TPTP Problem File: ITP042^2.p
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%------------------------------------------------------------------------------
% File : ITP042^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_99__7210728_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Coincidence/prob_99__7210728_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 430 ( 130 unt; 71 typ; 0 def)
% Number of atoms : 1008 ( 205 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 4766 ( 34 ~; 7 |; 123 &;4185 @)
% ( 0 <=>; 417 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 398 ( 398 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 63 usr; 8 con; 0-8 aty)
% Number of variables : 1333 ( 154 ^;1073 !; 23 ?;1333 :)
% ( 83 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:22:00.421
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
frechet_good_interp: $tType > $tType > $tType > $tType ).
thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
bounde2145540817linfun: $tType > $tType > $tType ).
thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
frechet_strm: $tType > $tType > $tType ).
thf(ty_t_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Syntax_Otrm,type,
trm: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_sz,type,
sz: $tType ).
thf(ty_tf_sf,type,
sf: $tType ).
thf(ty_tf_sc,type,
sc: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (54)
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo2133971006pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_Denotational__Semantics_OIagree,type,
denotational_Iagree:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_OVagree,type,
denotational_Vagree:
!>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_Ointerp_OFunctions,type,
denota593628964ctions:
!>[A: $tType,B: $tType,C: $tType,Z: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ Z ) > A > ( finite_Cartesian_vec @ real @ C ) > real ) ).
thf(sy_c_Denotational__Semantics_Ois__interp,type,
denota2077489681interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Denotational__Semantics_Osterm__sem,type,
denota126604975rm_sem:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > real ) ).
thf(sy_c_Frechet__Correctness_Oids_Oblin__frechet,type,
frechet_blin_frechet:
!>[Sf: $tType,Sc: $tType,Sz: $tType] : ( ( frechet_good_interp @ Sf @ Sc @ Sz ) > ( frechet_strm @ Sf @ Sz ) > ( finite_Cartesian_vec @ real @ Sz ) > ( bounde2145540817linfun @ ( finite_Cartesian_vec @ real @ Sz ) @ real ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
freche457001096interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
frechet_cr_strm:
!>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
freche227871258interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
freche229654227interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
frechet_raw_term:
!>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
frechet_simple_term:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lifting_Orel__pred__comp,type,
rel_pred_comp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Static__Semantics_OFVT,type,
static_FVT:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OSIGT,type,
static_SIGT:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ A ) ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_Syntax_Odfree,type,
dfree:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
thf(sy_c_Syntax_Otrm_OTimes,type,
times:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo2071040574ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I,type,
i: denota1663640101rp_ext @ a @ c @ b @ product_unit ).
thf(sy_v_J,type,
j: denota1663640101rp_ext @ a @ c @ b @ product_unit ).
thf(sy_v__092_060nu_062,type,
nu: product_prod @ ( finite_Cartesian_vec @ real @ b ) @ ( finite_Cartesian_vec @ real @ b ) ).
thf(sy_v__092_060nu_062_H,type,
nu2: product_prod @ ( finite_Cartesian_vec @ real @ b ) @ ( finite_Cartesian_vec @ real @ b ) ).
thf(sy_v__092_060theta_062_092_060_094sub_0621____,type,
theta_1: trm @ a @ b ).
thf(sy_v__092_060theta_062_092_060_094sub_0622____,type,
theta_2: trm @ a @ b ).
% Relevant facts (256)
thf(fact_0__092_060open_062_092_060And_062J_AI_O_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_ITimes_A_092_060theta_062_092_060_094sub_0621_A_092_060theta_062_092_060_094sub_0622_J_125_A_092_060Longrightarrow_062_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_092_060theta_062_092_060_094sub_0621_125_092_060close_062,axiom,
! [C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I: denota1663640101rp_ext @ a @ B @ C @ product_unit,J: denota1663640101rp_ext @ a @ B @ C @ product_unit] :
( ( denotational_Iagree @ a @ B @ C @ I @ J
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ B @ C ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ B @ C )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ B @ C ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ ( times @ a @ b @ theta_1 @ theta_2 ) ) ) ) ) )
=> ( denotational_Iagree @ a @ B @ C @ I @ J
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ B @ C ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ B @ C )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ B @ C ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_1 ) ) ) ) ) ) ) ).
% \<open>\<And>J I. Iagree I J {Inl x |x. x \<in> SIGT (Times \<theta>\<^sub>1 \<theta>\<^sub>2)} \<Longrightarrow> Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>1}\<close>
thf(fact_1__092_060open_062_092_060And_062J_AI_O_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_ITimes_A_092_060theta_062_092_060_094sub_0621_A_092_060theta_062_092_060_094sub_0622_J_125_A_092_060Longrightarrow_062_AIagree_AI_AJ_A_123Inl_Ax_A_124x_O_Ax_A_092_060in_062_ASIGT_A_092_060theta_062_092_060_094sub_0622_125_092_060close_062,axiom,
! [C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I: denota1663640101rp_ext @ a @ B @ C @ product_unit,J: denota1663640101rp_ext @ a @ B @ C @ product_unit] :
( ( denotational_Iagree @ a @ B @ C @ I @ J
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ B @ C ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ B @ C )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ B @ C ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ ( times @ a @ b @ theta_1 @ theta_2 ) ) ) ) ) )
=> ( denotational_Iagree @ a @ B @ C @ I @ J
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ B @ C ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ B @ C )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ B @ C ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_2 ) ) ) ) ) ) ) ).
% \<open>\<And>J I. Iagree I J {Inl x |x. x \<in> SIGT (Times \<theta>\<^sub>1 \<theta>\<^sub>2)} \<Longrightarrow> Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>2}\<close>
thf(fact_2_IA,axiom,
( denotational_Iagree @ a @ c @ b @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ c @ b ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ c @ b )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ c @ b ) @ X ) )
& ( ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_1 ) )
| ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_2 ) ) ) ) ) ) ).
% IA
thf(fact_3_dfree__Times_Oprems_I2_J,axiom,
( denotational_Iagree @ a @ c @ b @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ c @ b ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ c @ b )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ c @ b ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ ( times @ a @ b @ theta_1 @ theta_2 ) ) ) ) ) ) ).
