i_2E0),c_2Ebool_2EIN_2E2(A_27a,V4i_2E0,V3k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3k_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V3k_2E0)))))
& ! [V5i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V3k_2E0))
=> ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V5i_2E0)))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V5i_2E0))) ) )
& ! [V6i_2E0: A_27a,V7j_2E0: A_27a,V8x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V6i_2E0,V3k_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V7j_2E0,V3k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V6i_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V7j_2E0)))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6i_2E0),V8x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V7j_2E0),V8x_2E0) ) ) )
=> ? [V9g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V9g_2E0,V2s_2E0))
& ! [V10x_2E0: tyop_2Erealax_2Ereal,V11i_2E0: A_27a] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V11i_2E0,V3k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V11i_2E0)))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9g_2E0,V10x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V11i_2E0),V10x_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EPASTING__LEMMA__CLOSED,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4k_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V4k_2E0))
& ! [V5i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V4k_2E0))
=> ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3s_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V5i_2E0)))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V5i_2E0))) ) )
& ! [V6i_2E0: A_27a,V7j_2E0: A_27a,V8x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V6i_2E0,V4k_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V7j_2E0,V4k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V6i_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V7j_2E0)))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6i_2E0),V8x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V7j_2E0),V8x_2E0) ) )
& ! [V9x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V3s_2E0))
=> ? [V10j_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V10j_2E0,V4k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V10j_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V9x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V10j_2E0),V9x_2E0) ) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__OPEN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V6x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V3s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V4t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2g_2E0,V4t_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0))
& ~ p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CASES__LOCAL__OPEN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),V6x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V6x_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0)),V3s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0)),V4t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V2g_2E0,V4t_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0))
& ~ p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) )
| ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V4t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V0P_2E0),V1f_2E0),V2g_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__OPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__UNION__LOCAL__OPEN,axiom,
! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0t_2E0)),V2s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0t_2E0)),V0t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0t_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V0t_2E0))) ) ).
tff(thm_2Ereal__topology_2EPASTING__LEMMA__EXISTS,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3k_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V3k_2E0),V4i_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V4i_2E0),c_2Ebool_2EIN_2E2(A_27a,V4i_2E0,V3k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V3k_2E0)))))
& ! [V5i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V3k_2E0))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V5i_2E0)))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V5i_2E0))) ) )
& ! [V6i_2E0: A_27a,V7j_2E0: A_27a,V8x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V6i_2E0,V3k_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V7j_2E0,V3k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V6i_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V7j_2E0)))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6i_2E0),V8x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V7j_2E0),V8x_2E0) ) ) )
=> ? [V9g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V9g_2E0,V2s_2E0))
& ! [V10x_2E0: tyop_2Erealax_2Ereal,V11i_2E0: A_27a] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V11i_2E0,V3k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V11i_2E0)))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9g_2E0,V10x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V11i_2E0),V10x_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EPASTING__LEMMA,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4k_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ! [V5i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V4k_2E0))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3s_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V5i_2E0)))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V5i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V5i_2E0))) ) )
& ! [V6i_2E0: A_27a,V7j_2E0: A_27a,V8x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V6i_2E0,V4k_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V7j_2E0,V4k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V3s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V6i_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V7j_2E0)))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V6i_2E0),V8x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V7j_2E0),V8x_2E0) ) )
& ! [V9x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V3s_2E0))
=> ? [V10j_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V10j_2E0,V4k_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V10j_2E0)))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V9x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V10j_2E0),V9x_2E0) ) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3s_2E0)) ) ).
tff(thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__RIGHT,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0,V6x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0,V6x_2E0),V5k_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0,V6x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V2s_2E0),V0f_2E0),V7k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V7k_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V7k_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V2s_2E0),V0f_2E0),V7k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5k_2E0,V4u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V5k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0)))) ) )
=> ! [V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V7k_2E0,V3t_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V7k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V7k_2E0)))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPOSITION__LEFT,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0,V6x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0,V6x_2E0),V5k_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0,V6x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0),V7k_2E0)) ) )
=> ( ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7k_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5k_2E0,V4u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V5k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V5k_2E0)))) ) )
=> ! [V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V7k_2E0,V4u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V7k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7k_2E0)))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EPROPER__MAP__COMPOSE,axiom,
! [F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V10x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F5_2E0,V10x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V9k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0,V10x_2E0),V9k_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V10x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V9k_2E0),V10x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V10x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F5_2E0,V10x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V9k_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0),V7k_2E0)) ) )
=> ( ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7k_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7k_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5k_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5k_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5k_2E0,V3t_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V5k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5k_2E0)))) )
& ! [V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V7k_2E0,V4u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V7k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7k_2E0)))) ) )
=> ! [V9k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V9k_2E0,V4u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V9k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V9k_2E0)))) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EPROPER__MAP__FROM__COMPACT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V2s_2E0),V1f_2E0),V3k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),V3k_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V1f_2E0),V3k_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V2s_2E0),V1f_2E0),V3k_2E0)) )
=> ! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2s_2E0),V0t_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V2s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0t_2E0),V3k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V1f_2E0),V3k_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) ) )
=> ( V2x_2E0 = V3y_2E0 ) )
=> ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V4t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4t_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EPROPER__MAP,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6a_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6a_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0) = V6a_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6a_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6a_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6a_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3k_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3k_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3k_2E0,V2t_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V3k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3k_2E0)))) )
<=> ( ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
& ! [V6a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6a_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6a_2E0)))) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOSED__MONOTONE__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V4y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4y_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4y_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4y_2E0)) )
=> ( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6c_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0),V6c_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6c_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6c_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3c_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3c_2E0))) )
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4y_2E0)))) ) )
=> ! [V6c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V6c_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6c_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6c_2E0)))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__OPEN__MONOTONE__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = V4y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4y_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4y_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4y_2E0)) )
=> ( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6c_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0),V6c_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6c_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6c_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3c_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3c_2E0))) )
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4y_2E0)))) ) )
=> ! [V6c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V6c_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6c_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6c_2E0)))) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__POINT,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = V5y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V5y_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3k_2E0))) )
<=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V5y_2E0)),V4u_2E0)) )
=> ? [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V7v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0)),V4u_2E0)) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType,F9_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F8_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool))),F5_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F7_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F6_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),V14x_2E0: A_27f,V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V15y_2E0: A_27g,V16z_2E0: A_27h] : ( app_2E2(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F9_2E0,V13f_2E0),V14x_2E0),V12P_2E0),V15y_2E0),V16z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))),V13f_2E0,V14x_2E0),V15y_2E0),V16z_2E0),app_2E2(A_27h,tyop_2Emin_2Ebool,app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)),V12P_2E0,V14x_2E0),V15y_2E0),V16z_2E0)) )
=> ( ! [V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V14x_2E0: A_27f] : ( app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F4_2E0,V13f_2E0),V12P_2E0),V14x_2E0) = c_2Epair_2EUNCURRY_2E1(A_27g,A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F9_2E0,V13f_2E0),V14x_2E0),V12P_2E0)) )
=> ( ! [V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V17a_2E0: A_27i,V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool,app_2E2(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool)),F8_2E0,V12P_2E0),V17a_2E0),V13f_2E0))
<=> ? [V18x_2E0: A_27f,V19y_2E0: A_27g,V20z_2E0: A_27h] :
( p(app_2E2(A_27h,tyop_2Emin_2Ebool,app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)),V12P_2E0,V18x_2E0),V19y_2E0),V20z_2E0))
& p(c_2Ebool_2EIN_2E2(A_27i,V17a_2E0,app_2E2(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))),V13f_2E0,V18x_2E0),V19y_2E0),V20z_2E0))) ) )
=> ( ! [V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),V17a_2E0: A_27i] : ( app_2E2(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool))),F5_2E0,V12P_2E0),V13f_2E0),V17a_2E0) = c_2Epair_2E_2C_2E2(A_27i,tyop_2Emin_2Ebool,V17a_2E0,app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool,app_2E2(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool)),F8_2E0,V12P_2E0),V17a_2E0),V13f_2E0)) )
=> ( ! [V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V7x_2E0: A_27c,V8y_2E0: A_27d] : ( app_2E2(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0,V6f_2E0),V5P_2E0),V7x_2E0),V8y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),V6f_2E0,V7x_2E0),V8y_2E0),app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V5P_2E0,V7x_2E0),V8y_2E0)) )
=> ( ! [V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V9a_2E0: A_27e,V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool)),F7_2E0,V5P_2E0),V9a_2E0),V6f_2E0))
<=> ? [V10x_2E0: A_27c,V11y_2E0: A_27d] :
( p(app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V5P_2E0,V10x_2E0),V11y_2E0))
& p(c_2Ebool_2EIN_2E2(A_27e,V9a_2E0,app_2E2(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),V6f_2E0,V10x_2E0),V11y_2E0))) ) )
=> ( ! [V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),V9a_2E0: A_27e] : ( app_2E2(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool))),F3_2E0,V5P_2E0),V6f_2E0),V9a_2E0) = c_2Epair_2E_2C_2E2(A_27e,tyop_2Emin_2Ebool,V9a_2E0,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool)),F7_2E0,V5P_2E0),V9a_2E0),V6f_2E0)) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1f_2E0),V0P_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0,V2x_2E0),app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27b,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F6_2E0,V0P_2E0),V3a_2E0),V1f_2E0))
<=> ? [V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
& p(c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0,V4x_2E0))) ) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3a_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1f_2E0),V3a_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V3a_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F6_2E0,V0P_2E0),V3a_2E0),V1f_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EBIGUNION_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1f_2E0),V0P_2E0))) = c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1f_2E0)) )
& ! [V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)))] : ( c_2Epred__set_2EBIGUNION_2E1(A_27e,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0,V6f_2E0),V5P_2E0)))) = c_2Epred__set_2EGSPEC_2E1(A_27e,A_27e,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool))),F3_2E0,V5P_2E0),V6f_2E0)) )
& ! [V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))))] : ( c_2Epred__set_2EBIGUNION_2E1(A_27i,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h)),c_2Epair_2EUNCURRY_2E1(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F4_2E0,V13f_2E0),V12P_2E0)))) = c_2Epred__set_2EGSPEC_2E1(A_27i,A_27i,app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool))),F5_2E0,V12P_2E0),V13f_2E0)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBIGINTER__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType,F9_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F8_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool))),F5_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F7_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F6_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),V14x_2E0: A_27f,V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V15y_2E0: A_27g,V16z_2E0: A_27h] : ( app_2E2(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F9_2E0,V13f_2E0),V14x_2E0),V12P_2E0),V15y_2E0),V16z_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))),V13f_2E0,V14x_2E0),V15y_2E0),V16z_2E0),app_2E2(A_27h,tyop_2Emin_2Ebool,app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)),V12P_2E0,V14x_2E0),V15y_2E0),V16z_2E0)) )
=> ( ! [V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V14x_2E0: A_27f] : ( app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F4_2E0,V13f_2E0),V12P_2E0),V14x_2E0) = c_2Epair_2EUNCURRY_2E1(A_27g,A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F9_2E0,V13f_2E0),V14x_2E0),V12P_2E0)) )
=> ( ! [V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V17a_2E0: A_27i,V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool,app_2E2(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool)),F8_2E0,V12P_2E0),V17a_2E0),V13f_2E0))
<=> ! [V18x_2E0: A_27f,V19y_2E0: A_27g,V20z_2E0: A_27h] :
( p(app_2E2(A_27h,tyop_2Emin_2Ebool,app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)),V12P_2E0,V18x_2E0),V19y_2E0),V20z_2E0))
=> p(c_2Ebool_2EIN_2E2(A_27i,V17a_2E0,app_2E2(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),app_2E2(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))),V13f_2E0,V18x_2E0),V19y_2E0),V20z_2E0))) ) )
=> ( ! [V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),V17a_2E0: A_27i] : ( app_2E2(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool))),F5_2E0,V12P_2E0),V13f_2E0),V17a_2E0) = c_2Epair_2E_2C_2E2(A_27i,tyop_2Emin_2Ebool,V17a_2E0,app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool,app_2E2(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool)),F8_2E0,V12P_2E0),V17a_2E0),V13f_2E0)) )
=> ( ! [V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V7x_2E0: A_27c,V8y_2E0: A_27d] : ( app_2E2(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0,V6f_2E0),V5P_2E0),V7x_2E0),V8y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),V6f_2E0,V7x_2E0),V8y_2E0),app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V5P_2E0,V7x_2E0),V8y_2E0)) )
=> ( ! [V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V9a_2E0: A_27e,V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool)),F7_2E0,V5P_2E0),V9a_2E0),V6f_2E0))
<=> ! [V10x_2E0: A_27c,V11y_2E0: A_27d] :
( p(app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V5P_2E0,V10x_2E0),V11y_2E0))
=> p(c_2Ebool_2EIN_2E2(A_27e,V9a_2E0,app_2E2(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),V6f_2E0,V10x_2E0),V11y_2E0))) ) )
=> ( ! [V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),V9a_2E0: A_27e] : ( app_2E2(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool))),F3_2E0,V5P_2E0),V6f_2E0),V9a_2E0) = c_2Epair_2E_2C_2E2(A_27e,tyop_2Emin_2Ebool,V9a_2E0,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool)),F7_2E0,V5P_2E0),V9a_2E0),V6f_2E0)) )
=> ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1f_2E0),V0P_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0,V2x_2E0),app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27b,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F6_2E0,V0P_2E0),V3a_2E0),V1f_2E0))
<=> ! [V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0))
=> p(c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0,V4x_2E0))) ) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3a_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1f_2E0),V3a_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V3a_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F6_2E0,V0P_2E0),V3a_2E0),V1f_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EBIGINTER_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1f_2E0),V0P_2E0))) = c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1f_2E0)) )
& ! [V5P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V6f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)))] : ( c_2Epred__set_2EBIGINTER_2E1(A_27e,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0,V6f_2E0),V5P_2E0)))) = c_2Epred__set_2EGSPEC_2E1(A_27e,A_27e,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27e,tyop_2Emin_2Ebool))),F3_2E0,V5P_2E0),V6f_2E0)) )
& ! [V12P_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),V13f_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))))] : ( c_2Epred__set_2EBIGINTER_2E1(A_27i,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h)),c_2Epair_2EUNCURRY_2E1(A_27f,tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27g,A_27h),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F4_2E0,V13f_2E0),V12P_2E0)))) = c_2Epred__set_2EGSPEC_2E1(A_27i,A_27i,app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)))),tyop_2Emin_2Efun(A_27i,tyop_2Epair_2Eprod(A_27i,tyop_2Emin_2Ebool))),F5_2E0,V12P_2E0),V13f_2E0)) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__FORALL__IN,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V2a_2E0: A_27a,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1Q_2E0),V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1Q_2E0,V2a_2E0),V3x_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0P_2E0),V1Q_2E0),V4x_2E0))
<=> ! [V5a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V5a_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1Q_2E0,V5a_2E0),V4x_2E0)) ) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1Q_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0P_2E0),V1Q_2E0),V4x_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2a_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1Q_2E0),V2a_2E0)))) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1Q_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__FORALL,axiom,
! [A_27a: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V0Q_2E0),V3x_2E0))
<=> ! [V4a_2E0: A_27a] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0Q_2E0,V4a_2E0),V3x_2E0)) )
=> ( ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0Q_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V0Q_2E0),V3x_2E0)) )
=> ( ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1a_2E0: A_27a,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0Q_2E0),V1a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0Q_2E0,V1a_2E0),V2x_2E0)) )
=> ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V1a_2E0: A_27a] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0Q_2E0),V1a_2E0))))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0Q_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EOpen_2E1(V0s_2E0) = c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__DIFF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__DIFF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__BIGINTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0))
& ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) ) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__BIGUNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0))
& ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) ) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EIN__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0))) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),V2e_2E0) ) ).
tff(thm_2Ereal__topology_2EIN__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0))) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),V2e_2E0) ) ).
tff(thm_2Ereal__topology_2EIN__SPHERE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0))))
<=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) = V2e_2E0 ) ) ).
tff(thm_2Ereal__topology_2EIN__BALL__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1e_2E0) ) ).
tff(thm_2Ereal__topology_2EIN__CBALL__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),V1e_2E0) ) ).
tff(thm_2Ereal__topology_2EIN__SPHERE__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))))
<=> ( c_2Ereal_2Eabs_2E1(V0x_2E0) = V1e_2E0 ) ) ).
tff(thm_2Ereal__topology_2EBALL__TRIVIAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ).
