ITP001 Axioms: ITP086^5.ax
%------------------------------------------------------------------------------
% File : ITP086^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : ordinal^2.ax [Gau20]
% : HL4086^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 271 ( 24 unt; 31 typ; 0 def)
% Number of atoms : 3755 ( 198 equ; 0 cnn)
% Maximal formula atoms : 78 ( 13 avg)
% Number of connectives : 8119 ( 75 ~; 10 |; 147 &;7307 @)
% ( 60 <=>; 520 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 12 avg;7307 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 103 usr; 76 con; 0-2 aty)
% Number of variables : 718 ( 23 ^ 671 !; 24 ?; 718 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Eordinal_2Eordinal,type,
ty_2Eordinal_2Eordinal: del > del ).
thf(tp_c_2Eordinal_2EallOrds,type,
c_2Eordinal_2EallOrds: del > $i ).
thf(mem_c_2Eordinal_2EallOrds,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EallOrds @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(tp_c_2Eordinal_2Edclose,type,
c_2Eordinal_2Edclose: del > $i ).
thf(mem_c_2Eordinal_2Edclose,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Edclose @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eordinal_2Edownward__closed,type,
c_2Eordinal_2Edownward__closed: del > $i ).
thf(mem_c_2Eordinal_2Edownward__closed,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eordinal_2Eepsilon0,type,
c_2Eordinal_2Eepsilon0: del > $i ).
thf(mem_c_2Eordinal_2Eepsilon0,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(tp_c_2Eordinal_2Eeval__poly,type,
c_2Eordinal_2Eeval__poly: del > $i ).
thf(mem_c_2Eordinal_2Eeval__poly,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2EfromNat,type,
c_2Eordinal_2EfromNat: del > $i ).
thf(mem_c_2Eordinal_2EfromNat,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(tp_c_2Eordinal_2Eis__polyform,type,
c_2Eordinal_2Eis__polyform: del > del > $i ).
thf(mem_c_2Eordinal_2Eis__polyform,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ bool ) ) ) ).
thf(tp_c_2Eordinal_2Eoleast,type,
c_2Eordinal_2Eoleast: del > $i ).
thf(mem_c_2Eordinal_2Eoleast,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eoleast @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(tp_c_2Eordinal_2Eomax,type,
c_2Eordinal_2Eomax: del > $i ).
thf(mem_c_2Eordinal_2Eomax,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ty_2Eoption_2Eoption @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2Eomega,type,
c_2Eordinal_2Eomega: del > $i ).
thf(mem_c_2Eordinal_2Eomega,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eomega @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(tp_c_2Eordinal_2EordADD,type,
c_2Eordinal_2EordADD: del > $i ).
thf(mem_c_2Eordinal_2EordADD,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2EordDIV,type,
c_2Eordinal_2EordDIV: del > $i ).
thf(mem_c_2Eordinal_2EordDIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordDIV @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2EordEXP,type,
c_2Eordinal_2EordEXP: del > $i ).
thf(mem_c_2Eordinal_2EordEXP,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2EordMOD,type,
c_2Eordinal_2EordMOD: del > $i ).
thf(mem_c_2Eordinal_2EordMOD,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordMOD @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2EordMULT,type,
c_2Eordinal_2EordMULT: del > $i ).
thf(mem_c_2Eordinal_2EordMULT,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2EordSUC,type,
c_2Eordinal_2EordSUC: del > $i ).
thf(mem_c_2Eordinal_2EordSUC,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(tp_c_2Eordinal_2Eordinal__ABS,type,
c_2Eordinal_2Eordinal__ABS: del > $i ).
thf(mem_c_2Eordinal_2Eordinal__ABS,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(tp_c_2Eordinal_2Eordinal__ABS__CLASS,type,
c_2Eordinal_2Eordinal__ABS__CLASS: del > $i ).