% dfree_Times.prems(2)
thf(fact_4_dfree__Times_Ohyps_I1_J,axiom,
dfree @ a @ b @ theta_1 ).
% dfree_Times.hyps(1)
thf(fact_5_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X1: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X1 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_6_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A2: A,A3: A] :
( ( ( sum_Inl @ A @ B @ A2 )
= ( sum_Inl @ A @ B @ A3 ) )
= ( A2 = A3 ) ) ).
% old.sum.inject(1)
thf(fact_7_Isubs_I2_J,axiom,
! [E: $tType] :
( ord_less_eq @ ( set @ ( sum_sum @ a @ E ) )
@ ( collect @ ( sum_sum @ a @ E )
@ ^ [Uu: sum_sum @ a @ E] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ E @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_2 ) ) ) )
@ ( collect @ ( sum_sum @ a @ E )
@ ^ [Uu: sum_sum @ a @ E] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ E @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ ( times @ a @ b @ theta_1 @ theta_2 ) ) ) ) ) ) ).
% Isubs(2)
thf(fact_8_Isubs_I1_J,axiom,
! [D2: $tType] :
( ord_less_eq @ ( set @ ( sum_sum @ a @ D2 ) )
@ ( collect @ ( sum_sum @ a @ D2 )
@ ^ [Uu: sum_sum @ a @ D2] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ D2 @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_1 ) ) ) )
@ ( collect @ ( sum_sum @ a @ D2 )
@ ^ [Uu: sum_sum @ a @ D2] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ D2 @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ ( times @ a @ b @ theta_1 @ theta_2 ) ) ) ) ) ) ).
% Isubs(1)
thf(fact_9_Iagree__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A4: denota1663640101rp_ext @ A @ B @ C @ product_unit,B2: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
( ( denotational_Iagree @ A @ B @ C @ A4 @ B2 @ V )
=> ( denotational_Iagree @ A @ B @ C @ B2 @ A4 @ V ) ) ) ).
% Iagree_comm
thf(fact_10_Iagree__refl,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I @ I @ A4 ) ) ).
% Iagree_refl
thf(fact_11_Inl__inject,axiom,
! [B: $tType,A: $tType,X2: A,Y: A] :
( ( ( sum_Inl @ A @ B @ X2 )
= ( sum_Inl @ A @ B @ Y ) )
=> ( X2 = Y ) ) ).
% Inl_inject
thf(fact_12_IH1,axiom,
( ( denotational_Vagree @ b @ nu @ nu2 @ ( static_FVT @ a @ b @ theta_1 ) )
=> ( ( denotational_Iagree @ a @ c @ b @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ c @ b ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ c @ b )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ c @ b ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_1 ) ) ) ) )
=> ( ( denota126604975rm_sem @ a @ c @ b @ i @ theta_1 @ ( product_fst @ ( finite_Cartesian_vec @ real @ b ) @ ( finite_Cartesian_vec @ real @ b ) @ nu ) )
= ( denota126604975rm_sem @ a @ c @ b @ j @ theta_1 @ ( product_fst @ ( finite_Cartesian_vec @ real @ b ) @ ( finite_Cartesian_vec @ real @ b ) @ nu2 ) ) ) ) ) ).
% IH1
thf(fact_13_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R: A > B > $o,P: B > $o,X: A] :
? [Y2: B] :
( ( R @ X @ Y2 )
& ( P @ Y2 ) ) ) ) ).
% rel_pred_comp_def
thf(fact_14_IH2,axiom,
( ( denotational_Vagree @ b @ nu @ nu2 @ ( static_FVT @ a @ b @ theta_2 ) )
=> ( ( denotational_Iagree @ a @ c @ b @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ c @ b ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ c @ b )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ c @ b ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_2 ) ) ) ) )
=> ( ( denota126604975rm_sem @ a @ c @ b @ i @ theta_2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ b ) @ ( finite_Cartesian_vec @ real @ b ) @ nu ) )
= ( denota126604975rm_sem @ a @ c @ b @ j @ theta_2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ b ) @ ( finite_Cartesian_vec @ real @ b ) @ nu2 ) ) ) ) ) ).
% IH2
thf(fact_15_Iagree__Func,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),F: A] :
( ( denotational_Iagree @ A @ B @ C @ I @ J @ V )
=> ( ( member @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Inl @ A @ ( sum_sum @ B @ C ) @ F ) @ V )
=> ( ( denota593628964ctions @ A @ B @ C @ product_unit @ I @ F )
= ( denota593628964ctions @ A @ B @ C @ product_unit @ J @ F ) ) ) ) ) ).
% Iagree_Func
thf(fact_16_raw__interp__inject,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X2: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
( ( ( freche229654227interp @ A @ B @ C @ X2 )
= ( freche229654227interp @ A @ B @ C @ Y ) )
= ( X2 = Y ) ) ) ).
% raw_interp_inject
thf(fact_17_dfree__Times_Ohyps_I2_J,axiom,
dfree @ a @ b @ theta_2 ).
% dfree_Times.hyps(2)
thf(fact_18_coincidence__sterm,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Theta: trm @ B @ A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVT @ B @ A @ Theta ) )
=> ( ( denota126604975rm_sem @ B @ C @ A @ I @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) )
= ( denota126604975rm_sem @ B @ C @ A @ I @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu2 ) ) ) ) ) ).
% coincidence_sterm
thf(fact_19_dfree__Times_Oprems_I1_J,axiom,
denotational_Vagree @ b @ nu @ nu2 @ ( static_FVT @ a @ b @ ( times @ a @ b @ theta_1 @ theta_2 ) ) ).
% dfree_Times.prems(1)
thf(fact_20_VAs_I2_J,axiom,
denotational_Vagree @ b @ nu @ nu2 @ ( static_FVT @ a @ b @ theta_2 ) ).
% VAs(2)
thf(fact_21_VAs_I1_J,axiom,
denotational_Vagree @ b @ nu @ nu2 @ ( static_FVT @ a @ b @ theta_1 ) ).
% VAs(1)
thf(fact_22_VA,axiom,
denotational_Vagree @ b @ nu @ nu2 @ ( sup_sup @ ( set @ ( sum_sum @ b @ b ) ) @ ( static_FVT @ a @ b @ theta_1 ) @ ( static_FVT @ a @ b @ theta_2 ) ) ).