tff(thm_2Ereal__topology_2ECBALL__TRIVIAL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ).
tff(thm_2Ereal__topology_2ECENTRE__IN__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0))) = c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0) ) ).
tff(thm_2Ereal__topology_2EBALL__SUBSET__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2ESPHERE__SUBSET__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2ESUBSET__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1d_2E0,V2e_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1d_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESUBSET__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1d_2E0,V2e_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1d_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0)))) ) ).
tff(thm_2Ereal__topology_2EBALL__MAX__UNION,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal_2Emax_2E2(V1r_2E0,V2s_2E0))) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2EBALL__MIN__INTER,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal_2Emin_2E2(V1r_2E0,V2s_2E0))) = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECBALL__MAX__UNION,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal_2Emax_2E2(V1r_2E0,V2s_2E0))) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECBALL__MIN__INTER,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Emin_2E2(V1d_2E0,V2e_2E0))) = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1d_2E0)),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0))) ) ).
tff(thm_2Ereal__topology_2EBALL__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3y_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3y_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1x_2E0),V2r_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3y_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3y_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1x_2E0),V2r_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESPHERE__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V3y_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V3y_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0a_2E0,V1x_2E0),V2r_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBALL__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
? [V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V3y_2E0 )
& ! [V5x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0)) = c_2Ereal_2Eabs_2E1(V5x_2E0) ) )
=> ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),V2r_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
? [V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V3y_2E0 )
& ! [V5x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0)) = c_2Ereal_2Eabs_2E1(V5x_2E0) ) )
=> ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),V2r_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESPHERE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3y_2E0: tyop_2Erealax_2Ereal] :
? [V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V3y_2E0 )
& ! [V5x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0)) = c_2Ereal_2Eabs_2E1(V5x_2E0) ) )
=> ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),V2r_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBALL__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V3x_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0c_2E0))
=> ! [V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V1x_2E0),c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V2r_2E0))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V3x_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0c_2E0))
=> ! [V1x_2E0: tyop_2Erealax_2Ereal,V2r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V1x_2E0),c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V2r_2E0))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2r_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECBALL__DIFF__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2EBALL__UNION__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2ESPHERE__UNION__BALL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2ECBALL__DIFF__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)),c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0))) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EOpen_2E1(c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2ECENTRE__IN__BALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0))) = c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__BALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__BALL__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
<=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBALL__EQ__EMPTY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> p(c_2Ereal_2Ereal__lte_2E2(V1e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Ereal__topology_2EBALL__EMPTY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__CONTAINS__CBALL__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
<=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ESPHERE__EQ__EMPTY,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> p(c_2Erealax_2Ereal__lt_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPHERE__EMPTY,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V1r_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ( c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ENEGATIONS__BALL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0r_2E0))) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0r_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ENEGATIONS__CBALL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0r_2E0))) = c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0r_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ENEGATIONS__SPHERE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(V1x_2E0) )
=> ! [V0r_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0r_2E0))) = c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0r_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1u_2E0),V0s_2E0))
<=> ? [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
& ( V0s_2E0 = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1u_2E0,V2t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__INTER__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__OPEN__INTER,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0u_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__OPEN__IN__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1u_2E0),V0s_2E0))
<=> ? [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0))
& ( V0s_2E0 = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1u_2E0,V2t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SUBSET__EQ,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
=> ( c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0) = c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V0u_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__CLOSED__INTER,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0u_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SUBSET__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__SUBSET__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2Eopen__in,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
<=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V0u_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_27_2E0,V0u_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_27_2E0,V2x_2E0)),V3e_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_27_2E0,V1s_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__CONTAINS__BALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
<=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3e_2E0)),V1t_2E0),V0s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__CONTAINS__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
<=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3e_2E0)),V1t_2E0),V0s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V1t_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__TRANS__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V2u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) )
<=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__OPEN__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V1t_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__TRANS__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V2u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) )
<=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__CLOSED__TRANS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__INTER__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1u_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1u_2E0,V0s_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2v_2E0,V1u_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2v_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2v_2E0,V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__OPEN__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
<=> ( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__CLOSED__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
<=> ( p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SEGMENT__ALT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V2u_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V2u_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2u_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
=> ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2u_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0),V2u_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2u_2E0),V0a_2E0),c_2Erealax_2Ereal__mul_2E2(V2u_2E0,V1b_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V2u_2E0)) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ( V0a_2E0 != V1b_2E0 )
=> ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0a_2E0),V1b_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2Esegment,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V2u_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V2u_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2u_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
=> ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V0b_2E0: tyop_2Erealax_2Ereal,V2u_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1a_2E0),V0b_2E0),V2u_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2u_2E0),V1a_2E0),c_2Erealax_2Ereal__mul_2E2(V2u_2E0,V0b_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V2u_2E0)) )
=> ! [V0b_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1a_2E0),V0b_2E0)) )
& ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V0b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__REFL,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V0a_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) )
& ! [V1a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V1a_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EIN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
<=> ? [V3u_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3u_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V3u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ( V2x_2E0 = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3u_2E0),V0a_2E0),c_2Erealax_2Ereal__mul_2E2(V3u_2E0,V1b_2E0)) ) ) )
& ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
<=> ( ( V0a_2E0 != V1b_2E0 )
& ? [V4u_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4u_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V4u_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ( V2x_2E0 = c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V4u_2E0),V0a_2E0),c_2Erealax_2Ereal__mul_2E2(V4u_2E0,V1b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__SYM,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1b_2E0,V0a_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) )
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)) = c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3b_2E0,V2a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EENDS__IN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2EENDS__NOT__IN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__CLOSED__OPEN,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__OPEN__SUBSET__CLOSED,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ).
tff(thm_2Ereal__topology_2EMIDPOINT__IN__SEGMENT,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Emidpoint_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)),c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0))))
<=> ( V2a_2E0 != V3b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EBETWEEN__IN__SEGMENT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ebetween_2E2(V0x_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0)) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) ) ).
tff(thm_2Ereal__topology_2EREAL__CONVEX__BOUND__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3u_2E0: tyop_2Erealax_2Ereal,V4v_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V2a_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V2a_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3u_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4v_2E0))
& ( c_2Erealax_2Ereal__add_2E2(V3u_2E0,V4v_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(V3u_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V4v_2E0,V1y_2E0)),V2a_2E0)) ) ).
tff(thm_2Ereal__topology_2EIN__SEGMENT__COMPONENT,axiom,
! [A_27a: $tType,V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3i_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Emin_2E2(V0a_2E0,V1b_2E0),V2x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,c_2Ereal_2Emax_2E2(V0a_2E0,V1b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESEGMENT__TRANSLATION,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V4c_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V4c_2E0),V7x_2E0) = c_2Erealax_2Ereal__add_2E2(V4c_2E0,V7x_2E0) )
=> ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(V0c_2E0,V3x_2E0) )
=> ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0c_2E0,V1a_2E0),c_2Erealax_2Ereal__add_2E2(V0c_2E0,V2b_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) )
& ! [V4c_2E0: tyop_2Erealax_2Ereal,V5a_2E0: tyop_2Erealax_2Ereal,V6b_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V4c_2E0,V5a_2E0),c_2Erealax_2Ereal__add_2E2(V4c_2E0,V6b_2E0))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V4c_2E0),c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5a_2E0,V6b_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SEGMENT__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2b_2E0)),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SEGMENT__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4y_2E0) )
=> ( V3x_2E0 = V4y_2E0 ) ) )
=> ( c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2b_2E0))) = c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EIN__OPEN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ( V2x_2E0 != V0a_2E0 )
& ( V2x_2E0 != V1b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EIN__OPEN__SEGMENT__ALT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ( V2x_2E0 != V0a_2E0 )
& ( V2x_2E0 != V1b_2E0 )
& ( V0a_2E0 != V1b_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__CLOSED__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))))) ) ).
tff(thm_2Ereal__topology_2ECOLLINEAR__DIST__IN__OPEN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Ecollinear_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ).
tff(thm_2Ereal__topology_2EDIST__IN__OPEN__CLOSED__SEGMENT,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) )
& ! [V3a_2E0: tyop_2Erealax_2Ereal,V4b_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0))))
=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V3a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V4b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3a_2E0,V4b_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EDIST__IN__CLOSED__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EDIST__IN__OPEN__SEGMENT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1e1_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2e2_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0)))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0),V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V0s_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2e2_2E0,V0s_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__OPEN__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1e1_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2e2_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0)))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1e1_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2e2_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__OPEN__IN__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1e1_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2e2_2E0))
& ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = V0s_2E0 )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1e1_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2e2_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOSED__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1e1_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2e2_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0)))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1e1_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2e2_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOSED__IN__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1e1_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2e2_2E0))
& ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = V0s_2E0 )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V1e1_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2e2_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EEXISTS__DIFF,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ? [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,c_2Epred__set_2EUNIV_2E0(A_27a),V1s_2E0)))
<=> ? [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) )
=> ( ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V1t_2E0 = V0s_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOSED__SET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1e1_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2e2_2E0))
& ( V1e1_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2e2_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = V0s_2E0 )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__OPEN__SET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ~ ? [V1e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1e1_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2e2_2E0))
& ( V1e1_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( V2e2_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = V0s_2E0 )
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1e1_2E0,V2e2_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__IFF__CONNECTABLE__POINTS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
<=> ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0s_2E0)) )
=> ? [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V3t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EMPTY,axiom,
p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECONNECTED__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2ECONNECTED__REAL__LEMMA,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3e1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4e2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1a_2E0),V3e1_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2b_2E0),V4e2_2E0))
& ! [V5e_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V6x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V6x_2E0,V2b_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0)) )
=> ? [V7d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7d_2E0))
& ! [V8y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V8y_2E0,V6x_2E0)),V7d_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0))),V5e_2E0)) ) ) )
& ! [V9y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9y_2E0,V3e1_2E0))
=> ? [V10e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V10e_2E0))
& ! [V11y_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V11y_27_2E0,V9y_2E0)),V10e_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11y_27_2E0,V3e1_2E0)) ) ) )
& ! [V12y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V12y_2E0,V4e2_2E0))
=> ? [V13e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V13e_2E0))
& ! [V14y_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V14y_27_2E0,V12y_2E0)),V13e_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V14y_27_2E0,V4e2_2E0)) ) ) )
& ~ ? [V15x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V15x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V15x_2E0,V2b_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V15x_2E0),V3e1_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V15x_2E0),V4e2_2E0)) ) )
=> ? [V16x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V16x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V16x_2E0,V2b_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V16x_2E0),V3e1_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V16x_2E0),V4e2_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__SEGMENT,axiom,
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econnected_2E1(c_2Ereal__topology_2Eclosed__segment_2E1(c_2Elist_2ECONS_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1b_2E0),c_2Elist_2ENIL_2E0(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))))
& ! [V2a_2E0: tyop_2Erealax_2Ereal,V3b_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econnected_2E1(c_2Ereal__topology_2Eopen__segment_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2a_2E0,V3b_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__UNIV,axiom,
p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECLOPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) )
<=> ( ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
| ( V0s_2E0 = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__BIGUNION,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0P_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0)) )
& ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0P_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0P_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__DIFF__OPEN__FROM__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V2u_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__DISJOINT__BIGUNION__OPEN__UNIQUE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Epairwise_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(tyop_2Erealax_2Ereal),V0f_2E0))
& p(c_2Ereal__topology_2Epairwise_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(tyop_2Erealax_2Ereal),V1f_27_2E0))
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V0f_2E0))
=> ( p(c_2Ereal__topology_2EOpen_2E1(V2s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V2s_2E0))
& ( V2s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3s_2E0,V1f_27_2E0))
=> ( p(c_2Ereal__topology_2EOpen_2E1(V3s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V3s_2E0))
& ( V3s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0f_2E0) = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1f_27_2E0) ) )
=> ( V0f_2E0 = V1f_27_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__FROM__CLOSED__UNION__AND__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))
& p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) )
=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__FROM__OPEN__UNION__AND__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))
& p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) )
=> ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__INDUCTION,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V2s_2E0))
& ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4a_2E0,V3t_2E0)) )
=> ? [V5z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5z_2E0,V3t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5z_2E0)) ) )
& ! [V6a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6a_2E0,V2s_2E0))
=> ? [V7t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V7t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6a_2E0,V7t_2E0))
& ! [V8x_2E0: tyop_2Erealax_2Ereal,V9y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V7t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9y_2E0,V7t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V8x_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V9y_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1Q_2E0,V8x_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1Q_2E0,V9y_2E0)) ) ) ) )
=> ! [V10a_2E0: tyop_2Erealax_2Ereal,V11b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10a_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11b_2E0,V2s_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V10a_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V11b_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1Q_2E0,V10a_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1Q_2E0,V11b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION__GEN,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V2s_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V3x_2E0),V4y_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V4y_2E0),V3x_2E0)) )
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal,V7z_2E0: tyop_2Erealax_2Ereal] :
( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V5x_2E0),V6y_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V6y_2E0),V7z_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V5x_2E0),V7z_2E0)) )
& ! [V8t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V9a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V8t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9a_2E0,V8t_2E0)) )
=> ? [V10z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10z_2E0,V8t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V10z_2E0)) ) )
& ! [V11a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11a_2E0,V2s_2E0))
=> ? [V12t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V12t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11a_2E0,V12t_2E0))
& ! [V13x_2E0: tyop_2Erealax_2Ereal,V14y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V13x_2E0,V12t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V14y_2E0,V12t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V13x_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V14y_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V13x_2E0),V14y_2E0)) ) ) ) )
=> ! [V15a_2E0: tyop_2Erealax_2Ereal,V16b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V15a_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V16b_2E0,V2s_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V15a_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V16b_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1R_2E0,V15a_2E0),V16b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__INDUCTION__SIMPLE,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& ! [V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V1s_2E0))
=> ? [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V3t_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V3t_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5y_2E0)) ) ) ) )
=> ! [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6a_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7b_2E0,V1s_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V6a_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V7b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__EQUIVALENCE__RELATION,axiom,
! [V0R_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V2x_2E0),V3y_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V3y_2E0),V2x_2E0)) )
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal,V6z_2E0: tyop_2Erealax_2Ereal] :
( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V4x_2E0),V5y_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V5y_2E0),V6z_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V4x_2E0),V6z_2E0)) )
& ! [V7a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7a_2E0,V1s_2E0))
=> ? [V8t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V8t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7a_2E0,V8t_2E0))
& ! [V9x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V8t_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V7a_2E0),V9x_2E0)) ) ) ) )
=> ! [V10a_2E0: tyop_2Erealax_2Ereal,V11b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10a_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11b_2E0,V1s_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0R_2E0,V10a_2E0),V11b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__SUBSET,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0)) )
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIMPT__APPROACHABLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_27_2E0,V1s_2E0))
& ( V3x_27_2E0 != V0x_2E0 )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_27_2E0,V0x_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EAPPROACHABLE__LT__LE,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),V2d_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0)) ) )
<=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_2E0: A_27a] :
( p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0),V4d_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__APPROACHABLE__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_27_2E0,V1s_2E0))
& ( V3x_27_2E0 != V0x_2E0 )
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_27_2E0,V0x_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__CHOOSE__SIZE,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0c_2E0))
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(V1x_2E0) = V0c_2E0 ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__UNIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECLOSED__LIMPT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__EMPTY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ENO__LIMIT__POINT__IMP__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ~ ? [V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__POSITIVE__ORTHANT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0x_2E0,c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0))) ) ).
tff(thm_2Ereal__topology_2EFINITE__SET__AVOID,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
=> ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& ( V3x_2E0 != V0a_2E0 ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2d_2E0,c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V3x_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ELIMIT__POINT__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
=> ~ p(c_2Ereal__topology_2Elimit__point__of_2E2(V1a_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIMPT__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2ELIMIT__POINT__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))
<=> ( p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V0s_2E0))
| p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__INSERT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V0s_2E0)) = c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ELIMPT__OF__LIMPTS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1s_2E0),V2y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2y_2E0,c_2Ereal__topology_2Elimit__point__of_2E2(V2y_2E0,V1s_2E0)) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1s_2E0))))
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__LIMPTS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)))) ) ).
tff(thm_2Ereal__topology_2EDISCRETE__IMP__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3y_2E0,V2x_2E0)),V1e_2E0)) )
=> ( V3y_2E0 = V2x_2E0 ) ) )
=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIMPT__OF__UNIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ELIMPT__OF__OPEN__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
& p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIMPT__OF__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0)) )
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SING,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V0s_2E0))
& ~ p(c_2Ereal__topology_2Elimit__point__of_2E2(V1a_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = V0s_2E0 )
<=> p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = V0s_2E0 ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__EMPTY,axiom,
c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2EINTERIOR__UNIV,axiom,
c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2EOPEN__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2EOpen_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2EINTERIOR__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) = c_2Ereal__topology_2Einterior_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(V0s_2E0),V0s_2E0)) ).