thf(mem_c_2Eordinal_2Eordinal__ABS__CLASS,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( arr @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(tp_c_2Eordinal_2Eordinal__REP,type,
c_2Eordinal_2Eordinal__REP: del > $i ).
thf(mem_c_2Eordinal_2Eordinal__REP,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(tp_c_2Eordinal_2Eordinal__REP__CLASS,type,
c_2Eordinal_2Eordinal__REP__CLASS: del > $i ).
thf(mem_c_2Eordinal_2Eordinal__REP__CLASS,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) ) ).
thf(tp_c_2Eordinal_2Eordlt,type,
c_2Eordinal_2Eordlt: del > $i ).
thf(mem_c_2Eordinal_2Eordlt,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eordinal_2Epolyform,type,
c_2Eordinal_2Epolyform: del > $i ).
thf(mem_c_2Eordinal_2Epolyform,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Eordinal_2Epreds,type,
c_2Eordinal_2Epreds: del > $i ).
thf(mem_c_2Eordinal_2Epreds,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eordinal_2Esup,type,
c_2Eordinal_2Esup: del > $i ).
thf(mem_c_2Eordinal_2Esup,axiom,
! [A_27a: del] : ( mem @ ( c_2Eordinal_2Esup @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(ax_thm_2Eordinal_2Eordinal__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool )
@ ^ [V1c: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
@ ( lam @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V2r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) @ V2r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) @ V1c ) @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Eordinal_2Eordinal__bijections,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
=> ( ( p
@ ( ap
@ ( lam @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool )
@ ^ [V2c: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
@ ( lam @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V3r: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V3r ) @ V3r ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) @ V2c ) @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V3r ) ) ) ) ) )
@ V1r ) )
<=> ( ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordinal__ABS__REP__CLASS,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
=> ( ? [V2r: $i] :
( ( mem @ V2r @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) @ V2r ) )
& ( V1c
= ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2r ) ) )
<=> ( ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ V1c ) )
= V1c ) ) ) ) ).
thf(ax_thm_2Eordinal_2Eordinal__REP__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ V0a )
= ( ap @ ( c_2Emin_2E_40 @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP__CLASS @ A_27a ) @ V0a ) ) ) ) ).
thf(ax_thm_2Eordinal_2Eordinal__ABS__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V0r )
= ( ap @ ( c_2Eordinal_2Eordinal__ABS__CLASS @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0r ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordinal__QUOTIENT,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) ) @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) ) ) ).
thf(ax_thm_2Eordinal_2Eordlt__def,axiom,
! [A_27a: del,V0T1: $i] :
( ( mem @ V0T1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1T2: $i] :
( ( mem @ V1T2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0T1 ) @ V1T2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ V0T1 ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__REP @ A_27a ) @ V1T2 ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__REFL,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w ) @ V0w ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__TRANS,axiom,
! [A_27a: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2w3: $i] :
( ( mem @ V2w3 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w1 ) @ V1w2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1w2 ) @ V2w3 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w1 ) @ V2w3 ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__WF,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2Eordlt @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__trichotomy,axiom,
! [A_27a: del,V0w2: $i] :
( ( mem @ V0w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1w1: $i] :
( ( mem @ V1w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1w1 ) @ V0w2 ) )
| ( V1w1 = V0w2 )
| ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w2 ) @ V1w1 ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EallOrds__def,axiom,
! [A_27a: del] :
( ( c_2Eordinal_2EallOrds @ A_27a )
= ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0x: $i] :
( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ewellorder__allOrds,axiom,
! [A_27a: del] :
( p
@ ( ap @ ( c_2Ewellorder_2Ewellorder @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0x: $i] :
( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EWIN__allOrds,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EelsOf__allOrds,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Ewellorder_2EelsOf @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( c_2Eordinal_2EallOrds @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__mkOrdinal,axiom,
! [A_27a: del,V0o1: $i] :
( ( mem @ V0o1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1o2: $i] :
( ( mem @ V1o2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0o1 ) @ V1o2 ) )
<=> ! [V2w1: $i] :
( ( mem @ V2w1 @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
=> ! [V3w2: $i] :
( ( mem @ V3w2 @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ( ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V2w1 )
= V0o1 )
& ( ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V3w2 )
= V1o2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2w1 ) @ V3w2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eorderlt__iso__REFL,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eorderiso__wobound2,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V0x ) @ V1w ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V2y ) @ V1w ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ewellorder__ordinal__isomorphism,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0w ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Eordinal__ABS @ A_27a ) @ V0w ) ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2Epreds__def,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1w0: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ V1w0 ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1w0 ) @ V0w ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EIN__preds,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1w ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1w ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__11,axiom,
! [A_27a: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w1 )
= ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1w2 ) )
<=> ( V0w1 = V1w2 ) ) ) ) ).