% VA
thf(fact_23_agree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: set @ ( sum_sum @ A @ A ),B2: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A4 @ B2 )
=> ( ( denotational_Vagree @ A @ Nu @ Omega @ B2 )
=> ( denotational_Vagree @ A @ Nu @ Omega @ A4 ) ) ) ) ).
% agree_sub
thf(fact_24_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A4 @ B2 @ V )
=> ( denotational_Vagree @ A @ B2 @ A4 @ V ) ) ) ).
% agree_comm
thf(fact_25_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A4: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A4 ) ) ).
% agree_refl
thf(fact_26_agree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B2: set @ ( sum_sum @ A @ A ),A4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B2 @ A4 )
=> ( ( denotational_Vagree @ A @ Nu @ Nu2 @ A4 )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ B2 ) ) ) ) ).
% agree_supset
thf(fact_27_cr__good__interp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( freche457001096interp @ A @ B @ C )
= ( ^ [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y2: frechet_good_interp @ A @ B @ C] :
( X
= ( freche229654227interp @ A @ B @ C @ Y2 ) ) ) ) ) ).
% cr_good_interp_def
thf(fact_28_dfree__Times__simps,axiom,
! [B: $tType,A: $tType,A2: trm @ A @ B,B3: trm @ A @ B] :
( ( dfree @ A @ B @ ( times @ A @ B @ A2 @ B3 ) )
= ( ( dfree @ A @ B @ A2 )
& ( dfree @ A @ B @ B3 ) ) ) ).
% dfree_Times_simps
thf(fact_29_simple__term__inject,axiom,
! [C: $tType,A: $tType,X2: trm @ A @ C,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ X2 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( ( frechet_simple_term @ A @ C @ X2 )
= ( frechet_simple_term @ A @ C @ Y ) )
= ( X2 = Y ) ) ) ) ).
% simple_term_inject
thf(fact_30_simple__term__induct,axiom,
! [C: $tType,A: $tType,P2: ( frechet_strm @ A @ C ) > $o,X2: frechet_strm @ A @ C] :
( ! [Y3: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( P2 @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
=> ( P2 @ X2 ) ) ).
% simple_term_induct
thf(fact_31_simple__term__cases,axiom,
! [C: $tType,A: $tType,X2: frechet_strm @ A @ C] :
~ ! [Y3: trm @ A @ C] :
( ( X2
= ( frechet_simple_term @ A @ C @ Y3 ) )
=> ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ).
% simple_term_cases
thf(fact_32_raw__interp__inverse,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X2: frechet_good_interp @ A @ B @ C] :
( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X2 ) )
= X2 ) ) ).
% raw_interp_inverse
thf(fact_33_raw__term__induct,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C,P2: ( trm @ A @ C ) > $o] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ! [X3: frechet_strm @ A @ C] : ( P2 @ ( frechet_raw_term @ A @ C @ X3 ) )
=> ( P2 @ Y ) ) ) ).
% raw_term_induct
thf(fact_34_raw__term__cases,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ~ ! [X3: frechet_strm @ A @ C] :
( Y
!= ( frechet_raw_term @ A @ C @ X3 ) ) ) ).
% raw_term_cases
thf(fact_35_raw__term,axiom,
! [C: $tType,A: $tType,X2: frechet_strm @ A @ C] : ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X2 ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% raw_term
thf(fact_36_trm_Oinject_I5_J,axiom,
! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,Y51: trm @ A @ C,Y52: trm @ A @ C] :
( ( ( times @ A @ C @ X51 @ X52 )
= ( times @ A @ C @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_37_raw__term__inject,axiom,
! [C: $tType,A: $tType,X2: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
( ( ( frechet_raw_term @ A @ C @ X2 )
= ( frechet_raw_term @ A @ C @ Y ) )
= ( X2 = Y ) ) ).
% raw_term_inject
thf(fact_38_raw__term__inverse,axiom,
! [C: $tType,A: $tType,X2: frechet_strm @ A @ C] :
( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X2 ) )
= X2 ) ).
% raw_term_inverse
thf(fact_39_simple__term__inverse,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_40_agree__union,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A4: set @ ( sum_sum @ A @ A ),B2: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ Nu @ Omega @ A4 )
=> ( ( denotational_Vagree @ A @ Nu @ Omega @ B2 )
=> ( denotational_Vagree @ A @ Nu @ Omega @ ( sup_sup @ ( set @ ( sum_sum @ A @ A ) ) @ A4 @ B2 ) ) ) ) ) ).
% agree_union
thf(fact_41_Iagree__sub,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A4 @ B2 )
=> ( ( denotational_Iagree @ A @ B @ C @ I @ J @ B2 )
=> ( denotational_Iagree @ A @ B @ C @ I @ J @ A4 ) ) ) ) ).
% Iagree_sub
thf(fact_42_dfree_Odfree__Times,axiom,
! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
( ( dfree @ A @ C @ Theta_1 )
=> ( ( dfree @ A @ C @ Theta_2 )
=> ( dfree @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree.dfree_Times
thf(fact_43_cr__strm__def,axiom,
! [B: $tType,A: $tType] :
( ( frechet_cr_strm @ A @ B )
= ( ^ [X: trm @ A @ B,Y2: frechet_strm @ A @ B] :
( X
= ( frechet_raw_term @ A @ B @ Y2 ) ) ) ) ).
% cr_strm_def
thf(fact_44_fst__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( sup @ A )
& ( sup @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( product_fst @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
= ( sup_sup @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_sup
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_50_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A2 )
= ( ( ord_less_eq @ A @ B3 @ A2 )
& ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_51_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X2 @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_52_FVT_Osimps_I5_J,axiom,
! [C: $tType,A: $tType,F: trm @ A @ C,G: trm @ A @ C] :
( ( static_FVT @ A @ C @ ( times @ A @ C @ F @ G ) )
= ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVT @ A @ C @ F ) @ ( static_FVT @ A @ C @ G ) ) ) ).