tff(thm_2Ereal__topology_2ESUBSET__INTERIOR__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) = c_2Ereal__topology_2EOpen_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ESUBSET__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(V0s_2E0),c_2Ereal__topology_2Einterior_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__MAXIMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__MAXIMAL__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Einterior_2E1(V1t_2E0)) = c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__UNIQUE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& ! [V2t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_27_2E0,V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_27_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_27_2E0,V1t_2E0)) ) )
=> ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = V1t_2E0 ) ) ).
tff(thm_2Ereal__topology_2EIN__INTERIOR,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0)))
<=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2e_2E0)),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__SUBSET__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Einterior_2E1(V1t_2E0)) = c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(V0s_2E0),c_2Ereal__topology_2Einterior_2E1(V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__FINITE__BIGINTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0))
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0s_2E0)) = c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Einterior_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__BIGINTER__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0)),c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Einterior_2E0,V0f_2E0)))) ).
tff(thm_2Ereal__topology_2EUNION__INTERIOR__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(V0s_2E0),c_2Ereal__topology_2Einterior_2E1(V1t_2E0)),c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))) ).
tff(thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) )
=> ( V1t_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__EQ__EMPTY__ALT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__LIMIT__POINT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Einterior_2E1(V0s_2E0)))
=> p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CLOSED__UNION__EMPTY__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( c_2Ereal__topology_2Einterior_2E1(V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Ereal__topology_2Einterior_2E1(V0s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__UNION__EQ__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
| p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> ( ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2Einterior_2E1(V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__UNIONS__OPEN__SUBSETS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F1_2E0,V1t_2E0),V0s_2E0))
<=> ( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V1t_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F1_2E0,V1t_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))) = c_2Ereal__topology_2Einterior_2E1(V0s_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__ARCH__INV,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0e_2E0))
<=> ? [V1n_2E0: tyop_2Enum_2Enum] :
( ( V1n_2E0 != c_2Enum_2E0_2E0 )
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0)),V0e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EREAL__POW__LBOUND,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V1n_2E0),V0x_2E0)),c_2Ereal_2Epow_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),V1n_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__ARCH__POW,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0))
=> ? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,c_2Ereal_2Epow_2E2(V0x_2E0,V2n_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__ARCH__POW2,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Epow_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V1n_2E0))) ).
tff(thm_2Ereal__topology_2EREAL__ARCH__POW__INV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ? [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V2n_2E0),V1y_2E0)) ) ).
tff(thm_2Ereal__topology_2EFORALL__POS__MONO,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V1d_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V1d_2E0,V2e_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V1d_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V2e_2E0)) )
& ! [V3n_2E0: tyop_2Enum_2Enum] :
( ( V3n_2E0 != c_2Enum_2E0_2E0 )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__of__num_2E1(V3n_2E0)))) ) )
=> ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__SUC,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] :
( ( V1n_2E0 != c_2Enum_2E0_2E0 )
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V1n_2E0)) )
<=> ! [V2n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))) ) ).
tff(thm_2Ereal__topology_2ELT__NZ,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0))
<=> ( V0n_2E0 != c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Ereal__topology_2EREAL__ARCH__RDIV__EQ__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1c_2E0))
& ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2m_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V2m_2E0),V0x_2E0),V1c_2E0)) ) )
=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__APPROACHABLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3y_2E0,V0x_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__NONEMPTY__OPEN__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__INTERIOR,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2ECLOSURE__HULL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Etopology_2Ehull_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2EClosed_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = V0s_2E0 )
<=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = V0s_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2ESUBSET__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__INTER__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)))) ).
tff(thm_2Ereal__topology_2ECLOSURE__BIGINTER__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0)),c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal__topology_2Eclosure_2E0,V0f_2E0)))) ).
tff(thm_2Ereal__topology_2ECLOSURE__MINIMAL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__MINIMAL__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
=> ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),V1t_2E0) = c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__UNIQUE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& ! [V2t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2t_27_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2t_27_2E0)) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2t_27_2E0)) ) )
=> ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = V1t_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__EMPTY,axiom,
c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2ECLOSURE__UNIV,axiom,
c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2ECLOSURE__BIGUNION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V1s_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0))
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0f_2E0)) = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__EQ__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( V0s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__SUBSET__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),V0s_2E0) = c_2Ereal__topology_2EClosed_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__EQ__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__OPEN__IN__INTER__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0)) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) = c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) = c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTER__CLOSURE__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)),c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__OPEN__INTER__SUPERSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__COMPLEMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__COMPLEMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__INTERMEDIATE__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) )
=> p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__UNION__STRONG,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__DIFF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIMPT__OF__CLOSURE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)) = c_2Ereal__topology_2Elimit__point__of_2E2(V0x_2E0,V1s_2E0) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__LIMPT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0))
<=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1t_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__INTER__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
<=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) = V1t_2E0 ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CLOSURE__IDEMP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))) = c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__INTERIOR__IDEMP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0)))) = c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ENOWHERE__DENSE__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ( ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ENOWHERE__DENSE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& ( V2u_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V1t_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CLOSURE__INTER__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)),c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__INTERIOR__UNION__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0)),c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V1t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EREGULAR__OPEN__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = V0s_2E0 )
& ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) = V1t_2E0 ) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EREGULAR__CLOSED__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0)) = V0s_2E0 )
& ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V1t_2E0)) = V1t_2E0 ) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2EREGULAR__CLOSED__BIGUNION,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0))
& ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0f_2E0))
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V1t_2E0)) = V1t_2E0 ) ) )
=> ( c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) = c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0f_2E0) ) ) ).
tff(thm_2Ereal__topology_2EDIFF__CLOSURE__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)),c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))) ).
tff(thm_2Ereal__topology_2EDENSE__OPEN__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0)) )
| ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2u_2E0),V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2u_2E0)) ) )
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2u_2E0,c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))))
<=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2u_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2u_2E0,c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2EFRONTIER__CLOSURES,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efrontier_2E1(V0s_2E0) = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__STRADDLE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Efrontier_2E1(V1s_2E0)))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ( ? [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V3x_2E0)),V2e_2E0)) )
& ? [V4x_2E0: tyop_2Erealax_2Ereal] :
( ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V4x_2E0)),V2e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__SUBSET__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__EMPTY,axiom,
c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2EFRONTIER__UNIV,axiom,
c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2EFRONTIER__SUBSET__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),V0s_2E0) = c_2Ereal__topology_2EClosed_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__COMPLEMENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0)) = c_2Ereal__topology_2Efrontier_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__DISJOINT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
<=> p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0)))) ).
tff(thm_2Ereal__topology_2EFRONTIER__UNION__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0)))) ).
tff(thm_2Ereal__topology_2EFRONTIER__INTERIORS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efrontier_2E1(V0s_2E0) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),c_2Ereal__topology_2Einterior_2E1(V0s_2E0)),c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal),V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__FRONTIER__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2EINTERIOR__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)) = c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)),c_2Ereal__topology_2Eclosure_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__FRONTIER__EMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
| p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
| p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) )
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)) = c_2Ereal__topology_2Efrontier_2E1(V0s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__FRONTIER__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) = c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EUNION__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0)) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__INTER__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V0s_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Efrontier_2E1(V1t_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EINTERIOR__CLOSED__EQ__EMPTY__AS__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ( c_2Ereal__topology_2Einterior_2E1(V0s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
<=> ? [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& ( V0s_2E0 = c_2Ereal__topology_2Efrontier_2E1(V1t_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
=> ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__UNION__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Eclosure_2E1(V0s_2E0) = c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__INTERIOR__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Einterior_2E1(V0s_2E0)),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2EFRONTIER__CLOSURE__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)),c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2ESET__DIFF__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0)) = c_2Ereal__topology_2Einterior_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__INTER__SUBSET__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0),c_2Ereal__topology_2Efrontier_2E1(V1t_2E0)),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),c_2Ereal__topology_2Eclosure_2E1(V1t_2E0))))) ).
tff(thm_2Ereal__topology_2Enet__tybij,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Emk__net_2E1(A_27a,c_2Ereal__topology_2Enetord_2E1(A_27a,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ! [V2x_2E0: A_27b,V3y_2E0: A_27b] :
( ! [V4z_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1r_2E0,V4z_2E0),V2x_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1r_2E0,V4z_2E0),V3y_2E0)) )
| ! [V5z_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1r_2E0,V5z_2E0),V3y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1r_2E0,V5z_2E0),V2x_2E0)) ) )
<=> ( c_2Ereal__topology_2Enetord_2E1(A_27b,c_2Ereal__topology_2Emk__net_2E1(A_27b,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ENET,axiom,
! [A_27a: $tType,V0n_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( ! [V3z_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V3z_2E0,V1x_2E0))
=> p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V3z_2E0,V2y_2E0)) )
| ! [V4z_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V4z_2E0,V2y_2E0))
=> p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V4z_2E0,V1x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EOLDNET,axiom,
! [A_27a: $tType,V0n_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( ( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V1x_2E0,V1x_2E0))
& p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V2y_2E0,V2y_2E0)) )
=> ? [V3z_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V3z_2E0,V3z_2E0))
& ! [V4w_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V4w_2E0,V3z_2E0))
=> ( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V4w_2E0,V1x_2E0))
& p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V4w_2E0,V2y_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ENET__DILEMMA,axiom,
! [A_27a: $tType,V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( ( ? [V3a_2E0: A_27a] :
( ? [V4x_2E0: A_27a] : p(c_2Ereal__topology_2Enetord_2E3(A_27a,V2net_2E0,V4x_2E0,V3a_2E0))
& ! [V5x_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V2net_2E0,V5x_2E0,V3a_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V5x_2E0)) ) )
& ? [V6b_2E0: A_27a] :
( ? [V7x_2E0: A_27a] : p(c_2Ereal__topology_2Enetord_2E3(A_27a,V2net_2E0,V7x_2E0,V6b_2E0))
& ! [V8x_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V2net_2E0,V8x_2E0,V6b_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0Q_2E0,V8x_2E0)) ) ) )
=> ? [V9c_2E0: A_27a] :
( ? [V10x_2E0: A_27a] : p(c_2Ereal__topology_2Enetord_2E3(A_27a,V2net_2E0,V10x_2E0,V9c_2E0))
& ! [V11x_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V2net_2E0,V11x_2E0,V9c_2E0))
=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V11x_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0Q_2E0,V11x_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EAT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Enetord_2E3(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),V1x_2E0,V2y_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V0a_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2y_2E0,V0a_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EAT__INFINITY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Enetord_2E3(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat__infinity_2E0,V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Eabs_2E1(V1y_2E0)) ) ).
tff(thm_2Ereal__topology_2EAT__POSINFINITY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Enetord_2E3(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat__posinfinity_2E0,V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__ge_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal__topology_2EAT__NEGINFINITY,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Enetord_2E3(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat__neginfinity_2E0,V0x_2E0,V1y_2E0) = c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0) ) ).
tff(thm_2Ereal__topology_2ESEQUENTIALLY,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal__topology_2Enetord_2E3(tyop_2Enum_2Enum,c_2Ereal__topology_2Esequentially_2E0,V0m_2E0,V1n_2E0) = c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Ereal__topology_2EWITHIN,axiom,
! [A_27a: $tType,V0n_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a,V3y_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,c_2Ereal__topology_2Ewithin_2E2(A_27a,V0n_2E0,V1s_2E0),V2x_2E0,V3y_2E0))
<=> ( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0n_2E0,V2x_2E0,V3y_2E0))
& p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EIN__DIRECTION,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1v_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Enetord_2E3(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ein__direction_2E2(V0a_2E0,V1v_2E0),V2x_2E0,V3y_2E0))
<=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V0a_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3y_2E0,V0a_2E0))))
& ? [V4c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4c_2E0))
& ( c_2Ereal_2Ereal__sub_2E2(V2x_2E0,V0a_2E0) = c_2Erealax_2Ereal__mul_2E2(V4c_2E0,V1v_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2EWITHIN__UNIV,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0x_2E0),c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Ereal__topology_2Eat_2E1(V0x_2E0) ) ).
tff(thm_2Ereal__topology_2EWITHIN__WITHIN,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ewithin_2E2(A_27a,c_2Ereal__topology_2Ewithin_2E2(A_27a,V0net_2E0,V1s_2E0),V2t_2E0) = c_2Ereal__topology_2Ewithin_2E2(A_27a,V0net_2E0,c_2Epred__set_2EINTER_2E2(A_27a,V1s_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ETRIVIAL__LIMIT__WITHIN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1a_2E0),V0s_2E0)))
<=> ~ p(c_2Ereal__topology_2Elimit__point__of_2E2(V1a_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0))) ).
tff(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__INFINITY,axiom,
~ p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat__infinity_2E0)) ).
tff(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__POSINFINITY,axiom,
~ p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat__posinfinity_2E0)) ).
tff(thm_2Ereal__topology_2ETRIVIAL__LIMIT__AT__NEGINFINITY,axiom,
~ p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat__neginfinity_2E0)) ).
tff(thm_2Ereal__topology_2ETRIVIAL__LIMIT__SEQUENTIALLY,axiom,
~ p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Enum_2Enum,c_2Ereal__topology_2Esequentially_2E0)) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__CLOSED__TRIVIAL,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Etrivial__limit_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ENONTRIVIAL__LIMIT__WITHIN,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
=> p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,c_2Ereal__topology_2Ewithin_2E2(A_27a,V0net_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__HAPPENS,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V1p_2E0,V0net_2E0))
=> ( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
| ? [V2x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1p_2E0,V2x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__WITHIN__LE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V2p_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1a_2E0),V0s_2E0)))
<=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V1a_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V1a_2E0)),V3d_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2p_2E0,V4x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__WITHIN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V2p_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1a_2E0),V0s_2E0)))
<=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V1a_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V1a_2E0)),V3d_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2p_2E0,V4x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__AT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V1p_2E0,c_2Ereal__topology_2Eat_2E1(V0a_2E0)))
<=> ? [V2d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2d_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V0a_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V0a_2E0)),V2d_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1p_2E0,V3x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__SEQUENTIALLY,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Enum_2Enum,V0p_2E0,c_2Ereal__topology_2Esequentially_2E0))
<=> ? [V1N_2E0: tyop_2Enum_2Enum] :
! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1N_2E0,V2n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0p_2E0,V2n_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V0p_2E0,c_2Ereal__topology_2Eat__infinity_2E0))
<=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1b_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__AT__POSINFINITY,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V0p_2E0,c_2Ereal__topology_2Eat__posinfinity_2E0))
<=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__ge_2E2(V2x_2E0,V1b_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__AT__NEGINFINITY,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V0p_2E0,c_2Ereal__topology_2Eat__neginfinity_2E0))
<=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1b_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__AT__INFINITY__POS,axiom,
! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,V0p_2E0,c_2Ereal__topology_2Eat__infinity_2E0))
<=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1b_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1b_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EALWAYS__EVENTUALLY,axiom,
! [A_27a: $tType,V0p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( ! [V2x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0p_2E0,V2x_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V0p_2E0,V1net_2E0)) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__AND,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1p_2E0),V2q_2E0),V3x_2E0))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1p_2E0,V3x_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2q_2E0,V3x_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1p_2E0),V2q_2E0),V0net_2E0))
<=> ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V1p_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V2q_2E0,V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__MONO,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ! [V3x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1p_2E0,V3x_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2q_2E0,V3x_2E0)) )
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V1p_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V2q_2E0,V0net_2E0)) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__MP,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1p_2E0),V2q_2E0),V3x_2E0))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1p_2E0,V3x_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2q_2E0,V3x_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1p_2E0),V2q_2E0),V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V1p_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V2q_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__FALSE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ! [V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,F0_2E0,V1x_2E0) = c_2Ebool_2EF_2E0 )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Eeventually_2E2(A_27a,F0_2E0,V0net_2E0) = c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0) ) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__TRUE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ! [V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,F0_2E0,V1x_2E0) = c_2Ebool_2ET_2E0 )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] : ( c_2Ereal__topology_2Eeventually_2E2(A_27a,F0_2E0,V0net_2E0) = c_2Ebool_2ET_2E0 ) ) ).
