TPTP Problem File: ITP038^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP038^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_231__7211966_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_231__7211966_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 558 ( 170 unt; 200 typ; 0 def)
% Number of atoms : 896 ( 365 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2644 ( 54 ~; 1 |; 87 &;2190 @)
% ( 0 <=>; 312 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 45 ( 44 usr)
% Number of type conns : 550 ( 550 >; 0 *; 0 +; 0 <<)
% Number of symbols : 159 ( 156 usr; 14 con; 0-10 aty)
% Number of variables : 1070 ( 122 ^; 925 !; 23 ?;1070 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:01.725
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
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% Explicit typings (156)
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collec1767975749t_unit: ( denota610675952t_unit > $o ) > set_De208295462t_unit ).
thf(sy_c_Set_OCollect_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J,type,
collec1002602816eal_sz: ( finite824932053eal_sz > $o ) > set_Fi291318197eal_sz ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_J,type,
collec1199592193_sc_sz: ( produc1821101996_sc_sz > $o ) > set_Pr1169339874_sc_sz ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mt__Sum____Type__Osum_Itf__b_Mtf__c_J_J,type,
collec2086029184um_b_c: ( sum_su737395349um_b_c > $o ) > set_Su1783761653um_b_c ).
thf(sy_c_Set_OCollect_001t__Syntax__Otrm_Itf__a_Mtf__c_J,type,
collect_trm_a_c: ( trm_a_c > $o ) > set_trm_a_c ).
thf(sy_c_Set_OCollect_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J,type,
collect_trm_sf_sz: ( trm_sf_sz > $o ) > set_trm_sf_sz ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Static__Semantics_OFVDiff_001tf__a_001tf__c,type,
static_FVDiff_a_c: trm_a_c > set_Sum_sum_c_c ).
thf(sy_c_Static__Semantics_OFVDiff_001tf__sf_001tf__sz,type,
static_FVDiff_sf_sz: trm_sf_sz > set_Sum_sum_sz_sz ).
thf(sy_c_Static__Semantics_OFVT_001tf__a_001tf__c,type,
static_FVT_a_c: trm_a_c > set_Sum_sum_c_c ).
thf(sy_c_Static__Semantics_OFVT_001tf__sf_001tf__sz,type,
static_FVT_sf_sz: trm_sf_sz > set_Sum_sum_sz_sz ).
thf(sy_c_Static__Semantics_OSIGT_001tf__a_001tf__c,type,
static_SIGT_a_c: trm_a_c > set_a ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Sum____Type__Osum_Itf__b_Mtf__c_J,type,
sum_In2139678582um_b_c: a > sum_su737395349um_b_c ).
thf(sy_c_Syntax_OPredicational_001tf__sc_001tf__sf_001tf__sz,type,
predic913887884_sf_sz: sc > formula_sf_sc_sz ).
thf(sy_c_Syntax_Odfree_001tf__a_001tf__c,type,
dfree_a_c: trm_a_c > $o ).
thf(sy_c_Syntax_Odfree_001tf__sf_001tf__sz,type,
dfree_sf_sz: trm_sf_sz > $o ).
thf(sy_c_Syntax_Oids_OP_001tf__sc_001tf__sf_001tf__sz,type,
p_sc_sf_sz: sc > formula_sf_sc_sz ).
thf(sy_c_Syntax_Oids_Oempty_001tf__sz_001tf__sf,type,
empty_sz_sf: sz > trm_sf_sz ).
thf(sy_c_Syntax_Oids_Of0_001tf__sf_001tf__sz,type,
f0_sf_sz: sf > trm_sf_sz ).
thf(sy_c_Syntax_Otrm_OFunction_001tf__a_001tf__c,type,
function_a_c: a > ( c > trm_a_c ) > trm_a_c ).
thf(sy_c_Syntax_Otrm_OFunction_001tf__sf_001tf__sz,type,
function_sf_sz: sf > ( sz > trm_sf_sz ) > trm_sf_sz ).
thf(sy_c_Syntax_Otrm_OPlus_001tf__a_001tf__c,type,
plus_a_c: trm_a_c > trm_a_c > trm_a_c ).
thf(sy_c_Syntax_Otrm_OPlus_001tf__sf_001tf__sz,type,
plus_sf_sz: trm_sf_sz > trm_sf_sz > trm_sf_sz ).
thf(sy_c_Syntax_Otrm_OTimes_001tf__a_001tf__c,type,
times_a_c: trm_a_c > trm_a_c > trm_a_c ).