% FVT.simps(5)
thf(fact_53_SIGT_Osimps_I5_J,axiom,
! [C: $tType,A: $tType,T1: trm @ A @ C,T2: trm @ A @ C] :
( ( static_SIGT @ A @ C @ ( times @ A @ C @ T1 @ T2 ) )
= ( sup_sup @ ( set @ A ) @ ( static_SIGT @ A @ C @ T1 ) @ ( static_SIGT @ A @ C @ T2 ) ) ) ).
% SIGT.simps(5)
thf(fact_54_type__definition__strm,axiom,
! [C: $tType,A: $tType] : ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% type_definition_strm
thf(fact_55_good__interp__inverse,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
= Y ) ) ) ).
% good_interp_inverse
thf(fact_56_raw__interp__induct,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P2: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P2 @ ( freche229654227interp @ A @ B @ C @ X3 ) )
=> ( P2 @ Y ) ) ) ) ).
% raw_interp_induct
thf(fact_57_raw__interp__cases,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ~ ! [X3: frechet_good_interp @ A @ B @ C] :
( Y
!= ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ).
% raw_interp_cases
thf(fact_58_raw__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X2: frechet_good_interp @ A @ B @ C] : ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X2 ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% raw_interp
thf(fact_59_good__interp__inject,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X2: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X2 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( ( freche227871258interp @ A @ B @ C @ X2 )
= ( freche227871258interp @ A @ B @ C @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% good_interp_inject
thf(fact_60_good__interp__induct,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [P2: ( frechet_good_interp @ A @ B @ C ) > $o,X2: frechet_good_interp @ A @ B @ C] :
( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( P2 @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
=> ( P2 @ X2 ) ) ) ).
% good_interp_induct
thf(fact_61_good__interp__cases,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X2: frechet_good_interp @ A @ B @ C] :
~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( X2
= ( freche227871258interp @ A @ B @ C @ Y3 ) )
=> ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% good_interp_cases
thf(fact_62_subsetI,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% subsetI
thf(fact_63_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% subset_antisym
thf(fact_64_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X: A] : ( sup_sup @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% sup_apply
thf(fact_65_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_66_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ X2 )
= X2 ) ) ).
% sup_idem
thf(fact_67_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_68_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
= ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_left_idem
thf(fact_69_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_70_UnCI,axiom,
! [A: $tType,C3: A,B2: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ A4 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnCI
thf(fact_71_Un__iff,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( ( member @ A @ C3 @ A4 )
| ( member @ A @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_72_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_73_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_74_in__mono,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_75_subsetD,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_76_equalityE,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ) ).
% equalityE
thf(fact_77_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
! [X: A] :
( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_78_equalityD1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B2 ) ) ).
% equalityD1
thf(fact_79_equalityD2,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( A4 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A4 ) ) ).
% equalityD2
thf(fact_80_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A5 )
=> ( member @ A @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_81_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_82_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_83_subset__trans,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_84_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_85_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_86_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
= ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_87_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_88_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_89_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X: A,Y2: A] : ( sup_sup @ A @ Y2 @ X ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_90_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X: A] : ( sup_sup @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% sup_fun_def
thf(fact_91_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,K: A,A2: A,B3: A] :
( ( A4
= ( sup_sup @ A @ K @ A2 ) )
=> ( ( sup_sup @ A @ A4 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_92_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A2: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A2 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_93_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C3 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_94_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_95_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B5: A] : ( sup_sup @ A @ B5 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_96_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X: A,Y2: A] : ( sup_sup @ A @ Y2 @ X ) ) ) ) ).
% sup_commute
thf(fact_97_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A2: A,C3: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C3 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_98_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_99_UnE,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( ~ ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_100_UnI1,axiom,
! [A: $tType,C3: A,A4: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI1
thf(fact_101_UnI2,axiom,
! [A: $tType,C3: A,B2: set @ A,A4: set @ A] :
( ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ) ).
% UnI2
thf(fact_102_bex__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P2: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
& ( P2 @ X ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A4 )
& ( P2 @ X ) )
| ? [X: A] :
( ( member @ A @ X @ B2 )
& ( P2 @ X ) ) ) ) ).
% bex_Un
thf(fact_103_ball__Un,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,P2: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ( P2 @ X ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( P2 @ X ) )
& ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( P2 @ X ) ) ) ) ).
% ball_Un
thf(fact_104_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_105_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_106_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% Un_commute
thf(fact_107_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_left_absorb
thf(fact_108_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_109_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A4 )
& ( P2 @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_110_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
| ( member @ A @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_111_Collect__disj__eq,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
| ( Q @ X ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_112_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_113_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_114_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A,X2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X2 )
=> ~ ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ).
% le_supE
thf(fact_115_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ X2 )
=> ( ( ord_less_eq @ A @ B3 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ X2 ) ) ) ) ).
% le_supI
thf(fact_116_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge1
thf(fact_117_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge2
thf(fact_118_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_119_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,B3: A,A2: A] :
( ( ord_less_eq @ A @ X2 @ B3 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_120_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,D3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ( ord_less_eq @ A @ D3 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D3 ) @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_121_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C3: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ ( sup_sup @ A @ C3 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_122_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X2 ) ) ) ) ).
% sup_least
thf(fact_123_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y2: A] :
( ( sup_sup @ A @ X @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_124_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_125_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% sup.orderI
thf(fact_126_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X2: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_127_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B3 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_128_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( sup_sup @ A @ A2 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_129_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( sup_sup @ A @ X2 @ Y )
= X2 ) ) ) ).
% sup_absorb1
thf(fact_130_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( sup_sup @ A @ X2 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_131_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A2 )
=> ~ ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_132_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_133_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( A6
= ( sup_sup @ A @ A6 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_134_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B3: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_135_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A2: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_136_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( sup_sup @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_137_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( sup_sup @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_138_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_139_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B3: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_140_fst__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
=> ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_mono
thf(fact_141_Un__mono,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B2: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_142_Un__least,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_143_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper1
thf(fact_144_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) ) ).
% Un_upper2
thf(fact_145_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_146_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B2 )
= A4 ) ) ).