tff(thm_2Ereal__topology_2ENOT__EVENTUALLY,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: A_27a] : ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1p_2E0,V2x_2E0))
& ~ p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0)) )
=> ~ p(c_2Ereal__topology_2Eeventually_2E2(A_27a,V1p_2E0,V0net_2E0)) ) ).
tff(thm_2Ereal__topology_2EEVENTUALLY__FORALL,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V3x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V2s_2E0),V1p_2E0),V3x_2E0))
<=> ! [V4a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V4a_2E0,V2s_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0,V4a_2E0),V3x_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
& ( V2s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27b) ) )
=> ( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V2s_2E0),V1p_2E0),V0net_2E0))
<=> ! [V5a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V5a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0,V5a_2E0),V0net_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__EVENTUALLY,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V2s_2E0),V1p_2E0),V4x_2E0))
<=> ! [V5a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V5a_2E0,V2s_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0,V5a_2E0),V4x_2E0)) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
& ( V2s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27b) ) )
=> ( ! [V3a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0,V3a_2E0),V0net_2E0)) )
<=> p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V2s_2E0),V1p_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM,axiom,
! [A_27a: $tType,V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1l_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2f_2E0,V1l_2E0,V0net_2E0))
<=> ( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V0net_2E0))
| ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4y_2E0: A_27a] :
( ? [V5x_2E0: A_27a] : p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0net_2E0,V5x_2E0,V4y_2E0))
& ! [V6x_2E0: A_27a] :
( p(c_2Ereal__topology_2Enetord_2E3(A_27a,V0net_2E0,V6x_2E0,V4y_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V6x_2E0),V1l_2E0)),V3e_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V3s_2E0)))
<=> ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> ? [V5d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5d_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V3s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_2E0,V2a_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_2E0,V2a_2E0)),V5d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V1l_2E0)),V4e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V3s_2E0)))
<=> ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> ? [V5d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5d_2E0))
& ! [V6x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V3s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_2E0,V2a_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6x_2E0,V2a_2E0)),V5d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V1l_2E0)),V4e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat_2E1(V2a_2E0)))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V2a_2E0))))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V2a_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V1l_2E0)),V3e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat_2E1(V2a_2E0)))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V2a_2E0))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_2E0,V2a_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V1l_2E0)),V3e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__INFINITY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__infinity_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3b_2E0: tyop_2Erealax_2Ereal] :
! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(V4x_2E0),V3b_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V1l_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__INFINITY__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__infinity_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3b_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__ge_2E2(c_2Ereal_2Eabs_2E1(V4x_2E0),V3b_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V1l_2E0)),V2e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__POSINFINITY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__posinfinity_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3b_2E0: tyop_2Erealax_2Ereal] :
! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__ge_2E2(V4x_2E0,V3b_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V1l_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__NEGINFINITY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__neginfinity_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3b_2E0: tyop_2Erealax_2Ereal] :
! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V3b_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V1l_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__SEQUENTIALLY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3N_2E0: tyop_2Enum_2Enum] :
! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3N_2E0,V4n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V4n_2E0),V1l_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__EVENTUALLY,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2l_2E0),V3x_2E0))
<=> ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) = V2l_2E0 ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V1f_2E0),V2l_2E0),V0net_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V2l_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__POSINFINITY__SEQUENTIALLY,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2n_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__posinfinity_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__INFINITY__POSINFINITY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__infinity_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat__posinfinity_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__EMPTY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__SUBSET,axiom,
! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3a_2E0: tyop_2Erealax_2Ereal,V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3a_2E0),V4s_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0t_2E0,V4s_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3a_2E0),V0t_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__UNION,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2l_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V3s_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V4t_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0)))) ) ).
tff(thm_2Ereal__topology_2ELIM__UNION__UNIV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2l_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V3s_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V4t_2E0)))
& ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V2l_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPOSE__WITHIN,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4y_2E0: tyop_2Erealax_2Ereal,V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5z_2E0: tyop_2Erealax_2Ereal,V6w_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F0_2E0,V1f_2E0),V3s_2E0),V4y_2E0),V2g_2E0),V5z_2E0),V6w_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V6w_2E0),V3s_2E0))
& ( ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V6w_2E0) = V4y_2E0 )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V4y_2E0) = V5z_2E0 ) ) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4y_2E0: tyop_2Erealax_2Ereal,V5z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V4y_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))),F0_2E0,V1f_2E0),V3s_2E0),V4y_2E0),V2g_2E0),V5z_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V5z_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4y_2E0),V3s_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V1f_2E0),V5z_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__COMPOSE__AT,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal,V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4z_2E0: tyop_2Erealax_2Ereal,V5w_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1f_2E0),V3y_2E0),V2g_2E0),V4z_2E0),V5w_2E0))
<=> ( ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5w_2E0) = V3y_2E0 )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V3y_2E0) = V4z_2E0 ) ) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal,V4z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V1f_2E0,V3y_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Eeventually_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1f_2E0),V3y_2E0),V2g_2E0),V4z_2E0),V0net_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V2g_2E0,V4z_2E0,c_2Ereal__topology_2Eat_2E1(V3y_2E0))) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V1f_2E0),V4z_2E0,V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELIM__AT__WITHIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat_2E1(V2a_2E0)))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V3s_2E0))) ) ).
tff(thm_2Ereal__topology_2ELIM__WITHIN__OPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2a_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2a_2E0,V3s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V3s_2E0)) )
=> ( c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V3s_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Eat_2E1(V2a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EFROM__0,axiom,
c_2Ereal__topology_2Efrom_2E1(c_2Enum_2E0_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum) ).
tff(thm_2Ereal__topology_2EIN__FROM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V0m_2E0,c_2Ereal__topology_2Efrom_2E1(V1n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V0m_2E0) ) ).
tff(thm_2Ereal__topology_2EFROM__INTER__NUMSEG__GEN,axiom,
! [V0k_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,c_2Ereal__topology_2Efrom_2E1(V0k_2E0),c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V0k_2E0),c_2Eiterate_2E_2E_2E_2E2(V0k_2E0,V2n_2E0),c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0)) ) ).
tff(thm_2Ereal__topology_2EFROM__INTER__NUMSEG__MAX,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,c_2Ereal__topology_2Efrom_2E1(V2p_2E0),c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0)) = c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2EMAX_2E2(V2p_2E0,V0m_2E0),V1n_2E0) ) ).
tff(thm_2Ereal__topology_2EFROM__INTER__NUMSEG,axiom,
! [V0k_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2EINTER_2E2(tyop_2Enum_2Enum,c_2Ereal__topology_2Efrom_2E1(V0k_2E0),c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V1n_2E0)) = c_2Eiterate_2E_2E_2E_2E2(V0k_2E0,V1n_2E0) ) ).
tff(thm_2Ereal__topology_2EINFINITE__FROM,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Ereal__topology_2Efrom_2E1(V0n_2E0))) ).
tff(thm_2Ereal__topology_2EWLOG__LT,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1m_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V1m_2E0),V1m_2E0))
& ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V2m_2E0),V3n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V3n_2E0),V2m_2E0) )
& ! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V5n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V4m_2E0),V5n_2E0)) ) )
=> ! [V6m_2E0: tyop_2Enum_2Enum,V7y_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V6m_2E0),V7y_2E0)) ) ).
tff(thm_2Ereal__topology_2EWLOG__LE,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V1m_2E0),V2n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V2n_2E0),V1m_2E0) )
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V3m_2E0),V4n_2E0)) ) )
=> ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V5m_2E0),V6n_2E0)) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__BETWEEN,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V1b_2E0))
<=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V2x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDS__LINEAR__0,axiom,
! [V0A_2E0: tyop_2Enum_2Enum,V1B_2E0: tyop_2Enum_2Enum] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0A_2E0,V2n_2E0),V1B_2E0))
<=> ( V0A_2E0 = c_2Enum_2E0_2E0 ) ) ).
tff(thm_2Ereal__topology_2EBOUNDS__LINEAR,axiom,
! [V0A_2E0: tyop_2Enum_2Enum,V1B_2E0: tyop_2Enum_2Enum,V2C_2E0: tyop_2Enum_2Enum] :
( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2E_2A_2E2(V0A_2E0,V3n_2E0),c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(V1B_2E0,V3n_2E0),V2C_2E0)))
<=> p(c_2Earithmetic_2E_3C_3D_2E2(V0A_2E0,V1B_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__BIGINTER,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__UNIV,axiom,
p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECLOSED__EMPTY,axiom,
p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EOPEN__SUB__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EClosed_2E1(V0s_2E0) = c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V0s_2E0)) ).
tff(thm_2Ereal__topology_2EOPEN__IN__REFL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V0s_2E0)) ).
tff(thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN__SUBTOPOLOGY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Etopspace_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0)) = V0s_2E0 ) ).
tff(thm_2Ereal__topology_2ETOPSPACE__EUCLIDEAN,axiom,
c_2Etopology_2Etopspace_2E1(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ).
tff(thm_2Ereal__topology_2EOPEN__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2EOpen_2E1(V0s_2E0) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2EOPEN__EXISTS,axiom,
! [A_27a: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V0Q_2E0),V3x_2E0))
<=> ? [V4a_2E0: A_27a] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0Q_2E0,V4a_2E0),V3x_2E0)) )
=> ( ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0Q_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F2_2E0,V0Q_2E0),V3x_2E0)) )
=> ( ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V1a_2E0: A_27a,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0Q_2E0),V1a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0Q_2E0,V1a_2E0),V2x_2E0)) )
=> ! [V0Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V1a_2E0: A_27a] : p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0Q_2E0),V1a_2E0))))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0Q_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__EXISTS__IN,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V2a_2E0: A_27a,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1Q_2E0),V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1Q_2E0,V2a_2E0),V3x_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0P_2E0),V1Q_2E0),V4x_2E0))
<=> ? [V5a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V5a_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1Q_2E0,V5a_2E0),V4x_2E0)) ) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1Q_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0P_2E0),V1Q_2E0),V4x_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V2a_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2a_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V1Q_2E0),V2a_2E0)))) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0P_2E0),V1Q_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__BIGUNION,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__UNIV,axiom,
p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EOPEN__EMPTY,axiom,
p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__UNION,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V2t_2E0),V1s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V3u_2E0),V1s_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,c_2Epred__set_2EUNION_2E2(A_27a,V2t_2E0,V3u_2E0)),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__UNION,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V2t_2E0),V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V3u_2E0),V1s_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,c_2Epred__set_2EUNION_2E2(A_27a,V2t_2E0,V3u_2E0)),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__IMP__SUBSET,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1s_2E0),V2t_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__IMP__SUBSET,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1s_2E0),V2t_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBTOPOLOGY__UNIV,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a)] : ( c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,c_2Epred__set_2EUNIV_2E0(A_27a)) = V0top_2E0 ) ).
tff(thm_2Ereal__topology_2ESUBTOPOLOGY__TOPSPACE,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a)] : ( c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,c_2Etopology_2Etopspace_2E1(A_27a,V0top_2E0)) = V0top_2E0 ) ).
tff(thm_2Ereal__topology_2ESUBTOPOLOGY__SUPERSET,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(A_27a,c_2Etopology_2Etopspace_2E1(A_27a,V0top_2E0),V1s_2E0))
=> ( c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1s_2E0) = V0top_2E0 ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__REFL,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1u_2E0),V1u_2E0) = c_2Epred__set_2ESUBSET_2E2(A_27a,V1u_2E0,c_2Etopology_2Etopspace_2E1(A_27a,V0top_2E0)) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__REFL,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1u_2E0),V1u_2E0) = c_2Epred__set_2ESUBSET_2E2(A_27a,V1u_2E0,c_2Etopology_2Etopspace_2E1(A_27a,V0top_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY__EMPTY,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)),V1s_2E0))
<=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(A_27a) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY__EMPTY,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)),V1s_2E0))
<=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(A_27a) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1u_2E0),V2s_2E0))
<=> ? [V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(A_27a,V0top_2E0,V3t_2E0))
& ( V2s_2E0 = c_2Epred__set_2EINTER_2E2(A_27a,V3t_2E0,V1u_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ETOPSPACE__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Etopspace_2E1(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1u_2E0)) = c_2Epred__set_2EINTER_2E2(A_27a,c_2Etopology_2Etopspace_2E1(A_27a,V0top_2E0),V1u_2E0) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__SUBTOPOLOGY,axiom,
! [A_27a: $tType,V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Ereal__topology_2Esubtopology_2E2(A_27a,V0top_2E0,V1u_2E0),V2s_2E0))
<=> ? [V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(A_27a,V0top_2E0,V3t_2E0))
& ( V2s_2E0 = c_2Epred__set_2EINTER_2E2(A_27a,V3t_2E0,V1u_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EISTOPLOGY__SUBTOPOLOGY,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1u_2E0),V0top_2E0),V2s_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EINTER_2E2(A_27a,V2s_2E0,V1u_2E0),c_2Etopology_2Eopen__in_2E2(A_27a,V0top_2E0,V2s_2E0)) )
=> ! [V0top_2E0: tyop_2Etopology_2Etopology(A_27a),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : p(c_2Etopology_2Eistopology_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1u_2E0),V0top_2E0)))) ) ).
tff(thm_2Ereal__topology_2EINTER__BIGUNION,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))] :
( ! [V4t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V5x_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4t_2E0),V3s_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EINTER_2E2(A_27b,V4t_2E0,V5x_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V5x_2E0,V3s_2E0)) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EINTER_2E2(A_27a,V2x_2E0,V1t_2E0),c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0,V0s_2E0)) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V0s_2E0),V1t_2E0) = c_2Epred__set_2EBIGUNION_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V1t_2E0),V0s_2E0))) )
& ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27b,V4t_2E0,c_2Epred__set_2EBIGUNION_2E1(A_27b,V3s_2E0)) = c_2Epred__set_2EBIGUNION_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4t_2E0),V3s_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSET__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(A_27b,V1s_2E0,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2t_2E0)))
<=> ? [V3u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V3u_2E0,V2t_2E0))
& ( V1s_2E0 = c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V3u_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EDIM__SUBSET__UNIV,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal__topology_2Edim_2E1(V0s_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ).
tff(thm_2Ereal__topology_2EDIM__SUBSET,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal__topology_2Edim_2E1(V0s_2E0),c_2Ereal__topology_2Edim_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EDIM__UNIV,axiom,
c_2Ereal__topology_2Edim_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal)) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ).
tff(thm_2Ereal__topology_2EHAS__SIZE__STDBASIS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0i_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0i_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0i_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0i_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
=> ( ! [V0i_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0i_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0i_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0i_2E0)) )
=> p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__STDBASIS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0i_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0i_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0i_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0i_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
=> ( ! [V0i_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0i_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0i_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0i_2E0)) )
=> ( c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0)) = c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2EDIM__LE__CARD,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal__topology_2Edim_2E1(V0s_2E0),c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EDIM__UNIQUE,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V1v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1v_2E0,c_2Ereal__topology_2Espan_2E1(V2b_2E0)))
& p(c_2Ereal__topology_2Eindependent_2E1(V2b_2E0))
& p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V0n_2E0)) )
=> ( c_2Ereal__topology_2Edim_2E1(V1v_2E0) = V0n_2E0 ) ) ).
tff(thm_2Ereal__topology_2EBASIS__HAS__SIZE__DIM,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0))
& ( c_2Ereal__topology_2Espan_2E1(V1b_2E0) = V0v_2E0 ) )
=> p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Edim_2E1(V0v_2E0))) ) ).