thf(ax_thm_2Eordinal_2Edownward__closed__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
<=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__downward__closed,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__bij,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Epreds @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Edownward__closed @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__lt__PSUBSET,axiom,
! [A_27a: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0w1 ) @ V1w2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0w1 ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1w2 ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__wobound,axiom,
! [A_27a: del,V0ord: $i] :
( ( mem @ V0ord @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0ord )
= ( ap @ ( c_2Ewellorder_2EelsOf @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0ord ) @ ( c_2Eordinal_2EallOrds @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__inj__univ,axiom,
! [A_27a: del,V0ord: $i] :
( ( mem @ V0ord @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0ord ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ).
thf(stp_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,type,
tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone: $tType ).
thf(stp_inj_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,type,
inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone: tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone > $i ).
thf(stp_surj_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,type,
surj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone: $i > tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone ).
thf(stp_inj_surj_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone] :
( ( surj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ ( inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ X ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) ) ).
thf(stp_iso_mem_c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) )
=> ( X
= ( inj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ ( surj__c_ty_2Esum_2Esum_ty_2Enum_2Enum_ty_2Eone_2Eone @ X ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eunitinf__univnum,axiom,
p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ).
thf(stp_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,type,
tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone: $tType ).
thf(stp_inj_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,type,
inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone > $i ).
thf(stp_surj_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,type,
surj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone: $i > tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone ).
thf(stp_inj_surj_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone] :
( ( surj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ X ) @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) ).
thf(stp_iso_mem_c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) )
=> ( X
= ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ ( surj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ X ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ecord__countable__preds,axiom,
! [V0ord: tp__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone] : ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Eordinal_2Eordinal_ty_2Eone_2Eone @ V0ord ) ) ) ) ).
thf(conj_thm_2Eordinal_2Euniv__ord__greater__cardinal,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(conj_thm_2Eordinal_2Euniv__cord__uncountable,axiom,
~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ ty_2Eone_2Eone ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordle__lteq,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0b ) @ V1a ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) )
| ( V1a = V0b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordle__ANTISYM,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0b ) @ V1a ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) ) )
=> ( V1a = V0b ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordle__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V0x ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2z ) @ V1y ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2z ) @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlet__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V0x ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V2z ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlte__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2z ) @ V1y ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2Eoleast__def,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eordinal_2Eoleast @ A_27a ) @ V0P )
= ( ap @ ( c_2Emin_2E_40 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1x: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V1x ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2y ) @ V1x ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V0P @ V2y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eoleast__intro,axiom,
! [A_27a: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ? [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ V1P @ V2a ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ! [V4b: $i] :
( ( mem @ V4b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V4b ) @ V3a ) )
=> ~ ( p @ ( ap @ V1P @ V4b ) ) ) )
& ( p @ ( ap @ V1P @ V3a ) ) )
=> ( p @ ( ap @ V0Q @ V3a ) ) ) ) )
=> ( p @ ( ap @ V0Q @ ( ap @ ( c_2Eordinal_2Eoleast @ A_27a ) @ V1P ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EordSUC__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a )
= ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1b: $i] : ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EfromNat__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0a: $i] : c_2Ebool_2ET ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__def__compute,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0a: $i] : c_2Ebool_2ET ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__SUC,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__ZERO,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__surj,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0x ) )
& ( V0x
!= ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ? [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1y )
= V0x ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eno__maximal__ordinal,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ? [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__SUC,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordSUC__ZERO,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a )
!= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__DISCRETE1,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ~ ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__SUC__DISCRETE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
| ( V0a = V1b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordSUC__MONO,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordSUC__11,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) )
<=> ( V0a = V1b ) ) ) ) ).