% Un_absorb2
thf(fact_147_subset__UnE,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B2 ) )
=> ~ ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ A4 )
=> ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ B2 )
=> ( C2
!= ( sup_sup @ ( set @ A ) @ A7 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_148_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_149_type__definition__good__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% type_definition_good_interp
thf(fact_150_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_151_sterm__continuous_H,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,Theta: trm @ A @ C,S: set @ ( finite_Cartesian_vec @ real @ C )] :
( ( denota2077489681interp @ A @ B @ C @ I )
=> ( ( dfree @ A @ C @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ C ) @ real @ S @ ( denota126604975rm_sem @ A @ B @ C @ I @ Theta ) ) ) ) ) ).
% sterm_continuous'
thf(fact_152_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A2: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A2 ) )
= ( F1 @ A2 ) ) ).
% old.sum.simps(7)
thf(fact_153_strm_Orep__transfer,axiom,
! [C: $tType,A: $tType] :
( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
@ ^ [Y4: trm @ A @ C,Z3: trm @ A @ C] : Y4 = Z3
@ ^ [X: trm @ A @ C] : X
@ ( frechet_raw_term @ A @ C ) ) ).
% strm.rep_transfer
thf(fact_154_good__interp_Orep__transfer,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
@ ^ [Y4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z3: denota1663640101rp_ext @ A @ B @ C @ product_unit] : Y4 = Z3
@ ^ [X: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X
@ ( freche229654227interp @ A @ B @ C ) ) ) ).
% good_interp.rep_transfer
thf(fact_155_good__interp_Odomain,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
= ( denota2077489681interp @ A @ B @ C ) ) ) ).
% good_interp.domain
thf(fact_156_strm_Odomain,axiom,
! [C: $tType,A: $tType] :
( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
= ( dfree @ A @ C ) ) ).
% strm.domain
thf(fact_157_predicate1I,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% predicate1I
thf(fact_158_sup1CI,axiom,
! [A: $tType,B2: A > $o,X2: A,A4: A > $o] :
( ( ~ ( B2 @ X2 )
=> ( A4 @ X2 ) )
=> ( sup_sup @ ( A > $o ) @ A4 @ B2 @ X2 ) ) ).
% sup1CI
thf(fact_159_fun__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,C2: A > B > $o,A4: A > B > $o,B2: C > D > $o,D4: C > D > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ C2 @ A4 )
=> ( ( ord_less_eq @ ( C > D > $o ) @ B2 @ D4 )
=> ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C2 @ D4 ) ) ) ) ).
% fun_mono
thf(fact_160_predicate1D,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,X2: A] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ).
% predicate1D
thf(fact_161_rev__predicate1D,axiom,
! [A: $tType,P2: A > $o,X2: A,Q: A > $o] :
( ( P2 @ X2 )
=> ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( Q @ X2 ) ) ) ).
% rev_predicate1D
thf(fact_162_sup1I2,axiom,
! [A: $tType,B2: A > $o,X2: A,A4: A > $o] :
( ( B2 @ X2 )
=> ( sup_sup @ ( A > $o ) @ A4 @ B2 @ X2 ) ) ).
% sup1I2
thf(fact_163_sup1I1,axiom,
! [A: $tType,A4: A > $o,X2: A,B2: A > $o] :
( ( A4 @ X2 )
=> ( sup_sup @ ( A > $o ) @ A4 @ B2 @ X2 ) ) ).
% sup1I1
thf(fact_164_sup1E,axiom,
! [A: $tType,A4: A > $o,B2: A > $o,X2: A] :
( ( sup_sup @ ( A > $o ) @ A4 @ B2 @ X2 )
=> ( ~ ( A4 @ X2 )
=> ( B2 @ X2 ) ) ) ).
% sup1E
thf(fact_165_apply__rsp_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X2: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X2 @ Y )
=> ( R2 @ ( F @ X2 ) @ ( G @ Y ) ) ) ) ).
% apply_rsp'
thf(fact_166_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A4: set @ A,T4: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A4 )
=> ( ( T4
= ( ^ [X: A,Y2: B] :
( X
= ( Rep @ Y2 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T4
@ ^ [Y4: A,Z3: A] : Y4 = Z3
@ ^ [X: A] : X
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_167_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_168_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_169_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_170_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B3: A] :
( ! [A8: A,B7: A] :
( ( ord_less_eq @ A @ A8 @ B7 )
=> ( P2 @ A8 @ B7 ) )
=> ( ! [A8: A,B7: A] :
( ( P2 @ B7 @ A8 )
=> ( P2 @ A8 @ B7 ) )
=> ( P2 @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_171_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_172_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_173_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_174_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_175_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C3: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_176_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_177_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_178_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_179_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_180_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_181_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_182_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_183_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_184_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_185_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_186_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_187_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_188_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_189_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_190_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_191_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_192_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_193_sterm__continuous,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,Theta: trm @ A @ C] :
( ( denota2077489681interp @ A @ B @ C @ I )
=> ( ( dfree @ A @ C @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ C ) @ real @ ( top_top @ ( set @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( denota126604975rm_sem @ A @ B @ C @ I @ Theta ) ) ) ) ) ).
% sterm_continuous
thf(fact_194_sup__Un__eq,axiom,
! [A: $tType,R3: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R3 )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ^ [X: A] : ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_195_pred__subset__eq,axiom,
! [A: $tType,R3: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R3 )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_196_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_197_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_198_predicate2I,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% predicate2I
thf(fact_199_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_200_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X2 )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_201_fst__top,axiom,
! [B: $tType,A: $tType] :
( ( ( top @ A )
& ( top @ B ) )
=> ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
= ( top_top @ A ) ) ) ).
% fst_top
thf(fact_202_predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X2: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
=> ( ( P2 @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) ).
% predicate2D
thf(fact_203_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X2: A,Y: B,Q: A > B > $o] :
( ( P2 @ X2 @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
=> ( Q @ X2 @ Y ) ) ) ).
% rev_predicate2D
thf(fact_204_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_205_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_206_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_207_subset__UNIV,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_208_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_209_Un__UNIV__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_210_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_211_UNIV__eq__I,axiom,
! [A: $tType,A4: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A4 )
=> ( ( top_top @ ( set @ A ) )
= A4 ) ) ).
% UNIV_eq_I
thf(fact_212_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $true ) ) ).
% UNIV_def
thf(fact_213_DomainpE,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,A2: A] :
( ( domainp @ A @ B @ R4 @ A2 )
=> ~ ! [B7: B] :
~ ( R4 @ A2 @ B7 ) ) ).