tff(thm_2Ereal__topology_2EBASIS__CARD__EQ__DIM,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0v_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
& p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0)) )
=> ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1b_2E0))
& ( c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1b_2E0) = c_2Ereal__topology_2Edim_2E1(V0v_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__CARD__GE__DIM,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0v_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1b_2E0)) )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Ereal__topology_2Edim_2E1(V0v_2E0),c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1b_2E0))) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__CARD__LE__DIM,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0)) )
=> ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1b_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1b_2E0),c_2Ereal__topology_2Edim_2E1(V0v_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EBASIS__EXISTS,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
? [V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0v_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
& p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Edim_2E1(V0v_2E0))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INJECTIVE__LEFT__INVERSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0) )
=> ( V1x_2E0 = V2y_2E0 ) ) )
=> ? [V3g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V3g_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3g_2E0,V0f_2E0,V4x_2E0) = V4x_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__EQ__STDBASIS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0))
& ! [V2i_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2i_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2i_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2i_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V2i_2E0) ) ) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3s_2E0,c_2Ereal__topology_2Espan_2E1(V2b_2E0)))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2b_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0) ) ) )
=> ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__EQ__0,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1b_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__EQ__0__SPAN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1b_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
=> ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__KERNEL,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0f_2E0),V1x_2E0))
<=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F1_2E0,V0f_2E0),V1x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0))
=> ? [V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V2g_2E0))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1b_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INDEPENDENT__EXTEND__LEMMA,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1b_2E0))
=> ( p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0))
=> ? [V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0))) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,c_2Erealax_2Ereal__add_2E2(V3x_2E0,V4y_2E0)) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V4y_2E0)) ) )
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0)))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,c_2Erealax_2Ereal__mul_2E2(V6c_2E0,V5x_2E0)) = c_2Erealax_2Ereal__mul_2E2(V6c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V5x_2E0)) ) )
& ! [V7x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1b_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,V7x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__BREAKDOWN__EQ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V2s_2E0))))
<=> ? [V3k_2E0: tyop_2Erealax_2Ereal] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Erealax_2Ereal__mul_2E2(V3k_2E0,V1a_2E0)),c_2Ereal__topology_2Espan_2E1(V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET,axiom,
! [V0v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
? [V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V1b_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0v_2E0,c_2Ereal__topology_2Espan_2E1(V1b_2E0))) ) ).
tff(thm_2Ereal__topology_2EMAXIMAL__INDEPENDENT__SUBSET__EXTEND,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0)) )
=> ? [V2b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2b_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2b_2E0,V1v_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V2b_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1v_2E0,c_2Ereal__topology_2Espan_2E1(V2b_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__BOUND,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0))
=> ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V0s_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__SPAN__BOUND,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Espan_2E1(V1t_2E0))) )
=> ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V0s_2E0),c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECARD__STDBASIS,axiom,
c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ).
tff(thm_2Ereal__topology_2EEXCHANGE__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Espan_2E1(V1t_2E0))) )
=> ? [V2t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Erealax_2Ereal,V2t_27_2E0,c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,V1t_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V2t_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_27_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Espan_2E1(V2t_27_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__TRANS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2s_2E0)))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2EIN__SPAN__DELETE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0)))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V1b_2E0)))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V1b_2E0))))) ) ).
tff(thm_2Ereal__topology_2ESPANNING__SUBSET__INDEPENDENT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Eindependent_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Ereal__topology_2Espan_2E1(V1t_2E0))) )
=> ( V0s_2E0 = V1t_2E0 ) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__STDBASIS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0i_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0i_2E0))
<=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0i_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0i_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) )
=> ( ! [V0i_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0i_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0i_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0i_2E0)) )
=> p(c_2Ereal__topology_2Eindependent_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Eindependent_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))))
<=> ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EINDEPENDENT__EMPTY,axiom,
p(c_2Ereal__topology_2Eindependent_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ESPAN__EMPTY,axiom,
c_2Ereal__topology_2Espan_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ).
tff(thm_2Ereal__topology_2ESPAN__INDUCT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1h_2E0)) )
& p(c_2Ereal__topology_2Esubspace_2E1(V1h_2E0)) )
=> ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Espan_2E1(V0s_2E0)))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1h_2E0,V3x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESPAN__CLAUSES,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V2s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Ereal__topology_2Espan_2E1(V2s_2E0))) )
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Espan_2E1(V0s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal,V5s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Ereal__topology_2Espan_2E1(V5s_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,c_2Ereal__topology_2Espan_2E1(V5s_2E0))) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V3x_2E0,V4y_2E0),c_2Ereal__topology_2Espan_2E1(V5s_2E0))) )
& ! [V6x_2E0: tyop_2Erealax_2Ereal,V7c_2E0: tyop_2Erealax_2Ereal,V8s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,c_2Ereal__topology_2Espan_2E1(V8s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V7c_2E0,V6x_2E0),c_2Ereal__topology_2Espan_2E1(V8s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__SPAN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Esubspace_2E1(c_2Ereal__topology_2Espan_2E1(V0s_2E0))) ).
tff(thm_2Ereal__topology_2ESPAN__MONO,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Espan_2E1(V0s_2E0),c_2Ereal__topology_2Espan_2E1(V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ESPAN__SPAN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Espan_2E1(c_2Ereal__topology_2Espan_2E1(V0s_2E0)) = c_2Ereal__topology_2Espan_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__UNION__CHAIN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1t_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) )
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
| p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__INJECTIVE__0__SUBSPACE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0)) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3y_2E0) ) )
=> ( V2x_2E0 = V3y_2E0 ) )
<=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> ( V4x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__BIGINTER,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__TRIVIAL,axiom,
p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2ESUBSPACE__LINEAR__PREIMAGE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1s_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),V1s_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1s_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__SUM,axiom,
! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
& ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2t_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),V0s_2E0)) ) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Eiterate_2ESum_2E2(A_27a,V2t_2E0,V1f_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V2s_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__NEG,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V0x_2E0),V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2s_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V0x_2E0),V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Esubspace_2E1(V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1y_2E0,V2s_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1y_2E0),V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__0,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__IMP__NONEMPTY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Esubspace_2E1(V0s_2E0))
=> ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ESUBSPACE__UNIV,axiom,
p(c_2Ereal__topology_2Esubspace_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__PRE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V6k_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V6k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V6k_2E0))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V6k_2E0)) )
=> ( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V5k_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0)),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2D_2E2(V5k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0))
=> ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V4n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3m_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2D_2E2(V3m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2h_2E0),V0f_2E0),V1g_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__SUM__PARTIAL__SUC,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V6k_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V6k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V6k_2E0))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V6k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> ( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V5k_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0)),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V5k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5k_2E0))) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0))
=> ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V4n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V4n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3m_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3m_2E0))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V2h_2E0),V0f_2E0),V1g_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__BOUNDED__POS,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ? [V1B_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1B_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2x_2E0),V3y_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__mul_2E2(V1B_2E0,c_2Ereal_2Eabs_2E1(V2x_2E0)),c_2Ereal_2Eabs_2E1(V3y_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__BOUNDED,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ? [V1B_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2x_2E0),V3y_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Erealax_2Ereal__mul_2E2(V1B_2E0,c_2Ereal_2Eabs_2E1(V2x_2E0)),c_2Ereal_2Eabs_2E1(V3y_2E0)))) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)))))] :
( ! [V4h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V5i_2E0: A_27a,V6j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)))),F0_2E0,V4h_2E0),V3f_2E0),V2g_2E0),V5i_2E0),V6j_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4h_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V3f_2E0,V5i_2E0)),app_2E2(A_27b,tyop_2Erealax_2Ereal,V2g_2E0,V6j_2E0)) )
=> ! [V0t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V3f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ( p(c_2Ereal__topology_2Ebilinear_2E1(V4h_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27b,V0t_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4h_2E0,c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V3f_2E0)),c_2Eiterate_2ESum_2E2(A_27b,V0t_2E0,V2g_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epred__set_2ECROSS_2E2(A_27a,A_27b,V1s_2E0,V0t_2E0),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)))),F0_2E0,V4h_2E0),V3f_2E0),V2g_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__RSUB,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),c_2Ereal_2Ereal__sub_2E2(V2y_2E0,V3z_2E0)) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V2y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V3z_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__LSUB,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0)),V3z_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V3z_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2y_2E0),V3z_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__RZERO,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__LZERO,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),V1x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__RNEG,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),c_2Erealax_2Ereal__neg_2E1(V2y_2E0)) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V2y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__LNEG,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,c_2Erealax_2Ereal__neg_2E1(V1x_2E0)),V2y_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V2y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__RMUL,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2x_2E0),c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V3y_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2x_2E0),V3y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__LMUL,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0)),V3y_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2x_2E0),V3y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__RADD,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),c_2Erealax_2Ereal__add_2E2(V2y_2E0,V3z_2E0)) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V2y_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V3z_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__LADD,axiom,
! [V0h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ebilinear_2E1(V0h_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0)),V3z_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V1x_2E0),V3z_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0h_2E0,V2y_2E0),V3z_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__SWAP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0op_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0op_2E0),V1x_2E0),V2y_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0op_2E0,V2y_2E0),V1x_2E0) )
=> ! [V0op_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Ereal__topology_2Ebilinear_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0op_2E0)) = c_2Ereal__topology_2Ebilinear_2E1(V0op_2E0) ) ) ).
tff(thm_2Ereal__topology_2ESYMMETRIC__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V2x_2E0),V1s_2E0)) )
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0)) )
=> ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(V3x_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__BOUNDED__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ? [V1B_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1B_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)),c_2Erealax_2Ereal__mul_2E2(V1B_2E0,c_2Ereal_2Eabs_2E1(V2x_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__BOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ? [V1B_2E0: tyop_2Erealax_2Ereal] :
! [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)),c_2Erealax_2Ereal__mul_2E2(V1B_2E0,c_2Ereal_2Eabs_2E1(V2x_2E0)))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__SUM__MUL,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F1_2E0,V2c_2E0),V0f_2E0),V3v_2E0),V5i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V2c_2E0,V5i_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V3v_2E0,V5i_2E0))) )
=> ( ! [V2c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V3v_2E0),V4i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V2c_2E0,V4i_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V3v_2E0,V4i_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2c_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V3v_2E0))) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F1_2E0,V2c_2E0),V0f_2E0),V3v_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__SUM,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1g_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__SUB,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0)) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__ADD,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Erealax_2Ereal__add_2E2(V1x_2E0,V2y_2E0)) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2y_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__NEG,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Erealax_2Ereal__neg_2E1(V1x_2E0)) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__CMUL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Erealax_2Ereal__mul_2E2(V1c_2E0,V2x_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__0,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__MUL__COMPONENT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1v_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1v_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),V1v_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1v_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1v_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__COMPOSE__SUM,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3x_2E0: tyop_2Erealax_2Ereal,V4a_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V3x_2E0),V4a_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V4a_2E0),V3x_2E0) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0f_2E0),V3x_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
& ! [V2a_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V2a_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0,V2a_2E0))) ) )
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0f_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__NEGATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0x_2E0) = c_2Erealax_2Ereal__neg_2E1(V0x_2E0) )
=> p(c_2Ereal__topology_2Elinear_2E1(F0_2E0)) ) ).
tff(thm_2Ereal__topology_2ELINEAR__ZERO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
=> p(c_2Ereal__topology_2Elinear_2E1(F0_2E0)) ) ).
tff(thm_2Ereal__topology_2ELINEAR__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0x_2E0) = V0x_2E0 )
=> p(c_2Ereal__topology_2Elinear_2E1(F0_2E0)) ) ).
tff(thm_2Ereal__topology_2ELINEAR__COMPOSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0)) )
=> p(c_2Ereal__topology_2Elinear_2E1(c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__COMPOSE__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0)) )
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__COMPOSE__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0)) )
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__COMPOSE__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__COMPOSE__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__SCALING,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V1x_2E0) = c_2Erealax_2Ereal__mul_2E2(V0c_2E0,V1x_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elinear_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0))) ) ).
tff(thm_2Ereal__topology_2EREAL__CHOOSE__DIST,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ? [V2y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2y_2E0)) = V1e_2E0 ) ) ).
tff(thm_2Ereal__topology_2EDIST__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) = c_2Ereal_2Eabs_2E1(V0x_2E0) )
& ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0)) = c_2Ereal_2Eabs_2E1(V0x_2E0) ) ) ).
tff(thm_2Ereal__topology_2EDIST__EQ,axiom,
! [V0w_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3z_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0w_2E0,V1x_2E0)) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2y_2E0,V3z_2E0)) )
<=> ( c_2Ereal_2Epow_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0w_2E0,V1x_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) = c_2Ereal_2Epow_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2y_2E0,V3z_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EDIST__POS__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)))) ).
tff(thm_2Ereal__topology_2EDIST__LE__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__ADD__HALF,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V2x_27_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal,V4y_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2x_27_2E0)),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3y_2E0,V4y_27_2E0)),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V1x_2E0,V3y_2E0),c_2Erealax_2Ereal__add_2E2(V2x_27_2E0,V4y_27_2E0))),V0e_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__MUL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V1y_2E0))) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Eabs_2E1(V2c_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0))) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1x_27_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3y_27_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x_27_2E0,V3y_27_2E0))),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1x_27_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2y_2E0,V3y_27_2E0))))) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__R,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal,V1x1_2E0: tyop_2Erealax_2Ereal,V2x2_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3y_2E0,V1x1_2E0)),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3y_2E0,V2x2_2E0)),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x1_2E0,V2x2_2E0)),V0e_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__HALF__L,axiom,
! [V0e_2E0: tyop_2Erealax_2Ereal,V1x1_2E0: tyop_2Erealax_2Ereal,V2x2_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x1_2E0,V3y_2E0)),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x2_2E0,V3y_2E0)),c_2Ereal_2E_2F_2E2(V0e_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x1_2E0,V2x2_2E0)),V0e_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2z_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1y_2E0,V2z_2E0))),V3e_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),V3e_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal,V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2z_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1y_2E0,V2z_2E0))),V3e_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),V3e_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__NZ,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != V1y_2E0 )
<=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)))) ) ).
tff(thm_2Ereal__topology_2EDIST__POS__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != V1y_2E0 )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)))) ) ).
tff(thm_2Ereal__topology_2EDIST__EQ__0,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V0x_2E0 = V1y_2E0 ) ) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE__ALT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1y_2E0,V2z_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2z_2E0))))) ).
tff(thm_2Ereal__topology_2EDIST__TRIANGLE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V2z_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1y_2E0,V2z_2E0))))) ).
tff(thm_2Ereal__topology_2EDIST__SYM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Ereal__topology_2EDIST__REFL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V0x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Ereal__topology_2EREAL__WLOG__LT,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0),V1x_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V2x_2E0),V3y_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V3y_2E0),V2x_2E0) )
& ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(V4x_2E0,V5y_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V4x_2E0),V5y_2E0)) ) )
=> ! [V6x_2E0: tyop_2Erealax_2Ereal,V7y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V6x_2E0),V7y_2E0)) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__MONO,axiom,
! [A_27a: $tType,V0t_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0,V1s_2E0))
=> ? [V3y_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3y_2E0,V0t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2x_2E0,V3y_2E0)) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V1s_2E0),c_2Epred__set_2EBIGUNION_2E1(A_27a,V0t_2E0))) ) ).
tff(thm_2Ereal__topology_2EBIGUNION__MONO__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1g_2E0,V3x_2E0))) )
=> p(c_2Epred__set_2ESUBSET_2E2(A_27b,c_2Epred__set_2EBIGUNION_2E1(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0,V0s_2E0)),c_2Epred__set_2EBIGUNION_2E1(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1g_2E0,V0s_2E0)))) ) ).
tff(thm_2Ereal__topology_2EDEPENDENT__CHOICE,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ( ? [V2a_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2E0_2E0),V2a_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum,V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V3n_2E0),V4x_2E0))
=> ? [V5y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),V5y_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R_2E0,V3n_2E0),V4x_2E0),V5y_2E0)) ) ) )
=> ? [V6f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ! [V7n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V7n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,V7n_2E0)))
& ! [V8n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R_2E0,V8n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,V8n_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,c_2Enum_2ESUC_2E1(V8n_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EDEPENDENT__CHOICE__FIXED,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V2a_2E0: A_27a] :
( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2E0_2E0),V2a_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum,V4x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V3n_2E0),V4x_2E0))
=> ? [V5y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),V5y_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R_2E0,V3n_2E0),V4x_2E0),V5y_2E0)) ) ) )
=> ? [V6f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ( app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,c_2Enum_2E0_2E0) = V2a_2E0 )
& ! [V7n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0,V7n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,V7n_2E0)))
& ! [V8n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R_2E0,V8n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,V8n_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V6f_2E0,c_2Enum_2ESUC_2E1(V8n_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2EEXISTS__IN__INSERT,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1a_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ? [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1a_2E0,V2s_2E0)))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0)) )
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V1a_2E0))
| ? [V4x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EPERMUTES__INJECTIVE,axiom,
! [A_27a: $tType,V0p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Epermutes_2E2(A_27a,V0p_2E0,V1s_2E0))
=> ! [V2x_2E0: A_27a,V3y_2E0: A_27a] :
( ( app_2E2(A_27a,A_27a,V0p_2E0,V2x_2E0) = app_2E2(A_27a,A_27a,V0p_2E0,V3y_2E0) )
<=> ( V2x_2E0 = V3y_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2EPERMUTES__IMAGE,axiom,
! [A_27a: $tType,V0p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Epermutes_2E2(A_27a,V0p_2E0,V1s_2E0))
=> ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27a,V0p_2E0,V1s_2E0) = V1s_2E0 ) ) ).