thf(ax_thm_2Eordinal_2Esup__def,axiom,
! [A_27a: del,V0ordset: $i] :
( ( mem @ V0ordset @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0ordset )
= ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1a: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Epreds @ A_27a ) ) @ V0ordset ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eord__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ! [V1min: $i] :
( ( mem @ V1min @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1min ) )
=> ( p @ ( ap @ V0P @ V2b ) ) ) )
=> ( p @ ( ap @ V0P @ V1min ) ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3a ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__thm,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
<=> ? [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esuple__thm,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0b ) @ V1s ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__eq__sup,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s1 ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s2 ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2a ) @ V0s1 ) )
=> ? [V3b: $i] :
( ( mem @ V3b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V3b ) @ V1s2 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3b ) @ V2a ) ) ) ) )
& ! [V4b: $i] :
( ( mem @ V4b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V4b ) @ V1s2 ) )
=> ? [V5a: $i] :
( ( mem @ V5a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V5a ) @ V0s1 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V5a ) @ V4b ) ) ) ) ) )
=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s1 )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s2 ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EUnum__cle__Uinf,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ecsup__thm,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
<=> ? [V2d: $i] :
( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epredimage__sup__thm,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) ) ) ) )
<=> ? [V3d: $i] :
( ( mem @ V3d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3d ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ ( ap @ V0f @ V3d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epredimage__suplt__ELIM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) ) ) ) @ V2b ) )
=> ! [V3d: $i] :
( ( mem @ V3d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3d ) @ V1a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ ( ap @ V0f @ V3d ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esuppred__suplt__ELIM,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) @ V1b ) )
=> ! [V2d: $i] :
( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2d ) @ V0a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__SING,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
= V0a ) ) ).
thf(conj_thm_2Eordinal_2Esup__preds__SUC,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) )
= V0a ) ) ).
thf(conj_thm_2Eordinal_2Epreds__ordSUC,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) ) ).
thf(conj_thm_2Eordinal_2EcountableOrds__dclosed,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1b ) ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2Eomax__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
= ( ap @ ( c_2Eoption_2Esome @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1a: $i] :
( ap
@ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
@ ( ap @ ( ap @ ( c_2Eset__relation_2Emaximal__elements @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V2x: $i] :
( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2x ) @ V3y ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3y ) @ V2x ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomax__SOME,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) )
& ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomax__NONE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) )
=> ? [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomax__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__0,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Eordleq0,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
<=> ( V0x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__EQ__EMPTY,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ( V0x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomax__sup,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ V0s )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) )
=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
= V1a ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__omax__SOME__SUC,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1b ) )
<=> ( V0a
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esimple__ord__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
& ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3b ) @ V2a ) )
=> ( p @ ( ap @ V0P @ V3b ) ) ) ) )
=> ( p @ ( ap @ V0P @ V2a ) ) ) ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ V0P @ V4a ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__preds__omax__NONE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
= V0a ) ) ) ).