thf(sy_c_Syntax_Otrm_OTimes_001tf__sf_001tf__sz,type,
times_sf_sz: trm_sf_sz > trm_sf_sz > trm_sf_sz ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J,type,
topolo885751137z_real: set_Fi291318197eal_sz > ( finite824932053eal_sz > bounde472938360z_real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_001t__Product____Type__Oprod_It__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_Mt__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_J,type,
topolo1027477648z_real: set_Fi291318197eal_sz > ( finite824932053eal_sz > produc1892666919z_real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_001t__Real__Oreal,type,
topolo1348430467z_real: set_Fi291318197eal_sz > ( finite824932053eal_sz > real ) > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Frechet____Correctness__Oids__Ogood____interp_Itf__sf_Mtf__sc_Mtf__sz_J_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J,type,
type_d1092523951t_unit: ( freche2095075217_sc_sz > denota610675952t_unit ) > ( denota610675952t_unit > freche2095075217_sc_sz ) > set_De208295462t_unit > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Frechet____Correctness__Oids__Ostrm_Itf__a_Mtf__c_J_001t__Syntax__Otrm_Itf__a_Mtf__c_J,type,
type_d1238563341rm_a_c: ( frechet_strm_a_c > trm_a_c ) > ( trm_a_c > frechet_strm_a_c ) > set_trm_a_c > $o ).
thf(sy_c_Typedef_Otype__definition_001t__Frechet____Correctness__Oids__Ostrm_Itf__sf_Mtf__sz_J_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J,type,
type_d46389773_sf_sz: ( frechet_strm_sf_sz > trm_sf_sz ) > ( trm_sf_sz > frechet_strm_sf_sz ) > set_trm_sf_sz > $o ).
thf(sy_c_member_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__a_Mtf__b_Mtf__c_Mt__Product____Type__Ounit_J,type,
member1777622435t_unit: denota231621370t_unit > set_De455519194t_unit > $o ).
thf(sy_c_member_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J,type,
member754147591t_unit: denota610675952t_unit > set_De208295462t_unit > $o ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J,type,
member1693951742eal_sz: finite824932053eal_sz > set_Fi291318197eal_sz > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_J,type,
member1057565379_sc_sz: produc1821101996_sc_sz > set_Pr1169339874_sc_sz > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Sum____Type__Osum_Itf__b_Mtf__c_J_J,type,
member1893232190um_b_c: sum_su737395349um_b_c > set_Su1783761653um_b_c > $o ).
thf(sy_c_member_001t__Syntax__Otrm_Itf__a_Mtf__c_J,type,
member_trm_a_c: trm_a_c > set_trm_a_c > $o ).
thf(sy_c_member_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J,type,
member_trm_sf_sz: trm_sf_sz > set_trm_sf_sz > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: denota231621370t_unit ).
thf(sy_v_J,type,
j: denota231621370t_unit ).
thf(sy_v__092_060nu_062,type,
nu: produc190496183real_c ).
thf(sy_v__092_060nu_062_H,type,
nu2: produc190496183real_c ).
thf(sy_v_t1____,type,
t1: trm_a_c ).
thf(sy_v_t2____,type,
t2: trm_a_c ).
% Relevant facts (354)
thf(fact_0_agree__plus2,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( plus_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T2 ) ) ) ).
% agree_plus2
thf(fact_1_agree__plus2,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( plus_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T2 ) ) ) ).
% agree_plus2
thf(fact_2_agree,axiom,
denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ ( plus_a_c @ t1 @ t2 ) ) ).
% agree
thf(fact_3_agree1,axiom,
denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ t1 ) ).
% agree1
thf(fact_4_dfree2,axiom,
dfree_a_c @ t2 ).
% dfree2
thf(fact_5_agree__plus1,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( plus_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T1 ) ) ) ).
% agree_plus1
thf(fact_6_agree__plus1,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( plus_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T1 ) ) ) ).
% agree_plus1
thf(fact_7_agree__comm,axiom,
! [A: produc1149990247eal_sz,B: produc1149990247eal_sz,V: set_Sum_sum_sz_sz] :
( ( denota102713844ree_sz @ A @ B @ V )
=> ( denota102713844ree_sz @ B @ A @ V ) ) ).
% agree_comm
thf(fact_8_agree__comm,axiom,
! [A: produc190496183real_c,B: produc190496183real_c,V: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ A @ B @ V )
=> ( denota1997846518gree_c @ B @ A @ V ) ) ).