tff(thm_2Ereal__topology_2EPAIRWISE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V1r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V3x_2E0: A_27b,V4y_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1r_2E0),V3x_2E0),V4y_2E0))
<=> ( ( app_2E2(A_27b,A_27a,V2f_2E0,V3x_2E0) != app_2E2(A_27b,A_27a,V2f_2E0,V4y_2E0) )
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1r_2E0,app_2E2(A_27b,A_27a,V2f_2E0,V3x_2E0)),app_2E2(A_27b,A_27a,V2f_2E0,V4y_2E0))) ) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] : ( c_2Ereal__topology_2Epairwise_2E2(A_27a,V1r_2E0,c_2Epred__set_2EIMAGE_2E2(A_27b,A_27a,V2f_2E0,V0s_2E0)) = c_2Ereal__topology_2Epairwise_2E2(A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1r_2E0),V0s_2E0) ) ) ).
tff(thm_2Ereal__topology_2EPAIRWISE__INSERT,axiom,
! [A_27a: $tType,V0r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1x_2E0,V2s_2E0)))
<=> ( ! [V3y_2E0: A_27a] :
( ( p(c_2Ebool_2EIN_2E2(A_27a,V3y_2E0,V2s_2E0))
& ( V3y_2E0 != V1x_2E0 ) )
=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0r_2E0,V1x_2E0),V3y_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0r_2E0,V3y_2E0),V1x_2E0)) ) )
& p(c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EPAIRWISE__MONO,axiom,
! [A_27a: $tType,V0r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2EPAIRWISE__SING,axiom,
! [A_27a: $tType,V0r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1x_2E0: A_27a] : ( c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a))) = c_2Ebool_2ET_2E0 ) ).
tff(thm_2Ereal__topology_2EPAIRWISE__EMPTY,axiom,
! [A_27a: $tType,V0r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Ereal__topology_2Epairwise_2E2(A_27a,V0r_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)) = c_2Ebool_2ET_2E0 ) ).
tff(thm_2Ereal__topology_2EREAL__WLOG__LE,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0),V2y_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V2y_2E0),V1x_2E0) )
& ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V4y_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V3x_2E0),V4y_2E0)) ) )
=> ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V5x_2E0),V6y_2E0)) ) ).
tff(thm_2Ereal__topology_2EIMAGE__SING,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1x_2E0: A_27a] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a))) = c_2Epred__set_2EINSERT_2E2(A_27b,app_2E2(A_27a,A_27b,V0f_2E0,V1x_2E0),c_2Epred__set_2EEMPTY_2E0(A_27b)) ) ).
tff(thm_2Ereal__topology_2ESUM__ABS__TRIANGLE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3a_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3a_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3a_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0)),V2b_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0)),V2b_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ESUMS__SYM,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V4y_2E0),V1t_2E0),V5x_2E0),V0s_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V1t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0)) ) )
=> ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4y_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1t_2E0),V0s_2E0),V4y_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V4y_2E0,V5x_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V4y_2E0),V1t_2E0),V5x_2E0),V0s_2E0)) )
=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal,V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V1t_2E0)) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0),V2x_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,c_2Erealax_2Ereal__add_2E2(V2x_2E0,V3y_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V2x_2E0),V0s_2E0),V3y_2E0),V1t_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V0s_2E0),V1t_2E0))) = c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1t_2E0),V0s_2E0))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ESUM__GP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0))
<=> ( V0x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) )
=> ( ! [V0x_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3i_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V3i_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V2n_2E0,V1m_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,F1_2E0,V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1m_2E0)),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V1m_2E0),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ESUM__GP__MULTIPLIED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V3i_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V3i_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0))) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V1m_2E0),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ESUM__GP__BASIC,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V2i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2i_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V2i_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V1n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0))) = c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,c_2Enum_2ESUC_2E1(V1n_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EREAL__INV__1__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V0x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Erealax_2Einv_2E1(V0x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE__EQ,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V8x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5w_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5w_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V5w_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5w_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3k_2E0))) )
<=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5w_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V5w_2E0)),V4u_2E0)) )
=> ? [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5w_2E0,V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0)),V4u_2E0)) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__OPEN__SUPERSET__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V8x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V4w_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V4w_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4w_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V4w_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4w_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4w_2E0)),V3u_2E0)) )
=> ? [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4w_2E0,V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0)),V3u_2E0)) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE__EQ,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V8x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5w_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5w_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V5w_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5w_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3k_2E0))) )
<=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5w_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V5w_2E0)),V4u_2E0)) )
=> ? [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5w_2E0,V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0)),V4u_2E0)) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__CLOSED__SUPERSET__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V8x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V7v_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V4w_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V4w_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4w_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V4w_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4w_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4w_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4w_2E0)),V3u_2E0)) )
=> ? [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4w_2E0,V7v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V7v_2E0)),V3u_2E0)) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6y_2E0,V3t_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6y_2E0) ) )
=> ( V5x_2E0 = V6y_2E0 ) )
& ! [V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V7k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V7k_2E0))) ) )
=> ! [V8k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V8k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8k_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__INJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal,V6y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6y_2E0,V3t_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6y_2E0) ) )
=> ( V5x_2E0 = V6y_2E0 ) )
& ! [V7k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V7k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V7k_2E0))) ) )
=> ! [V8k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V8k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8k_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V5k_2E0))) ) )
=> ! [V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V6k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6k_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__FROM__COMPOSITION__SURJECTIVE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V5k_2E0))) ) )
=> ! [V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V6k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V6k_2E0))) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__IMP__CLOSED__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4u_2E0))) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3u_2E0))) )
& ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0)))) ) )
=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6u_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__IMP__OPEN__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4u_2E0))) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3u_2E0))) )
& ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0)))) ) )
=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6u_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__CLOSED__MAP__EQ,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6k_2E0))) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6k_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6k_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) ) )
=> ( ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
<=> ! [V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6k_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6k_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__MAP__EQ,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6k_2E0))) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6k_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6k_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) ) )
=> ( ! [V5k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
<=> ! [V6k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V6k_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6k_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__RESTRICT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V3t_27_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V3t_27_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_27_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V3t_27_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4u_2E0))) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_27_2E0,V2t_2E0)) )
=> ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_27_2E0))),V5u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_27_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__RESTRICT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V3t_27_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V3t_27_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_27_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V3t_27_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_27_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V4u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4u_2E0))) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_27_2E0,V2t_2E0)) )
=> ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_27_2E0))),V5u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_27_2E0),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__INTERIORS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) )
<=> ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Einterior_2E1(V2s_2E0)),c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__CLOSURES,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) )
<=> ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eclosure_2E1(V2s_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE__GEN,axiom,
! [F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0) = V6y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6y_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6y_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V6y_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V3c_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8x_2E0),V3c_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1s_2E0),V0f_2E0),V3c_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V1s_2E0),V0f_2E0),V3c_2E0)) )
=> ( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V4u_2E0) ) )
& ! [V6y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6y_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V6y_2E0)))) )
& ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3c_2E0))
| p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3c_2E0)) )
& p(c_2Ereal__topology_2Econnected_2E1(V3c_2E0)) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V1s_2E0),V0f_2E0),V3c_2E0)))) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__MONOTONE__QUOTIENT__PREIMAGE,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0) = V5y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5y_2E0: tyop_2Erealax_2Ereal,V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V5y_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5y_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V2t_2E0 )
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) )
& ! [V5y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5y_2E0,V2t_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V5y_2E0)))) )
& p(c_2Ereal__topology_2Econnected_2E1(V2t_2E0)) )
=> p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__RESTRICT,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V3c_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0),V3c_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V3c_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V3c_2E0)) )
=> ( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V4u_2E0) ) )
& ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3c_2E0))
| p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3c_2E0)) ) )
=> ! [V6u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6u_2E0,V3c_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V3c_2E0))),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V3c_2E0))),V0f_2E0),V6u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3c_2E0),V6u_2E0) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__SUBSET,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V2t_2E0),V0f_2E0),V6v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0),V6v_2E0)) ) )
=> ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2t_2E0),V0f_2E0),V6v_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V7x_2E0),V2t_2E0),V0f_2E0),V6v_2E0)) )
=> ( ! [V5x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V4v_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4v_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V1s_2E0),V0f_2E0),V4v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0),V3u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2t_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0) = V3u_2E0 )
& ! [V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4v_2E0,V3u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V4v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3u_2E0),V4v_2E0) ) ) )
=> ! [V6v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V6v_2E0,V3u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V2t_2E0),V0f_2E0),V6v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3u_2E0),V6v_2E0) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__FROM__COMPOSITION,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0,V6x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V5v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0,V6x_2E0),V5v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V5v_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0,V6x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V5v_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5v_2E0,V4u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F0_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V5v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),V5v_2E0) ) ) )
=> ! [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V7v_2E0,V4u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),V7v_2E0) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__COMPOSE,axiom,
! [F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V10x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F5_2E0,V10x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V9v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E3(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0,V10x_2E0),V9v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V10x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V9v_2E0),V10x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V10x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F5_2E0,V10x_2E0),V2s_2E0),V1g_2E0),V0f_2E0),V9v_2E0)) )
=> ( ! [V8x_2E0: tyop_2Erealax_2Ereal,V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V8x_2E0),V7v_2E0)) ) )
=> ( ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V8x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7v_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V3t_2E0),V1g_2E0),V7v_2E0)) )
=> ( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& ! [V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5v_2E0,V3t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V5v_2E0) ) )
& ! [V7v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V7v_2E0,V4u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V3t_2E0),V1g_2E0),V7v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),V7v_2E0) ) ) )
=> ! [V9v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V9v_2E0,V4u_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0,V2s_2E0),V1g_2E0),V0f_2E0),V9v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V4u_2E0),V9v_2E0) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIFT__TO__QUOTIENT__SPACE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0) = V3t_2E0 )
& ! [V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5v_2E0,V3t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V5v_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1h_2E0,V2s_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,V2s_2E0) = V4u_2E0 )
& ! [V7x_2E0: tyop_2Erealax_2Ereal,V8y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8y_2E0,V2s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V8y_2E0) ) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,V7x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,V8y_2E0) ) ) )
=> ? [V9g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V9g_2E0,V3t_2E0))
& ( c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9g_2E0,V3t_2E0) = V4u_2E0 )
& ! [V10x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V10x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1h_2E0,V10x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V9g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V10x_2E0)) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EFUNCTION__FACTORS__LEFT__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2g_2E0: tyop_2Emin_2Efun(A_27a,A_27c)] :
( ! [V3x_2E0: A_27a,V4y_2E0: A_27a] :
( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4y_2E0))
& ( app_2E2(A_27a,A_27c,V2g_2E0,V3x_2E0) = app_2E2(A_27a,A_27c,V2g_2E0,V4y_2E0) ) )
=> ( app_2E2(A_27a,A_27b,V1f_2E0,V3x_2E0) = app_2E2(A_27a,A_27b,V1f_2E0,V4y_2E0) ) )
<=> ? [V5h_2E0: tyop_2Emin_2Efun(A_27c,A_27b)] :
! [V6x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0))
=> ( app_2E2(A_27a,A_27b,V1f_2E0,V6x_2E0) = app_2E2(A_27c,A_27b,V5h_2E0,app_2E2(A_27a,A_27c,V2g_2E0,V6x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE__QUOTIENT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5v_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5v_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V4u_2E0))
& ! [V5v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5v_2E0,V3t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5v_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V5v_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__OPEN__CLOSED,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V5u_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5u_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) )
<=> ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V5u_2E0))) = c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V5u_2E0) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__LEFT__INVERSE__IMP__QUOTIENT__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V5x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),V4u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)))
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) = V3x_2E0 ) ) )
=> ! [V4u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4u_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V4u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)),V4u_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__RIGHT__INVERSE__IMP__QUOTIENT__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5u_2E0)) ) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5u_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V6x_2E0),V2s_2E0),V0f_2E0),V5u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0),V3t_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3t_2E0),V2s_2E0))
& ! [V4y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4y_2E0,V3t_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4y_2E0)) = V4y_2E0 ) ) )
=> ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5u_2E0,V3t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V0f_2E0),V5u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V5u_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__MAP__IMP__QUOTIENT__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0))) ) )
=> ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),V3t_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__MAP__IMP__QUOTIENT__MAP,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2t_2E0))) ) )
=> ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3t_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),V3t_2E0) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__CLOSED,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EQUOTIENT__MAP__IMP__CONTINUOUS__OPEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))) = c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0) ) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EQUASICOMPACT__OPEN__CLOSED,axiom,
! [F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V6x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V6x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),V5u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V5u_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V6x_2E0),V1s_2E0),V0f_2E0),V5u_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0))))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0)) ) )
<=> ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V5u_2E0,V2t_2E0))
=> ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V5u_2E0))))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V5u_2E0)) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0)) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__TRANSLATION__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) = c_2Ereal__topology_2Econnected_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONNECTED__CONTINUOUS__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Econnected_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Econnected_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__LINEAR__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__TRANSLATION__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) = c_2Ereal__topology_2Ecompact_2E1(V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__TRANSLATION,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V1a_2E0,V2x_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1a_2E0),V0s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CONTINUOUS__IMAGE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__DOT2,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__DOT2,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V2g_2E0,V0net_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V2g_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__DOT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1y_2E0) = c_2Erealax_2Ereal__mul_2E2(V0x_2E0,V1y_2E0) )
=> p(c_2Ereal__topology_2Ebilinear_2E1(F0_2E0)) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__ON__COMPOSE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))] :
( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V3s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__CONTINUOUS__COMPOSE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))))] :
( ! [V3h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V3h_2E0),V1f_2E0),V2g_2E0),V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3h_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V2g_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Ebilinear_2E1(V3h_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V3h_2E0),V1f_2E0),V2g_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPONENT__COMPOSE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__COMPONENT__COMPOSE,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V3x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2i_2E0: A_27b] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1f_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON__COMPOSE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1g_2E0),V0f_2E0),V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V1g_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1g_2E0),V0f_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__COMPOSE,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2g_2E0),V1f_2E0),V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2g_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1f_2E0,V0net_2E0))
& p(c_2Ereal__topology_2Elinear_2E1(V2g_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2g_2E0),V1f_2E0),V0net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__ON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__WITHIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__CONTINUOUS__AT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1a_2E0))) ) ).
tff(thm_2Ereal__topology_2ELINEAR__LIM__0,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Eat_2E1(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__EQ,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__EQ,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE__UNIV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1s_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V1s_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0)))
& p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1s_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE__UNIV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1s_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V1s_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0)))
& p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0f_2E0),V1s_2E0)))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__PREIMAGE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0)) )
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__OPEN__PREIMAGE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__CLOSED__IN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),V2t_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CLOSED__IN__PREIMAGE__GEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),V2t_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CLOSED__GEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0))
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0)))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__IMP__OPEN__IN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),V2t_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__OPEN__IN__PREIMAGE__GEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2t_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2t_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)),V2t_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2t_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__OPEN__GEN,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V3u_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3u_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
=> ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V3u_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3u_2E0)))) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__OPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0)))
<=> ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),V2t_2E0)) )
=> ? [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V3s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V3s_2E0))
& ! [V4x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_27_2E0,V3s_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_27_2E0),V2t_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__OPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V2u_2E0)))
<=> ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V3t_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),V3t_2E0)) )
=> ? [V4s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V4s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V4s_2E0))
& ! [V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V4s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V2u_2E0)) )
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_27_2E0),V3t_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__COMPOSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__COMPOSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0)))
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Eat_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),c_2Ereal__topology_2Eat_2E1(V2x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPOSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal,V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V3s_2E0)))
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3s_2E0)))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V3s_2E0))) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(F0_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Econtinuous__on_2E2(F0_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2Eat_2E1(V0a_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__ID,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V2x_2E0) = V2x_2E0 )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,F0_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0a_2E0),V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUM,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal,V5a_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V4x_2E0),V5a_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V5a_2E0),V4x_2E0) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V1f_2E0),V4x_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V4x_2E0)) )
=> ! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
& ! [V3a_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V3a_2E0),V0t_2E0)) ) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V1f_2E0),V0t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__VMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2v_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1c_2E0,V3x_2E0),V2v_2E0) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2v_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1c_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2v_2E0),V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__MUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1g_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0)))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V2s_2E0))) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBILINEAR__UNIFORMLY__CONTINUOUS__ON__COMPOSE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))))] :
( ! [V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1g_2E0,V3s_2E0))
& p(c_2Ereal__topology_2Ebilinear_2E1(V2h_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3s_2E0)))
& p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3s_2E0))) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F0_2E0,V2h_2E0),V0f_2E0),V1g_2E0),V3s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__COMPOSE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1g_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2s_2E0))) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V0f_2E0),V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ELINEAR__UNIFORMLY__CONTINUOUS__ON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elinear_2E1(V0f_2E0))
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = V1c_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__SUM,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal,V5a_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V4x_2E0),V5a_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V5a_2E0),V4x_2E0) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V1f_2E0),V4x_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V4x_2E0)) )
=> ! [V0t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
& ! [V3a_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1f_2E0,V3a_2E0),V0t_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V1f_2E0),V0t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__MIN,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Emin_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__MAX,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Emax_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__ABS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V2s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = V1c_2E0 )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1c_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4x_2E0: A_27a,V5a_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V4x_2E0),V5a_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1f_2E0,V5a_2E0),V4x_2E0) )
=> ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V1f_2E0),V4x_2E0) = c_2Eiterate_2ESum_2E2(A_27b,V2s_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F1_2E0,V1f_2E0),V4x_2E0)) )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
& ! [V3a_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,V2s_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1f_2E0,V3a_2E0),V0net_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2s_2E0),V1f_2E0),V0net_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__MIN,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Emin_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V2net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1g_2E0,V2net_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__MAX,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Emax_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V2net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1g_2E0,V2net_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ABS,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V1net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__SUB,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V2net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1g_2E0,V2net_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ADD,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V2net_2E0))
& p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V1g_2E0,V2net_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__NEG,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V1net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CMUL,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V2net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V2net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__CONST,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ! [V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V2x_2E0) = V1c_2E0 )
=> ! [V0net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V1c_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V1c_2E0),V0net_2E0)) ) ).