% DomainpE
thf(fact_214_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,A2: A] :
( ( domainp @ A @ B @ R4 @ A2 )
=> ~ ! [B7: B] :
~ ( R4 @ A2 @ B7 ) ) ).
% Domainp.cases
thf(fact_215_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R5: A > B > $o,A6: A] :
? [B5: A,C4: B] :
( ( A6 = B5 )
& ( R5 @ B5 @ C4 ) ) ) ) ).
% Domainp.simps
thf(fact_216_Domainp_ODomainI,axiom,
! [B: $tType,A: $tType,R4: A > B > $o,A2: A,B3: B] :
( ( R4 @ A2 @ B3 )
=> ( domainp @ A @ B @ R4 @ A2 ) ) ).
% Domainp.DomainI
thf(fact_217_Domainp_Oinducts,axiom,
! [B: $tType,A: $tType,R4: A > B > $o,X2: A,P2: A > $o] :
( ( domainp @ A @ B @ R4 @ X2 )
=> ( ! [A8: A,B7: B] :
( ( R4 @ A8 @ B7 )
=> ( P2 @ A8 ) )
=> ( P2 @ X2 ) ) ) ).
% Domainp.inducts
thf(fact_218_fun_Orel__mono,axiom,
! [D: $tType,B: $tType,A: $tType,R3: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ Ra )
=> ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z3: D] : Y4 = Z3
@ R3 )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z3: D] : Y4 = Z3
@ Ra ) ) ) ).
% fun.rel_mono
thf(fact_219_continuous__on__cases__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S2: set @ A,H: A > real,A2: real,F: A > B,G: A > B] :
( ( topolo2071040574ous_on @ A @ B
@ ( collect @ A
@ ^ [T3: A] :
( ( member @ A @ T3 @ S2 )
& ( ord_less_eq @ real @ ( H @ T3 ) @ A2 ) ) )
@ F )
=> ( ( topolo2071040574ous_on @ A @ B
@ ( collect @ A
@ ^ [T3: A] :
( ( member @ A @ T3 @ S2 )
& ( ord_less_eq @ real @ A2 @ ( H @ T3 ) ) ) )
@ G )
=> ( ( topolo2071040574ous_on @ A @ real @ S2 @ H )
=> ( ! [T5: A] :
( ( member @ A @ T5 @ S2 )
=> ( ( ( H @ T5 )
= A2 )
=> ( ( F @ T5 )
= ( G @ T5 ) ) ) )
=> ( topolo2071040574ous_on @ A @ B @ S2
@ ^ [T3: A] : ( if @ B @ ( ord_less_eq @ real @ ( H @ T3 ) @ A2 ) @ ( F @ T3 ) @ ( G @ T3 ) ) ) ) ) ) ) ) ).
% continuous_on_cases_le
thf(fact_220_continuous__on__cases__1,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S2: set @ real,A2: real,F: real > A,G: real > A] :
( ( topolo2071040574ous_on @ real @ A
@ ( collect @ real
@ ^ [T3: real] :
( ( member @ real @ T3 @ S2 )
& ( ord_less_eq @ real @ T3 @ A2 ) ) )
@ F )
=> ( ( topolo2071040574ous_on @ real @ A
@ ( collect @ real
@ ^ [T3: real] :
( ( member @ real @ T3 @ S2 )
& ( ord_less_eq @ real @ A2 @ T3 ) ) )
@ G )
=> ( ( ( member @ real @ A2 @ S2 )
=> ( ( F @ A2 )
= ( G @ A2 ) ) )
=> ( topolo2071040574ous_on @ real @ A @ S2
@ ^ [T3: real] : ( if @ A @ ( ord_less_eq @ real @ T3 @ A2 ) @ ( F @ T3 ) @ ( G @ T3 ) ) ) ) ) ) ) ).
% continuous_on_cases_1
thf(fact_221_agree__UNIV__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( denotational_Vagree @ A @ Nu @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
=> ( Nu = Omega ) ) ) ).
% agree_UNIV_eq
thf(fact_222_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_223_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_224_fun_Orel__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,E2: $tType,D: $tType,Sa: A > C > $o,Sc2: B > E2 > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > E2 > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E2 ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E2 > $o ) @ Sa
@ ( bNF_rel_fun @ B @ E2 @ $o @ $o @ Sc2
@ ^ [Y4: $o,Z3: $o] : Y4 = Z3 ) )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E2 ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y4: D,Z3: D] : Y4 = Z3
@ Sa )
@ ( bNF_rel_fun @ ( D > B ) @ ( D > E2 ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ B @ E2
@ ^ [Y4: D,Z3: D] : Y4 = Z3
@ Sc2 )
@ ^ [Y4: $o,Z3: $o] : Y4 = Z3 ) )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z3: D] : Y4 = Z3 )
@ ( bNF_rel_fun @ D @ D @ C @ E2
@ ^ [Y4: D,Z3: D] : Y4 = Z3 ) ) ).
% fun.rel_transfer
thf(fact_225_fun_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X2: D > B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( bNF_rel_fun @ D @ D @ B @ B
@ ^ [Y4: D,Z3: D] : Y4 = Z3
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl
thf(fact_226_fun_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y4: D,Z3: D] : Y4 = Z3
@ ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [Y4: D > A,Z3: D > A] : Y4 = Z3 ) ) ).
% fun.rel_eq
thf(fact_227_frechet__continuous,axiom,
! [I: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,Theta: trm @ sf @ sz] :
( ( denota2077489681interp @ sf @ sc @ sz @ I )
=> ( ( dfree @ sf @ sz @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ sz ) @ ( bounde2145540817linfun @ ( finite_Cartesian_vec @ real @ sz ) @ real ) @ ( top_top @ ( set @ ( finite_Cartesian_vec @ real @ sz ) ) ) @ ( frechet_blin_frechet @ sf @ sc @ sz @ ( freche227871258interp @ sf @ sc @ sz @ I ) @ ( frechet_simple_term @ sf @ sz @ Theta ) ) ) ) ) ).