thf(ax_thm_2Eordinal_2Edclose__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V1x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ V1x )
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2y ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__sup,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) )
= ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ecsup__lesup,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s ) )
=> ! [V1d: $i] :
( ( mem @ V1d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1d ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) @ V1d ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ecsup__suple,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0s ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
<=> ! [V2d: $i] :
( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__sup__thm,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
<=> ? [V2d: $i] :
( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__lesup,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ! [V1d: $i] :
( ( mem @ V1d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1d ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) @ V1d ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epreds__suple,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Eordinal_2Edownward__closed @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
<=> ! [V2d: $i] :
( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2d ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__11,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) )
<=> ( V0x = V1y ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__fromNat,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> ? [V2m: tp__ty_2Enum_2Enum] :
( ( V1x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__ordlt,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).
thf(ax_thm_2Eordinal_2Eomega__def,axiom,
! [A_27a: del] :
( ( c_2Eordinal_2Eomega @ A_27a )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ V0i ) ) @ c_2Ebool_2ET ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Elt__omega,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
<=> ? [V1m: tp__ty_2Enum_2Enum] :
( V0a
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__lt__omega,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__eq__omega,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
!= ( c_2Eordinal_2Eomega @ A_27a ) ) ).
thf(conj_thm_2Eordinal_2Eord__RECURSION,axiom,
! [A_27a: del,A_27b: del,V0z: $i] :
( ( mem @ V0z @ A_27b )
=> ! [V1sf: $i] :
( ( mem @ V1sf @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V2lf: $i] :
( ( mem @ V2lf @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( arr @ A_27b @ bool ) @ A_27b ) ) )
=> ? [V3h: $i] :
( ( mem @ V3h @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ A_27b ) )
& ( ( ap @ V3h @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0z )
& ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ V3h @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V4a ) )
= ( ap @ ( ap @ V1sf @ V4a ) @ ( ap @ V3h @ V4a ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V5a ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V5a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ V3h @ V5a )
= ( ap @ ( ap @ V2lf @ V5a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ A_27b ) @ V3h ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V5a ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EordADD__def,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0b )
& ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ V1a ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) @ V2a )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0b ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__0L,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a )
= V0a ) ) ).
thf(conj_thm_2Eordinal_2Eubsup__thm,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2a ) @ V1b ) ) ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) )
<=> ? [V4d: $i] :
( ( mem @ V4d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V4d ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ V4d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__fromNat,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomax__preds__omega,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Eomega__islimit,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__fromNat__omega,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eomega @ A_27a ) )
= ( c_2Eordinal_2Eomega @ A_27a ) ) ).
thf(conj_thm_2Eordinal_2Elt__suppreds,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) )
<=> ? [V2d: $i] :
( ( mem @ V2d @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2d ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V2d ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomax__preds__SUC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0x ) ) )
= ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0x ) ) ) ).
thf(conj_thm_2Eordinal_2EORD__ONE,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordSUC__NUMERAL,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordZERO__ltSUC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0x ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__CANCEL__ADDR,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0b ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__CANCEL__ADDL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b ) ) @ V0a ) )
<=> $false ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__CANCEL1,axiom,
! [A_27a: del] :
( ! [V0c: $i] :
( ( mem @ V0c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( V1a
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0c ) )
<=> ( V0c
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V3a ) @ V2c )
= V3a )
<=> ( V2c
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__MONO,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V0b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__CANCEL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2c ) @ V0b ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__RIGHT__CANCEL,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0b )
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V2c ) )
<=> ( V0b = V2c ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EleqLEFT__CANCEL,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0x ) ) @ V0x ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordlt__EXISTS__ADD,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
<=> ? [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( V2c
!= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ( V1b
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordle__EXISTS__ADD,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V0a ) )
<=> ? [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( V1b
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eordle__CANCEL__ADDR,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0x ) @ V1a ) ) @ V0x ) ) ) ) ).