tff(thm_2Ereal__topology_2ELIM__CONTINUOUS__FUNCTION,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2g_2E0),V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V2g_2E0,V4x_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a),V2g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V3l_2E0)))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,V2g_2E0,V3l_2E0,V1net_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V2g_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3l_2E0),V1net_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SEQUENTIALLY,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0f_2E0),V2x_2E0),V3y_2E0),V7n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V7n_2E0)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3y_2E0,V7n_2E0))) )
=> ( ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2x_2E0),V3y_2E0),V6n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V6n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3y_2E0,V6n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V4n_2E0),V1s_2E0))
& ! [V5n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3y_2E0,V5n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2x_2E0),V3y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0f_2E0),V2x_2E0),V3y_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__SEQUENTIALLY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3a_2E0,V1s_2E0))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2x_2E0,V4n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2x_2E0,V3a_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3a_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__SEQUENTIALLY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1a_2E0)))
<=> ! [V2x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V2x_2E0,V1a_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1a_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SEQUENTIALLY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2a_2E0),V1s_2E0)))
<=> ! [V3x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V3x_2E0,V4n_2E0),V1s_2E0))
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V3x_2E0,V2a_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2a_2E0),c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EFORALL__POS__MONO__1,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V1d_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V1d_2E0,V2e_2E0))
& p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V1d_2E0)) )
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V2e_2E0)) )
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Einv_2E1(c_2Erealax_2Ereal__add_2E2(c_2Ereal_2Ereal__of__num_2E1(V3n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))))) )
=> ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4e_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EISOMETRY__ON__IMP__CONTINUOUS__ON,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0)) )
=> ( c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3y_2E0))) = c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)) ) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTRACTION__IMP__CONTINUOUS__ON,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V3y_2E0))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V3y_2E0)))) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1f_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__FINITE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__NO__LIMPT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ~ ? [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__EMPTY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__SING,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) )
& p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V1g_2E0,V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V1g_2E0,V2s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__INTERIOR,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,c_2Ereal__topology_2Einterior_2E1(V1s_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0))) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__ON__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V2t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__SUBSET,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V1s_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2t_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V2t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__AT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1s_2E0))
=> ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__ON__EQ__CONTINUOUS__WITHIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V1s_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__IMP__CONTINUOUS__ON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONTINUOUS__IMP__CONTINUOUS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Euniformly__continuous__on_2E2(V0f_2E0,V1s_2E0))
=> p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__COMPARISON,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3a_2E0),V2s_2E0)))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0))),c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V3a_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0))))) ) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3a_2E0),V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__BALL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0)))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V3d_2E0))),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),V2e_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__BALL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V1s_2E0)))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2x_2E0,V4d_2E0)),V1s_2E0)),c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),V3e_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2Econtinuous__at,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1f_2E0,c_2Ereal__topology_2Eat_2E1(V0x_2E0)))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_27_2E0,V0x_2E0)),V3d_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V4x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V0x_2E0))),V2e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2Econtinuous__within,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V2f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V0x_2E0),V1s_2E0)))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& ! [V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V1s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_27_2E0,V0x_2E0)),V4d_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V5x_27_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2f_2E0,V0x_2E0))),V3e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__SET__IMP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4x_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V3t_2E0),V2s_2E0),V4x_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V3t_2E0))
=> p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0)) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Eeventually_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V3t_2E0),V2s_2E0),c_2Ereal__topology_2Eat_2E1(V1a_2E0)))
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1a_2E0),V2s_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1a_2E0),V3t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN__IN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V3t_2E0),V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4a_2E0,V2s_2E0))
& ! [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4a_2E0),V3t_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V4a_2E0),V3t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN__OPEN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2EOpen_2E1(V2s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3a_2E0,V2s_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V2s_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Eat_2E1(V3a_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__AT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal,V3d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3d_2E0))
& ! [V4x_27_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_27_2E0,V2x_2E0)),V3d_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_27_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V4x_27_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Eat_2E1(V2x_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__TRANSFORM__WITHIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal,V4d_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4d_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V2s_2E0))
& ! [V5x_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_27_2E0,V2s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5x_27_2E0,V3x_2E0)),V4d_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5x_27_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1g_2E0,V5x_27_2E0) ) )
& p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V2s_2E0))) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1g_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V3x_2E0),V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN__CLOSED__NONTRIVIAL,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V2s_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1a_2E0,V2s_2E0)) )
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1a_2E0),V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT__WITHIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0)))
=> p(c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V1x_2E0),V2s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__AT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V0f_2E0,c_2Ereal__topology_2Eat_2E1(V1x_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V0f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0),c_2Ereal__topology_2Eat_2E1(V1x_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__WITHIN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Econtinuous_2E2(tyop_2Erealax_2Ereal,V1f_2E0,c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V0s_2E0)) = c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Erealax_2Ereal,V1f_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1f_2E0,V2x_2E0),c_2Ereal__topology_2Ewithin_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eat_2E1(V2x_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECONTINUOUS__TRIVIAL__LIMIT,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1net_2E0: tyop_2Ereal__topology_2Enet(A_27a)] :
( p(c_2Ereal__topology_2Etrivial__limit_2E1(A_27a,V1net_2E0))
=> p(c_2Ereal__topology_2Econtinuous_2E2(A_27a,V0f_2E0,V1net_2E0)) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CAUCHY__IMP__UNIFORMLY__CONVERGENT,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V2l_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
( ( ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4N_2E0: tyop_2Enum_2Enum] :
! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum,V7x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V4N_2E0,V5m_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V4N_2E0,V6n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V7x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V5m_2E0),V7x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V6n_2E0),V7x_2E0))),V3e_2E0)) ) )
& ! [V8x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V8x_2E0))
=> ! [V9e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V9e_2E0))
=> ? [V10N_2E0: tyop_2Enum_2Enum] :
! [V11n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V10N_2E0,V11n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V11n_2E0),V8x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2l_2E0,V8x_2E0))),V9e_2E0)) ) ) ) )
=> ! [V12e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V12e_2E0))
=> ? [V13N_2E0: tyop_2Enum_2Enum] :
! [V14n_2E0: tyop_2Enum_2Enum,V15x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V13N_2E0,V14n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V15x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V14n_2E0),V15x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2l_2E0,V15x_2E0))),V12e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY__ALT,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ? [V2l_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4N_2E0: tyop_2Enum_2Enum] :
! [V5n_2E0: tyop_2Enum_2Enum,V6x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V4N_2E0,V5n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V5n_2E0),V6x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2l_2E0,V6x_2E0))),V3e_2E0)) ) )
<=> ! [V7e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7e_2E0))
=> ? [V8N_2E0: tyop_2Enum_2Enum] :
! [V9m_2E0: tyop_2Enum_2Enum,V10n_2E0: tyop_2Enum_2Enum,V11x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V8N_2E0,V9m_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V8N_2E0,V10n_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V9m_2E0,V10n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V11x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V9m_2E0),V11x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V10n_2E0),V11x_2E0))),V7e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EUNIFORMLY__CONVERGENT__EQ__CAUCHY,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
( ? [V2l_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4N_2E0: tyop_2Enum_2Enum] :
! [V5n_2E0: tyop_2Enum_2Enum,V6x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V4N_2E0,V5n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V6x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V5n_2E0),V6x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2l_2E0,V6x_2E0))),V3e_2E0)) ) )
<=> ! [V7e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7e_2E0))
=> ? [V8N_2E0: tyop_2Enum_2Enum] :
! [V9m_2E0: tyop_2Enum_2Enum,V10n_2E0: tyop_2Enum_2Enum,V11x_2E0: A_27a] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V8N_2E0,V9m_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V8N_2E0,V10n_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V11x_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V9m_2E0),V11x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0,V10n_2E0),V11x_2E0))),V7e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__NEST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V4n_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V4n_2E0),c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4n_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal__topology_2Ecompact_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0)))
& ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
& ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2m_2E0))) ) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CHAIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> ( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V0f_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0,V0f_2E0)) )
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2s_2E0,V3t_2E0))
| p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3t_2E0,V2s_2E0)) ) ) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CLOSED__CHAIN,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ! [V2s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2s_2E0,V0f_2E0))
=> ( p(c_2Ereal__topology_2EClosed_2E1(V2s_2E0))
& ( V2s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
& ! [V3s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3s_2E0,V0f_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4t_2E0,V0f_2E0)) )
=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3s_2E0,V4t_2E0))
| p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V3s_2E0)) ) )
& p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V1b_2E0)) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EDECREASING__CLOSED__NEST__SING,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))] :
( ! [V10t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),F1_2E0,V10t_2E0),V0s_2E0))
<=> ? [V11n_2E0: tyop_2Enum_2Enum] : ( V10t_2E0 = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V11n_2E0) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),V10t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V10t_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V10t_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),F1_2E0,V10t_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2EClosed_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0)))
& ! [V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2n_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3m_2E0))) )
& ! [V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> ? [V6n_2E0: tyop_2Enum_2Enum] :
! [V7x_2E0: tyop_2Erealax_2Ereal,V8y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V6n_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8y_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V6n_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7x_2E0,V8y_2E0)),V5e_2E0)) ) ) )
=> ? [V9a_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0))) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V9a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ) ) ).
tff(thm_2Ereal__topology_2EDECREASING__CLOSED__NEST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2EClosed_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0)))
& ! [V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2n_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3m_2E0))) )
& ! [V5e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5e_2E0))
=> ? [V6n_2E0: tyop_2Enum_2Enum] :
! [V7x_2E0: tyop_2Erealax_2Ereal,V8y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V6n_2E0)))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8y_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V6n_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7x_2E0,V8y_2E0)),V5e_2E0)) ) ) )
=> ? [V9a_2E0: tyop_2Erealax_2Ereal] :
! [V10n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V10n_2E0))) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CLOSED__NEST,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal__topology_2EClosed_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V1n_2E0)))
& ! [V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V2n_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V3m_2E0))) )
& p(c_2Ereal__topology_2Ebounded__def_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,c_2Enum_2E0_2E0))) )
=> ? [V5a_2E0: tyop_2Erealax_2Ereal] :
! [V6n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5a_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0,V6n_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__FIP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
& ! [V2f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2f_27_2E0,V0f_2E0)) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V2f_27_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__FIP,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
& ? [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V0f_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V2t_2E0)) )
& ! [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V0f_2E0)) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IMP__FIP__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(V2t_2E0)) )
& ! [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V1f_2E0)) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V1f_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IMP__FIP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0)) )
& ? [V3t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3t_2E0,V1f_2E0))
& p(c_2Ereal__topology_2Ebounded__def_2E1(V3t_2E0)) )
& ! [V4f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4f_27_2E0,V1f_2E0)) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V4f_27_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V1f_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__IMP__FIP,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V2t_2E0)) )
& ! [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V1f_2E0)) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V1f_2E0)) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Efrontier_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)))) ).
tff(thm_2Ereal__topology_2EBOUNDED__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)))) ).
tff(thm_2Ereal__topology_2ECOMPACT__SPHERE,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1r_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Esphere_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0a_2E0,V1r_2E0)))) ).
tff(thm_2Ereal__topology_2ECOMPACT__DIFF,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EDIFF_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__BIGUNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0s_2E0))
& ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) ) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__BIGINTER__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
<=> ! [V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0)),V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__DELETE,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),V1s_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V1s_2E0,V2a_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__DELETE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1x_2E0))) ) ).
tff(thm_2Ereal__topology_2EFRONTIER__SUBSET__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Efrontier_2E1(V0s_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__FRONTIER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__FRONTIER__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Efrontier_2E1(V0s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CBALL,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1e_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1e_2E0)))) ).
tff(thm_2Ereal__topology_2ECLOSED__INSERT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSURE__SING,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__SING,axiom,
! [V0u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1x_2E0: tyop_2Erealax_2Ereal] : ( c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0u_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0u_2E0) ) ).
tff(thm_2Ereal__topology_2ECLOSED__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2ECOMPACT__INSERT,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__SING,axiom,
! [V0a_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)))) ).
tff(thm_2Ereal__topology_2EFINITE__IMP__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EFINITE__IMP__CLOSED__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) )
=> p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EFINITE__IMP__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__BIGINTER,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0f_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(V1s_2E0)) )
& ( V0f_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0f_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__INTER__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__INTER__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__EMPTY,axiom,
p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2EBOUNDED__EQ__BOLZANO__WEIERSTRASS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
<=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0)) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0)) )
=> ? [V1x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V1x_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EDISCRETE__BOUNDED__IMP__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1e_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3y_2E0,V2x_2E0)),V1e_2E0)) )
=> ( V3y_2E0 = V2x_2E0 ) )
& p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0)) )
=> p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__CONTRAPOS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> ~ p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) ) )
=> p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) = c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL__SUBTOPOLOGY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2t_2E0)) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1f_2E0))) )
=> ? [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V1f_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__COMPACT__EQ,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
<=> ( p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IN__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__SEQUENCE__WITH__LIMIT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V1l_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))))) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__IMP__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__IMP__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__EQ__BOUNDED__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
<=> ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__EQ__BOLZANO__WEIERSTRASS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
<=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__EQ__HEINE__BOREL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
<=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1f_2E0))) )
=> ? [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V1f_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) ) )
=> p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ESEQUENCE__UNIQUE__LIMPT,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V3y_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),F1_2E0,V3y_2E0),V0f_2E0))
<=> ? [V4n_2E0: tyop_2Enum_2Enum] : ( V3y_2E0 = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0) ) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3y_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),F1_2E0,V3y_2E0),V0f_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal,V2l_27_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
& p(c_2Ereal__topology_2Elimit__point__of_2E2(V2l_27_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0)))) )
=> ( V2l_27_2E0 = V1l_2E0 ) ) ) ) ).
tff(thm_2Ereal__topology_2ELIMPT__OF__SEQUENCE__SUBSEQUENCE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elimit__point__of_2E2(V1l_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))))
=> ? [V2r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2r_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2r_2E0,V4n_2E0))) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0f_2E0,V2r_2E0),V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELE__1,axiom,
( ! [V0n_2E0: tyop_2Enum_2Enum] :
( ( V0n_2E0 != c_2Enum_2E0_2E0 )
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V0n_2E0)) )
& ! [V1n_2E0: tyop_2Enum_2Enum] :
( ( V1n_2E0 != c_2Enum_2E0_2E0 )
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0)) )
& ! [V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
=> ( V2n_2E0 != c_2Enum_2E0_2E0 ) )
& ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V3n_2E0))
=> p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V3n_2E0)) )
& ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V4n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V4n_2E0)) )
& ! [V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V5n_2E0))
=> ( V5n_2E0 != c_2Enum_2E0_2E0 ) ) ) ).