% frechet_continuous
thf(fact_228_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o,F: A > C,G: B > D] :
( ! [X3: A,Y3: B] :
( ( A4 @ X3 @ Y3 )
=> ( B2 @ ( F @ X3 ) @ ( G @ Y3 ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F @ G ) ) ).
% rel_funI
thf(fact_229_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_230_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F @ G )
=> ( ( A4 @ X2 @ Y )
=> ( B2 @ ( F @ X2 ) @ ( G @ Y ) ) ) ) ).
% rel_funD
thf(fact_231_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A4: C > D > $o,F: A > C,G: B > D,Y5: A > B > $o,B2: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A4 @ F @ G )
=> ( ! [X3: A,Y3: B] :
( ( Y5 @ X3 @ Y3 )
=> ( X4 @ X3 @ Y3 ) )
=> ( ! [X3: C,Y3: D] :
( ( A4 @ X3 @ Y3 )
=> ( B2 @ X3 @ Y3 ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_232_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A4: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
( ! [X3: A,Y3: B] :
( ( Y5 @ X3 @ Y3 )
=> ( X4 @ X3 @ Y3 ) )
=> ( ! [X3: C,Y3: D] :
( ( A4 @ X3 @ Y3 )
=> ( B2 @ X3 @ Y3 ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A4 @ F @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_233_ge__eq__refl,axiom,
! [A: $tType,R3: A > A > $o,X2: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z3: A] : Y4 = Z3
@ R3 )
=> ( R3 @ X2 @ X2 ) ) ).
% ge_eq_refl
thf(fact_234_refl__ge__eq,axiom,
! [A: $tType,R3: A > A > $o] :
( ! [X3: A] : ( R3 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z3: A] : Y4 = Z3
@ R3 ) ) ).
% refl_ge_eq
thf(fact_235_let__rsp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
@ ^ [S3: A,F2: A > C] : ( F2 @ S3 )
@ ^ [S3: B,F2: B > D] : ( F2 @ S3 ) ) ).
% let_rsp
thf(fact_236_Let__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A4 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 ) @ B2 )
@ ^ [S3: A,F2: A > C] : ( F2 @ S3 )
@ ^ [S3: B,F2: B > D] : ( F2 @ S3 ) ) ).
% Let_transfer
thf(fact_237_rel__funD2,axiom,
! [B: $tType,C: $tType,A: $tType,A4: A > A > $o,B2: B > C > $o,F: A > B,G: A > C,X2: A] :
( ( bNF_rel_fun @ A @ A @ B @ C @ A4 @ B2 @ F @ G )
=> ( ( A4 @ X2 @ X2 )
=> ( B2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% rel_funD2
thf(fact_238_rel__funE,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A4 @ B2 @ F @ G )
=> ( ( A4 @ X2 @ Y )
=> ( B2 @ ( F @ X2 ) @ ( G @ Y ) ) ) ) ).
% rel_funE
thf(fact_239_Domainp__refl,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( domainp @ A @ B ) ) ).
% Domainp_refl
thf(fact_240_Domainp__iff,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [T6: A > B > $o,X: A] :
? [X5: B] : ( T6 @ X @ X5 ) ) ) ).
% Domainp_iff
thf(fact_241_Domain__eq__top,axiom,
! [A: $tType] :
( ( domainp @ A @ A
@ ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( top_top @ ( A > $o ) ) ) ).
% Domain_eq_top
thf(fact_242_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_243_continuous__on__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B )
& ( topolo503727757_space @ C ) )
=> ! [S2: set @ A,F: A > ( product_prod @ B @ C )] :
( ( topolo2071040574ous_on @ A @ ( product_prod @ B @ C ) @ S2 @ F )
=> ( topolo2071040574ous_on @ A @ B @ S2
@ ^ [X: A] : ( product_fst @ B @ C @ ( F @ X ) ) ) ) ) ).
% continuous_on_fst
thf(fact_244_continuous__on__cong,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S2: set @ A,T7: set @ A,F: A > B,G: A > B] :
( ( S2 = T7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ T7 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( topolo2071040574ous_on @ A @ B @ S2 @ F )
= ( topolo2071040574ous_on @ A @ B @ T7 @ G ) ) ) ) ) ).
% continuous_on_cong
thf(fact_245_continuous__on__discrete,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo2133971006pology @ A )
& ( topolo503727757_space @ B ) )
=> ! [A4: set @ A,F: A > B] : ( topolo2071040574ous_on @ A @ B @ A4 @ F ) ) ).
% continuous_on_discrete
thf(fact_246_continuous__on__id,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S2: set @ A] :
( topolo2071040574ous_on @ A @ A @ S2
@ ^ [X: A] : X ) ) ).
% continuous_on_id
thf(fact_247_continuous__on__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S2: set @ A,C3: B] :
( topolo2071040574ous_on @ A @ B @ S2
@ ^ [X: A] : C3 ) ) ).
% continuous_on_const
thf(fact_248_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S2: set @ A,F: A > B,T7: set @ A] :
( ( topolo2071040574ous_on @ A @ B @ S2 @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ T7 @ S2 )
=> ( topolo2071040574ous_on @ A @ B @ T7 @ F ) ) ) ) ).
% continuous_on_subset
thf(fact_249_continuous__on__product__then__coordinatewise__UNIV,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ C ) )
=> ! [F: A > B > C,I2: B] :
( ( topolo2071040574ous_on @ A @ ( B > C ) @ ( top_top @ ( set @ A ) ) @ F )
=> ( topolo2071040574ous_on @ A @ C @ ( top_top @ ( set @ A ) )
@ ^ [X: A] : ( F @ X @ I2 ) ) ) ) ).
% continuous_on_product_then_coordinatewise_UNIV
thf(fact_250_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F3: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ? [X5: B] : ( F3 @ X5 ) )
= ( ? [B5: A] : ( F3 @ ( Rep @ B5 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_251_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P2 ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_252_continuous__on__product__then__coordinatewise,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ C ) )
=> ! [S: set @ A,F: A > B > C,I2: B] :
( ( topolo2071040574ous_on @ A @ ( B > C ) @ S @ F )
=> ( topolo2071040574ous_on @ A @ C @ S
@ ^ [X: A] : ( F @ X @ I2 ) ) ) ) ).