thf(conj_thm_2Eordinal_2Edclose__BIGUNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) @ ( c_2Eordinal_2Epreds @ A_27a ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Eordinal_2EcountableOrds__uncountable,axiom,
! [A_27a: del] :
~ ( p
@ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ V0a ) @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Edclose__cardleq__univinf,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2Edclose @ A_27a ) @ V0s ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__lt__implies,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s ) ) @ V1a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__eq__max,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1a ) @ V0s ) ) )
=> ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
= V1a ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__eq__SUC,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Egeneric__continuity,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1a ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ V0f @ V1a )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) ) ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3y ) @ V2x ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V3y ) ) @ ( ap @ V0f @ V2x ) ) ) ) ) ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V4s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( V4s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V4s ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ V4s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eord__CASES,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( V0a
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
| ? [V1a0: $i] :
( ( mem @ V1a0 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ( V0a
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a0 ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eislimit__0,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Eordinal__IVT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2a2: $i] :
( ( mem @ V2a2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V4a ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ V0f @ V4a )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a ) ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V6y: $i] :
( ( mem @ V6y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V6y ) @ V5x ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V6y ) ) @ ( ap @ V0f @ V5x ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a1 ) @ V2a2 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ V1a1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ V2a2 ) ) ) )
=> ? [V7b: $i] :
( ( mem @ V7b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V7b ) @ V1a1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V7b ) @ V2a2 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ V7b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3c ) @ ( ap @ V0f @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V7b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__continuous,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__ASSOC,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1b ) @ V2c ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b ) ) @ V2c ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EADD1R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0a ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__weak__MONO,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V2b ) @ V0c ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ V0c ) ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EordMULT__def,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ V1a ) ) @ V0b ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ V2a )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__0L,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__0R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__1L,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a )
= V0a ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__1R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V0a ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__2R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V0a ) ) ) ).
thf(conj_thm_2Eordinal_2Eislimit__SUC__lt,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0b ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V0b ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__lt__MONO__R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2c ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__le__MONO__R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V0a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V1b ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__lt__MONO__R__EQN,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0c ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0c ) @ V2b ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0c ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__le__MONO__L,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0y ) @ V1x ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0y ) @ V2z ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1x ) @ V2z ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__le__MONO__L,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ V0a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2c ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ V2c ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__CANCEL__R,axiom,
! [A_27a: del,V0z: $i] :
( ( mem @ V0z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0z ) @ V1x )
= ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0z ) @ V2y ) )
<=> ( ( V0z
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
| ( V1x = V2y ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__continuous,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__fromNat,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomega__MUL__fromNat,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eomega @ A_27a ) )
= ( c_2Eordinal_2Eomega @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__LDISTRIB,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V2c ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__ASSOC,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2c ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ V1b ) ) @ V2c ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EordDIVISION,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1b ) )
=> ( ( V0a
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordDIV @ A_27a ) @ V0a ) @ V1b ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMOD @ A_27a ) @ V0a ) @ V1b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMOD @ A_27a ) @ V0a ) @ V1b ) ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordDIV__UNIQUE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1b ) )
& ( V0a
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2q ) ) @ V3r ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3r ) @ V1b ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordDIV @ A_27a ) @ V0a ) @ V1b )
= V2q ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMOD__UNIQUE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1b ) )
& ( V0a
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1b ) @ V2q ) ) @ V3r ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3r ) @ V1b ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordMOD @ A_27a ) @ V0a ) @ V1b )
= V3r ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2EordEXP__def,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2a_27: $i] :
( ( mem @ V2a_27 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V2a_27 ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1a ) @ V2a_27 ) ) @ V1a ) ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V4a_27: $i] :
( ( mem @ V4a_27 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V4a_27 ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a_27 ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V3a ) @ V4a_27 )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V3a ) ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V4a_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__1R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= V0a ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__1L,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__2R,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ V0a ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__fromNat,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__le__MONO__L,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2b ) @ V1a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2b ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1a ) @ V0x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EIFF__ZERO__lt,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( V0x
!= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) )
& ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eislimit__SUC,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V0x ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
<=> $false ) ) ).