tff(thm_2Ereal__topology_2ESEQUENCE__INFINITE__LEMMA,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V3y_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),F1_2E0,V3y_2E0),V0f_2E0))
<=> ? [V4n_2E0: tyop_2Enum_2Enum] : ( V3y_2E0 = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0) ) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V3y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3y_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),F1_2E0,V3y_2E0),V0f_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0) != V1l_2E0 )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0f_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__INF__LT__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1a_2E0))
<=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V1a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LT__INF__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0)))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,V2x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__INF__LE__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1a_2E0))
<=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1a_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EREAL__LE__INF__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0)))
<=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINF__FINITE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Eiterate_2Einf_2E1(V0s_2E0),V0s_2E0))
& ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EINF__FINITE__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
& ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
=> ? [V1b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0s_2E0))
& ! [V2x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2x_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBOLZANO__WEIERSTRASS__IMP__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0)) )
=> ? [V2x_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2Elimit__point__of_2E2(V2x_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EHEINE__BOREL__IMP__BOLZANO__WEIERSTRASS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1f_2E0))) )
=> ? [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V1f_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0))) ) )
=> ! [V4t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V4t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4t_2E0,V0s_2E0)) )
=> ? [V5x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Elimit__point__of_2E2(V5x_2E0,V4t_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__IMP__HEINE__BOREL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V1f_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0)) )
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1f_2E0))) )
=> ? [V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0,V1f_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3f_27_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V3f_27_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EHEINE__BOREL__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> ! [V1t_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V1t_2E0)))
& ! [V2b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2b_2E0,V1t_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2b_2E0)) ) )
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
=> ? [V5b_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5b_2E0,V1t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4x_2E0,V3e_2E0)),V5b_2E0)) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__IMP__TOTALLY__BOUNDED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1e_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1e_2E0),V3x_2E0) = c_2Ereal__topology_2Eball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V3x_2E0,V1e_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> ! [V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ? [V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V2k_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2k_2E0,V0s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V1e_2E0),V2k_2E0)))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONVERGENT__IMP__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum)))) ) ).
tff(thm_2Ereal__topology_2ECONVERGENT__EQ__CAUCHY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ? [V1l_2E0: tyop_2Erealax_2Ereal] : p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
<=> p(c_2Ereal__topology_2Ecauchy_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPLETE__EQ__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Ecomplete_2E1(V0s_2E0) = c_2Ereal__topology_2EClosed_2E1(V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ECOMPLETE__UNIV,axiom,
p(c_2Ereal__topology_2Ecomplete_2E1(c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ECOMPACT__IMP__COMPLETE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ecomplete_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECAUCHY__IMP__BOUNDED,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal,V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),F1_2E0,V1y_2E0),V0s_2E0))
<=> ? [V2n_2E0: tyop_2Enum_2Enum] : ( V1y_2E0 = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2n_2E0) ) )
=> ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0),V1y_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1y_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),F1_2E0,V1y_2E0),V0s_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Ecauchy_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Ebounded__def_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0s_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
=> ? [V2a_2E0: tyop_2Erealax_2Ereal] :
! [V3x_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2a_2E0)) ) ) ).
tff(thm_2Ereal__topology_2EGREATER__EQ__REFL,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3E_3D_2E2(V0m_2E0,V0m_2E0)) ).
tff(thm_2Ereal__topology_2ECONVERGENT__IMP__CAUCHY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V1l_2E0,c_2Ereal__topology_2Esequentially_2E0))
=> p(c_2Ereal__topology_2Ecauchy_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECAUCHY,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Ereal__topology_2Ecauchy_2E1(V0s_2E0))
<=> ! [V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V2N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal__topology_2EDist_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2N_2E0))),V1e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EBOUNDED__CLOSED__IMP__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V0s_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__LEMMA,axiom,
! [A_27a: $tType,V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ebounded__def_2E1(V1s_2E0))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V2n_2E0),V1s_2E0)) )
=> ? [V3l_2E0: tyop_2Erealax_2Ereal,V4r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V5m_2E0,V6n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V5m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V6n_2E0))) )
& ! [V7e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7e_2E0))
=> ? [V8N_2E0: tyop_2Enum_2Enum] :
! [V9n_2E0: tyop_2Enum_2Enum,V10i_2E0: A_27a] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V8N_2E0,V9n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V9n_2E0)),V3l_2E0)),V7e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECOMPACT__REAL__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Erealax_2Ereal] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2n_2E0)),V1b_2E0))
=> ? [V3l_2E0: tyop_2Erealax_2Ereal,V4r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V5m_2E0,V6n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V5m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V6n_2E0))) )
& ! [V7e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V7e_2E0))
=> ? [V8N_2E0: tyop_2Enum_2Enum] :
! [V9n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V8N_2E0,V9n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V4r_2E0,V9n_2E0)),V3l_2E0)),V7e_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ECONVERGENT__BOUNDED__MONOTONE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2n_2E0)),V1b_2E0))
& ( ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V4n_2E0))) )
| ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V5m_2E0,V6n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V6n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V5m_2E0))) ) ) )
=> ? [V7l_2E0: tyop_2Erealax_2Ereal] :
! [V8e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V8e_2E0))
=> ? [V9N_2E0: tyop_2Enum_2Enum] :
! [V10n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V9N_2E0,V10n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V10n_2E0),V7l_2E0)),V8e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ECONVERGENT__BOUNDED__INCREASING,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V3n_2E0))) )
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V4n_2E0)),V1b_2E0)) )
=> ? [V5l_2E0: tyop_2Erealax_2Ereal] :
! [V6e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V6e_2E0))
=> ? [V7N_2E0: tyop_2Enum_2Enum] :
! [V8n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V7N_2E0,V8n_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V8n_2E0),V5l_2E0)),V6e_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EMONOTONE__SUBSEQUENCE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
? [V1r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V2m_2E0,V3n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V2m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V3n_2E0))) )
& ( ! [V4m_2E0: tyop_2Enum_2Enum,V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V4m_2E0,V5n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V4m_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V5n_2E0)))) )
| ! [V6m_2E0: tyop_2Enum_2Enum,V7n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V6m_2E0,V7n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V7n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V6m_2E0)))) ) ) ) ).
tff(thm_2Ereal__topology_2ELIM__SUBSEQUENCE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2l_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1r_2E0,V4n_2E0))) )
& p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,V0s_2E0,V2l_2E0,c_2Ereal__topology_2Esequentially_2E0)) )
=> p(c_2Ereal__topology_2E_2D_2D_3E_2E3(tyop_2Enum_2Enum,c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0s_2E0,V1r_2E0),V2l_2E0,c_2Ereal__topology_2Esequentially_2E0)) ) ).
tff(thm_2Ereal__topology_2EMONOTONE__BIGGER,axiom,
! [V0r_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0r_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0r_2E0,V2n_2E0))) )
=> ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V3n_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0r_2E0,V3n_2E0))) ) ).
tff(thm_2Ereal__topology_2ECLOSED__AS__GDELTA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Egdelta_2E1(V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3v_2E0,V0s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V3v_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__ALT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V2u_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Eclosure_2E1(V2u_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V2u_2E0),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER__CBALL,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)),V0s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER__CBALLS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
=> ? [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
& ! [V3d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V3d_2E0,V2e_2E0))
=> p(c_2Ereal__topology_2EClosed_2E1(c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Ecball_2E1(c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1x_2E0,V3d_2E0)),V0s_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1k_2E0)) )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V2u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2u_2E0,V3v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3v_2E0,V0s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V3v_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT__ALT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1k_2E0)) )
=> ? [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V2u_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V2u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(c_2Ereal__topology_2Eclosure_2E1(V2u_2E0)))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eclosure_2E1(V2u_2E0),V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__COMPACT__SUBOPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ! [V1k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V1k_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V2t_2E0)) )
=> ? [V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1k_2E0,V3u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V4v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V4v_2E0,V0s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V3u_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V4v_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2EOPEN__IMP__LOCALLY__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ECLOSED__IMP__LOCALLY__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2EIS__INTERVAL__IMP__LOCALLY__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Eis__interval_2E1(V0s_2E0))
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0)) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__UNIV,axiom,
p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__INTER,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1t_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__IN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1t_2E0)) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__DELETE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EDELETE_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1a_2E0))) ) ).
tff(thm_2Ereal__topology_2EHOMEOMORPHIC__LOCAL__COMPACTNESS,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Ehomeomorphic_2E2(V0s_2E0,V1t_2E0))
=> ( c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0) = c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1t_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__TRANSLATION__EQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(V0a_2E0,V2x_2E0) )
=> ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0a_2E0),V1s_2E0)) = c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1s_2E0) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__CLOSED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2EClosed_2E0,V0s_2E0) = c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1t_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),V0s_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),V1t_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__UNION,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1t_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),V0s_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)),V1t_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))) ) ).
tff(thm_2Ereal__topology_2EOPEN__IN__LOCALLY__COMPACT,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V0s_2E0),V1t_2E0))
<=> ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
& ! [V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Ecompact_2E1(V2k_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2k_2E0,V0s_2E0)) )
=> p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2k_2E0),c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V2k_2E0,V1t_2E0))) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__PROPER__IMAGE__EQ,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2k_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2k_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ! [V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2k_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))
& p(c_2Ereal__topology_2Ecompact_2E1(V2k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2k_2E0)))) ) )
=> ( c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1s_2E0) = c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__PROPER__IMAGE,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V3x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2k_2E0)) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2k_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V1s_2E0),V0f_2E0),V2k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& ! [V2k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V2k_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0)))
& p(c_2Ereal__topology_2Ecompact_2E1(V2k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V2k_2E0)))) )
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1s_2E0)) )
=> p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0))) ) ) ) ).
tff(thm_2Ereal__topology_2EMUMFORD__LEMMA,axiom,
! [F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))))] :
( ! [V7x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V3y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V7x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V7x_2E0) = V3y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V3y_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V7x_2E0),V1s_2E0),V0f_2E0),V3y_2E0)) )
=> ( ! [V4x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal] :
( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3y_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1s_2E0))
& ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0) = V3y_2E0 ) ) )
=> ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V3y_2E0: tyop_2Erealax_2Ereal,V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3y_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V4x_2E0),V1s_2E0),V0f_2E0),V3y_2E0)) )
=> ( ! [V9x_2E0: tyop_2Erealax_2Ereal,V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V9x_2E0),V5u_2E0),V0f_2E0),V8k_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V5u_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V9x_2E0),V8k_2E0)) ) )
=> ( ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V9x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V5u_2E0),V0f_2E0),V8k_2E0),V9x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V9x_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F3_2E0,V9x_2E0),V5u_2E0),V0f_2E0),V8k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3y_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal__topology_2Econtinuous__on_2E2(V0f_2E0,V1s_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V1s_2E0),V2t_2E0))
& p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1s_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V2t_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0f_2E0),V3y_2E0)))) )
=> ? [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V5u_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V2t_2E0),V6v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V0f_2E0),V3y_2E0)),V5u_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V6v_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,c_2Epred__set_2EIMAGE_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0f_2E0,V5u_2E0),V6v_2E0))
& ! [V8k_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V8k_2E0,V6v_2E0))
& p(c_2Ereal__topology_2Ecompact_2E1(V8k_2E0)) )
=> p(c_2Ereal__topology_2Ecompact_2E1(c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F2_2E0,V5u_2E0),V0f_2E0),V8k_2E0)))) ) ) ) ) ) ) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__OPEN__INTER__CLOSURE,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
=> ? [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& ( V0s_2E0 = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1t_2E0,c_2Ereal__topology_2Eclosure_2E1(V0s_2E0)) ) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__IN__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
=> ? [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V1t_2E0))
& p(c_2Etopology_2Eclosed__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1t_2E0),V0s_2E0)) ) ) ).
tff(thm_2Ereal__topology_2ELOCALLY__COMPACT__CLOSED__INTER__OPEN,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V0s_2E0))
<=> ? [V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EClosed_2E1(V1t_2E0))
& p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0))
& ( V0s_2E0 = c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V2u_2E0) ) ) ) ).
tff(thm_2Ereal__topology_2EBAIRE,axiom,
! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1s_2E0))
& p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0g_2E0))
& ! [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V0g_2E0))
=> ( p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V2t_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,c_2Ereal__topology_2Eclosure_2E1(V2t_2E0))) ) ) )
=> p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,c_2Ereal__topology_2Eclosure_2E1(c_2Epred__set_2EBIGINTER_2E1(tyop_2Erealax_2Ereal,V0g_2E0)))) ) ).
tff(thm_2Ereal__topology_2EBAIRE__ALT,axiom,
! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ereal__topology_2Elocally_2E2(c_2Ereal__topology_2Ecompact_2E0,V1s_2E0))
& ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
& p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0g_2E0))
& ( c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0g_2E0) = V1s_2E0 ) )
=> ? [V2t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2t_2E0,V0g_2E0))
& p(c_2Etopology_2Eopen__in_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Esubtopology_2E2(tyop_2Erealax_2Ereal,c_2Ereal__topology_2Eeuclidean_2E0,V1s_2E0),V3u_2E0))
& p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V3u_2E0,c_2Ereal__topology_2Eclosure_2E1(V2t_2E0))) ) ) ).
tff(thm_2Ereal__topology_2ENOWHERE__DENSE__COUNTABLE__BIGUNION__CLOSED,axiom,
! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0g_2E0))
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0g_2E0))
=> ( p(c_2Ereal__topology_2EClosed_2E1(V1s_2E0))
& ( c_2Ereal__topology_2Einterior_2E1(V1s_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0g_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ENOWHERE__DENSE__COUNTABLE__BIGUNION,axiom,
! [V0g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2Ecountable_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0g_2E0))
& ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1s_2E0,V0g_2E0))
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Ereal__topology_2Eclosure_2E1(V1s_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( c_2Ereal__topology_2Einterior_2E1(c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0g_2E0)) = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) ).
tff(thm_2Ereal__topology_2ESUBORDINATE__PARTITION__OF__UNITY,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal))),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))))] :
( ! [V6f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V11x_2E0: tyop_2Erealax_2Ereal,V12u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal)),F1_2E0,V6f_2E0),V11x_2E0),V12u_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6f_2E0,V12u_2E0),V11x_2E0) )
=> ( ! [V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0c_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V5u_2E0),V0c_2E0),V4v_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5u_2E0,V0c_2E0))
& ( c_2Epred__set_2EINTER_2E2(tyop_2Erealax_2Ereal,V5u_2E0,V4v_2E0) != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) ) )
=> ( ! [V0c_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V5u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V0c_2E0),V4v_2E0),V5u_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V5u_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V5u_2E0),V0c_2E0),V4v_2E0)) )
=> ( ! [V15u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0c_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V14n_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool))),F3_2E0,V15u_2E0),V0c_2E0),V14n_2E0),V6f_2E0))
<=> ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V15u_2E0,V0c_2E0))
& ~ ! [V16x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V16x_2E0,V14n_2E0))
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6f_2E0,V15u_2E0),V16x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) )
=> ( ! [V0c_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V14n_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V6f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),V15u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0,V0c_2E0),V14n_2E0),V6f_2E0),V15u_2E0) = c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V15u_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool))),F3_2E0,V15u_2E0),V0c_2E0),V14n_2E0),V6f_2E0)) )
=> ! [V0c_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1s_2E0,c_2Epred__set_2EBIGUNION_2E1(tyop_2Erealax_2Ereal,V0c_2E0)))
& ! [V2u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V2u_2E0,V0c_2E0))
=> p(c_2Ereal__topology_2EOpen_2E1(V2u_2E0)) )
& ! [V3x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1s_2E0))
=> ? [V4v_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V4v_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V4v_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0,V0c_2E0),V4v_2E0)))) ) ) )
=> ? [V6f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V7u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V7u_2E0,V0c_2E0))
=> ( p(c_2Ereal__topology_2Econtinuous__on_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6f_2E0,V7u_2E0),V1s_2E0))
& ! [V8x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V8x_2E0,V1s_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6f_2E0,V7u_2E0),V8x_2E0))) ) ) )
& ! [V9x_2E0: tyop_2Erealax_2Ereal,V10u_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V10u_2E0,V0c_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V1s_2E0))
& ~ p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V9x_2E0,V10u_2E0)) )
=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6f_2E0,V10u_2E0),V9x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
& ! [V11x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V11x_2E0,V1s_2E0))
=> ( c_2Eiterate_2ESum_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal)),F1_2E0,V6f_2E0),V11x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) )
& ! [V13x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V13x_2E0,V1s_2E0))
=> ? [V14n_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
( p(c_2Ereal__topology_2EOpen_2E1(V14n_2E0))
& p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V13x_2E0,V14n_2E0))
& p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),F2_2E0,V0c_2E0),V14n_2E0),V6f_2E0)))) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------