% agree_comm
thf(fact_9_agree__refl,axiom,
! [Nu: produc1149990247eal_sz,A: set_Sum_sum_sz_sz] : ( denota102713844ree_sz @ Nu @ Nu @ A ) ).
% agree_refl
thf(fact_10_agree__refl,axiom,
! [Nu: produc190496183real_c,A: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu @ Nu @ A ) ).
% agree_refl
thf(fact_11_dfree1,axiom,
dfree_a_c @ t1 ).
% dfree1
thf(fact_12_raw__interp__inject,axiom,
! [X: freche2095075217_sc_sz,Y: freche2095075217_sc_sz] :
( ( ( freche1421597129_sc_sz @ X )
= ( freche1421597129_sc_sz @ Y ) )
= ( X = Y ) ) ).
% raw_interp_inject
thf(fact_13_raw__term__inject,axiom,
! [X: frechet_strm_sf_sz,Y: frechet_strm_sf_sz] :
( ( ( freche782854530_sf_sz @ X )
= ( freche782854530_sf_sz @ Y ) )
= ( X = Y ) ) ).
% raw_term_inject
thf(fact_14_raw__term__inject,axiom,
! [X: frechet_strm_a_c,Y: frechet_strm_a_c] :
( ( ( frechet_raw_term_a_c @ X )
= ( frechet_raw_term_a_c @ Y ) )
= ( X = Y ) ) ).
% raw_term_inject
thf(fact_15_agree__times1,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( times_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T1 ) ) ) ).
% agree_times1
thf(fact_16_agree__times1,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( times_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T1 ) ) ) ).
% agree_times1
thf(fact_17_agree__times2,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( times_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T2 ) ) ) ).
% agree_times2
thf(fact_18_agree__times2,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( times_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T2 ) ) ) ).
% agree_times2
thf(fact_19_agree__func,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,Var: sf,Args: sz > trm_sf_sz,I: sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( function_sf_sz @ Var @ Args ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( Args @ I ) ) ) ) ).
% agree_func
thf(fact_20_agree__func,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,Var: a,Args: c > trm_a_c,I: c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( function_a_c @ Var @ Args ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( Args @ I ) ) ) ) ).
% agree_func
thf(fact_21_P__def,axiom,
p_sc_sf_sz = predic913887884_sf_sz ).
% P_def
thf(fact_22_IH2,axiom,
( ( denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ t2 ) )
=> ( ( denota69106024_a_b_c @ i @ j
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ t2 ) ) ) ) )
=> ( ( denota229585092_a_b_c @ i @ t2 @ ( produc2010422875real_c @ nu ) @ ( produc314122909real_c @ nu ) )
= ( denota229585092_a_b_c @ j @ t2 @ ( produc2010422875real_c @ nu2 ) @ ( produc314122909real_c @ nu2 ) ) ) ) ) ).
% IH2
thf(fact_23_seq__sem_Ocases,axiom,
! [X: produc1821101996_sc_sz] :
~ ! [I2: denota610675952t_unit,S: produc866628903_sc_sz] :
( X
!= ( produc789536734_sc_sz @ I2 @ S ) ) ).
% seq_sem.cases
thf(fact_24_raw__term,axiom,
! [X: frechet_strm_a_c] : ( member_trm_a_c @ ( frechet_raw_term_a_c @ X ) @ ( collect_trm_a_c @ dfree_a_c ) ) ).
% raw_term
thf(fact_25_raw__term,axiom,
! [X: frechet_strm_sf_sz] : ( member_trm_sf_sz @ ( freche782854530_sf_sz @ X ) @ ( collect_trm_sf_sz @ dfree_sf_sz ) ) ).
% raw_term
thf(fact_26_raw__term__cases,axiom,
! [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_27_raw__term__cases,axiom,
! [Y: trm_sf_sz] :
( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ~ ! [X3: frechet_strm_sf_sz] :
( Y
!= ( freche782854530_sf_sz @ X3 ) ) ) ).
% raw_term_cases
thf(fact_28_raw__term__induct,axiom,
! [Y: trm_a_c,P: trm_a_c > $o] :
( ( member_trm_a_c @ Y @ ( collect_trm_a_c @ dfree_a_c ) )
=> ( ! [X3: frechet_strm_a_c] : ( P @ ( frechet_raw_term_a_c @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_term_induct
thf(fact_29_raw__term__induct,axiom,
! [Y: trm_sf_sz,P: trm_sf_sz > $o] :
( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ! [X3: frechet_strm_sf_sz] : ( P @ ( freche782854530_sf_sz @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_term_induct
thf(fact_30_coincidence__frechet,axiom,
! [Theta: trm_a_c,Nu: produc190496183real_c,Nu2: produc190496183real_c,I3: denota231621370t_unit] :
( ( dfree_a_c @ Theta )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ Theta ) )
=> ( ( denota229585092_a_b_c @ I3 @ Theta @ ( produc2010422875real_c @ Nu ) @ ( produc314122909real_c @ Nu ) )
= ( denota229585092_a_b_c @ I3 @ Theta @ ( produc2010422875real_c @ Nu2 ) @ ( produc314122909real_c @ Nu2 ) ) ) ) ) ).