thf(conj_thm_2Eordinal_2Eislimit__fromNat,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
<=> ( V0x = fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__ZERO__limit,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0x ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__ZERO__nonlimit,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0x ) )
!= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Esup__EQ__0,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
=> ( ( ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V0s )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
| ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__EQ__0,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1x ) @ V0y )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( ( V1x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ( V0y
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EIMAGE__EQ__SING,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V0f ) @ V1s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( ? [V3y: $i] :
( ( mem @ V3y @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1s ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V1s ) )
=> ( ( ap @ V0f @ V4y )
= V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMULT__EQ__0,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0x ) @ V1y )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( ( V0x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
| ( V1y
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__EQ__0,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V1x ) @ V0y )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( ( V1x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V0y ) )
!= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EZERO__lt__ordEXP__I,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EZERO__lt__ordEXP,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1x ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
| ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1x ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__lt__MONO__R,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ V0y ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V1x ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V0y ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__lt__IFF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V1y ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__le__MONO__R,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ V0x ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V2a ) @ V0x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__continuous,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__ADD,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1y ) @ V2z ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__MUL,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V1y ) @ V2z ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0x ) @ V1y ) ) @ V2z ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Efixpoints__exist,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) )
=> ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1s ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ V1s ) ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V2x ) ) @ V2x ) ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ? [V4b: $i] :
( ( mem @ V4b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V4b ) @ V3a ) )
& ( ( ap @ V0f @ V4b )
= V4b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ex__le__ordEXP__x,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1x ) ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2Eepsilon0__def,axiom,
! [A_27a: del] :
( ( c_2Eordinal_2Eepsilon0 @ A_27a )
= ( ap @ ( c_2Eordinal_2Eoleast @ A_27a )
@ ( lam @ ( ty_2Eordinal_2Eordinal @ A_27a )
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0x ) ) @ V0x ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eepsilon0__fixpoint,axiom,
! [A_27a: del] :
( ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) )
= ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ).
thf(conj_thm_2Eordinal_2Eepsilon0__least__fixpoint,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomega__lt__epsilon0,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2EfromNat__lt__epsilon0,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ).
thf(conj_thm_2Eordinal_2Eadd__nat__islimit,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
<=> $false ) ) ) ).
thf(conj_thm_2Eordinal_2Estrict__continuity__preserves__islimit,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V2s ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V2s ) )
= ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ V2s ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ V0f @ V4y ) ) ) ) ) )
& ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
& ( V1a
!= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ V0f @ V1a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eadd__omega__islimit,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eislimit__mul__R,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V1a ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0b ) @ V1a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Emul__omega__islimit,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eomega__exp__islimit,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0a ) )
=> ( ( ap @ ( c_2Eordinal_2Eomax @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
= ( c_2Eoption_2ENONE @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eexpbound__add,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EaddL__fixpoint__iff,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0a ) @ V1b )
= V1b )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordADD__under__epsilon0,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ V0x ) @ V1y ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordMUL__under__epsilon0,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1y ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0x ) @ V1y ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordEXP__under__epsilon0,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0a ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1b ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V1b ) ) @ ( c_2Eordinal_2Eepsilon0 @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eeval__poly__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ bool ) ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V1a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V4e: $i] :
( ( mem @ V4e @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( p @ ( ap @ ( ap @ V0P @ V2a ) @ V5t ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V3c ) @ V4e ) ) @ V5t ) ) ) ) ) ) ) ) )
=> ! [V6v: $i] :
( ( mem @ V6v @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V7v1: $i] :