% continuous_on_product_then_coordinatewise
thf(fact_253_continuous__on__coordinatewise__then__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ C ) )
=> ! [S: set @ A,F: A > B > C] :
( ! [I3: B] :
( topolo2071040574ous_on @ A @ C @ S
@ ^ [X: A] : ( F @ X @ I3 ) )
=> ( topolo2071040574ous_on @ A @ ( B > C ) @ S @ F ) ) ) ).
% continuous_on_coordinatewise_then_product
thf(fact_254_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S2: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X3: B] :
( S2
!= ( Abs @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_255_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E2: $tType,F4: $tType,A: $tType,R3: A > B > $o,S: C > E2 > $o,T4: D > F4 > $o,F: A > C > D,G: B > E2 > F4] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E2 > F4 ) @ R3 @ ( bNF_rel_fun @ C @ E2 @ D @ F4 @ S @ T4 ) @ F @ G )
= ( ! [X: A,Y2: B] :
( ( R3 @ X @ Y2 )
=> ( bNF_rel_fun @ C @ E2 @ D @ F4 @ S @ T4 @ ( F @ X ) @ ( G @ Y2 ) ) ) ) ) ).
% rel_fun_def_butlast
% Subclasses (5)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( order @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (87)
thf(tcon_Bounded__Linear__Function_Oblinfun___Real__Vector__Spaces_Oreal__normed__vector,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( real_V55928688vector @ A10 ) )
=> ( real_V55928688vector @ ( bounde2145540817linfun @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_1,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( finite_finite @ A10 ) )
=> ( real_V55928688vector @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Oprod___Lattices_Obounded__lattice_2,axiom,
! [A9: $tType,A10: $tType] :
( ( ( bounded_lattice @ A9 )
& ( bounded_lattice @ A10 ) )
=> ( bounded_lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_3,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A10 ) )
=> ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_4,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( real_V55928688vector @ A10 ) )
=> ( real_V55928688vector @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_5,axiom,
ordere890947078_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector_6,axiom,
real_V55928688vector @ real ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_7,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_8,axiom,
! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_9,axiom,
! [A9: $tType,A10: $tType] :
( ( bounded_lattice @ A10 )
=> ( bounded_lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A9: $tType,A10: $tType] :
( ( topolo503727757_space @ A10 )
=> ( topolo503727757_space @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A9: $tType,A10: $tType] :
( ( bounded_lattice @ A10 )
=> ( bounded_lattice_top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( semilattice_sup @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A9: $tType,A10: $tType] :
( ( order_top @ A10 )
=> ( order_top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A9: $tType,A10: $tType] :
( ( lattice @ A10 )
=> ( lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 )
=> ( top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( sup @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_10,axiom,
! [A9: $tType] : ( bounded_lattice_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_11,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_12,axiom,
! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_14,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_15,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_16,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_17,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_18,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_19,axiom,
! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_20,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology,axiom,
topolo2133971006pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_21,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_22,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_23,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_24,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_25,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_26,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_27,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_28,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_29,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_30,axiom,
sup @ $o ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_31,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_32,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_33,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_34,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_35,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_36,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_37,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Lattices_Osup_38,axiom,
sup @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_39,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_40,axiom,
! [A9: $tType,A10: $tType] :
( ( ( topolo503727757_space @ A9 )
& ( topolo503727757_space @ A10 ) )
=> ( topolo503727757_space @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Obounded__lattice__top_41,axiom,
! [A9: $tType,A10: $tType] :
( ( ( bounded_lattice @ A9 )
& ( bounded_lattice @ A10 ) )
=> ( bounded_lattice_top @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_42,axiom,
! [A9: $tType,A10: $tType] :
( ( ( semilattice_sup @ A9 )
& ( semilattice_sup @ A10 ) )
=> ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_43,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order_top @ A9 )
& ( order_top @ A10 ) )
=> ( order_top @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_44,axiom,
! [A9: $tType,A10: $tType] :
( ( ( preorder @ A9 )
& ( preorder @ A10 ) )
=> ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_45,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Olattice_46,axiom,
! [A9: $tType,A10: $tType] :
( ( ( lattice @ A9 )
& ( lattice @ A10 ) )
=> ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_47,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( order @ A10 ) )
=> ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_48,axiom,
! [A9: $tType,A10: $tType] :
( ( ( top @ A9 )
& ( top @ A10 ) )
=> ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_49,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( ord @ A10 ) )
=> ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osup_50,axiom,
! [A9: $tType,A10: $tType] :
( ( ( sup @ A9 )
& ( sup @ A10 ) )
=> ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_51,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_52,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_53,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_54,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_55,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_56,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_57,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_58,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_59,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_60,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_61,axiom,
sup @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_62,axiom,
! [A9: $tType,A10: $tType] :
( ( ( topolo503727757_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( topolo503727757_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_63,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_64,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A10 ) )
=> ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_65,axiom,
! [A9: $tType,A10: $tType] :
( ( ( linorder @ A9 )
& ( cARD_1 @ A10 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_66,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Olattice_67,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( lattice @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_68,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A10 ) )
=> ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_69,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( finite_finite @ A10 ) )
=> ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osup_70,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Bounded__Linear__Function_Oblinfun___Topological__Spaces_Otopological__space_71,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( real_V55928688vector @ A10 ) )
=> ( topolo503727757_space @ ( bounde2145540817linfun @ A9 @ A10 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Free types (7)
thf(tfree_0,hypothesis,
finite_finite @ b ).
thf(tfree_1,hypothesis,
finite_finite @ a ).
thf(tfree_2,hypothesis,
finite_finite @ c ).
thf(tfree_3,hypothesis,
finite_finite @ sc ).
thf(tfree_4,hypothesis,
finite_finite @ sz ).
thf(tfree_5,hypothesis,
linorder @ sz ).
thf(tfree_6,hypothesis,
finite_finite @ sf ).
% Conjectures (1)
thf(conj_0,conjecture,
( denotational_Iagree @ a @ c @ b @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ c @ b ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ c @ b )] :
? [X: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ c @ b ) @ X ) )
& ( member @ a @ X @ ( static_SIGT @ a @ b @ theta_1 ) ) ) ) ) ).
%------------------------------------------------------------------------------