( ( mem @ V7v1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V6v ) @ V7v1 ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eeval__poly__def,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V4a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V3c ) @ V2e ) ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Eordinal_2EordADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V4a ) @ V2e ) ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V4a ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eis__polyform__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ bool ) ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V1a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V4e: $i] :
( ( mem @ V4e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V3c ) @ V4e ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V6c1: $i] :
( ( mem @ V6c1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V7e1: $i] :
( ( mem @ V7e1 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ! [V8c2: $i] :
( ( mem @ V8c2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V9e2: $i] :
( ( mem @ V9e2 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ! [V10t: $i] :
( ( mem @ V10t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
=> ( ( p @ ( ap @ ( ap @ V0P @ V5a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V8c2 ) @ V9e2 ) ) @ V10t ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V5a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V6c1 ) @ V7e1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V8c2 ) @ V9e2 ) ) @ V10t ) ) ) ) ) ) ) ) ) ) ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eis__polyform__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) )
<=> $true ) )
& ! [V1e: $i] :
( ( mem @ V1e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V3a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V2c ) @ V1e ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V2c ) @ V3a ) ) ) ) ) ) )
& ! [V4t: $i] :
( ( mem @ V4t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
=> ! [V5e2: $i] :
( ( mem @ V5e2 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ! [V6e1: $i] :
( ( mem @ V6e1 @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ! [V7c2: $i] :
( ( mem @ V7c2 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V8c1: $i] :
( ( mem @ V8c1 @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V9a: $i] :
( ( mem @ V9a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V9a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V8c1 ) @ V6e1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V7c2 ) @ V5e2 ) ) @ V4t ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V8c1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V8c1 ) @ V9a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27b ) @ V5e2 ) @ V6e1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V9a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V7c2 ) @ V5e2 ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eis__polyform__ELthm,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1ces: $i] :
( ( mem @ V1ces @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ V1ces ) )
<=> ( ! [V2i: tp__ty_2Enum_2Enum,V3j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ V1ces ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V1ces ) ) ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1ces ) ) ) ) )
& ! [V4c: $i] :
( ( mem @ V4c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V5e: $i] :
( ( mem @ V5e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V4c ) @ V5e ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ V1ces ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V4c ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V4c ) @ V0a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epolyform__exists,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
=> ? [V2ces: $i] :
( ( mem @ V2ces @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ V2ces ) )
& ( V1b
= ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V2ces ) ) ) ) ) ) ).
thf(ax_thm_2Eordinal_2Epolyform__def,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1b ) ) )
& ( V1b
= ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2ECNF__thm,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0b ) ) )
& ( V0b
= ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0b ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epolyform__0,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epolyform__EQ__NIL,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
=> ( ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1x )
= ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
<=> ( V1x
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eis__polyform__CONS__E,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ ( ty_2Eordinal_2Eordinal @ A_27b ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V1c ) @ V2e ) ) @ V3t ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1c ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27b ) @ V0a ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eis__polyform__head__dominates__tail,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V1c ) @ V2e ) ) @ V3t ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V3t ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0a ) @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Ecx__lt__x,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordMULT @ A_27a ) @ V0x ) @ V1c ) ) @ V0x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
& ( V1c
= ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epolyform__UNIQUE,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V2ces: $i] :
( ( mem @ V2ces @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ V2ces ) )
& ( V1b
= ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V2ces ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ V1b )
= V2ces ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Epolyform__eval__poly,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0a ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eis__polyform @ A_27a @ A_27a ) @ V0a ) @ V1b ) ) )
=> ( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Eordinal_2Eeval__poly @ A_27a ) @ V0a ) @ V1b ) )
= V1b ) ) ) ) ).
thf(conj_thm_2Eordinal_2ECNF__nat,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2EordLOG__correct,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0b ) )
& ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V1x ) ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0b ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0b ) @ V1x ) ) ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ V0b ) @ V1x ) ) ) ) @ V2a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ V0b ) @ V2a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eordinal_2Eolog__correct,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0x ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0x ) ) ) ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ ( c_2Elist_2EHD @ ( ty_2Epair_2Eprod @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Eordinal_2Epolyform @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V0x ) ) ) ) @ V1a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eordinal_2EordEXP @ A_27a ) @ ( c_2Eordinal_2Eomega @ A_27a ) ) @ V1a ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------