% coincidence_frechet
thf(fact_31_coincidence__frechet,axiom,
! [Theta: trm_sf_sz,Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,I3: denota610675952t_unit] :
( ( dfree_sf_sz @ Theta )
=> ( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ Theta ) )
=> ( ( denota1979904720_sc_sz @ I3 @ Theta @ ( produc1111759555eal_sz @ Nu ) @ ( produc1269210629eal_sz @ Nu ) )
= ( denota1979904720_sc_sz @ I3 @ Theta @ ( produc1111759555eal_sz @ Nu2 ) @ ( produc1269210629eal_sz @ Nu2 ) ) ) ) ) ).
% coincidence_frechet
thf(fact_32_IA,axiom,
( denota69106024_a_b_c @ i @ j
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ ( plus_a_c @ t1 @ t2 ) ) ) ) ) ) ).
% IA
thf(fact_33_IA1,axiom,
( denota69106024_a_b_c @ i @ j
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ t1 ) ) ) ) ) ).
% IA1
thf(fact_34_IA2,axiom,
( denota69106024_a_b_c @ i @ j
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ t2 ) ) ) ) ) ).
% IA2
thf(fact_35_IH1,axiom,
( ( denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ t1 ) )
=> ( ( denota69106024_a_b_c @ i @ j
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ t1 ) ) ) ) )
=> ( ( denota229585092_a_b_c @ i @ t1 @ ( produc2010422875real_c @ nu ) @ ( produc314122909real_c @ nu ) )
= ( denota229585092_a_b_c @ j @ t1 @ ( produc2010422875real_c @ nu2 ) @ ( produc314122909real_c @ nu2 ) ) ) ) ) ).
% IH1
thf(fact_36_Iagree__comm,axiom,
! [A: denota231621370t_unit,B: denota231621370t_unit,V: set_Su1783761653um_b_c] :
( ( denota69106024_a_b_c @ A @ B @ V )
=> ( denota69106024_a_b_c @ B @ A @ V ) ) ).
% Iagree_comm
thf(fact_37_Iagree__refl,axiom,
! [I3: denota231621370t_unit,A: set_Su1783761653um_b_c] : ( denota69106024_a_b_c @ I3 @ I3 @ A ) ).
% Iagree_refl
thf(fact_38_subs_I2_J,axiom,
( ord_le929608341um_b_c
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ t2 ) ) ) )
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ ( plus_a_c @ t1 @ t2 ) ) ) ) ) ) ).
% subs(2)
thf(fact_39_subs_I1_J,axiom,
( ord_le929608341um_b_c
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ t1 ) ) ) )
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ ( plus_a_c @ t1 @ t2 ) ) ) ) ) ) ).
% subs(1)
thf(fact_40_simple__term__inverse,axiom,
! [Y: trm_a_c] :
( ( member_trm_a_c @ Y @ ( collect_trm_a_c @ dfree_a_c ) )
=> ( ( frechet_raw_term_a_c @ ( freche665377492rm_a_c @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_41_simple__term__inverse,axiom,
! [Y: trm_sf_sz] :
( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ( freche782854530_sf_sz @ ( freche1046279700_sf_sz @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_42_prod_Ocollapse,axiom,
! [Prod: produc866628903_sc_sz] :
( ( produc1822718231_sc_sz @ ( produc548504323_sc_sz @ Prod ) @ ( produc4753477_sc_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_43_prod_Ocollapse,axiom,
! [Prod: produc1241881959eal_sz] :
( ( produc1235791063eal_sz @ ( produc286845635eal_sz @ Prod ) @ ( produc1746886149eal_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_44_prod_Ocollapse,axiom,
! [Prod: produc1892666919z_real] :
( ( produc784663575z_real @ ( produc870616579z_real @ Prod ) @ ( produc1422611269z_real @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_45_prod_Ocollapse,axiom,
! [Prod: produc1149990247eal_sz] :
( ( produc1308130519eal_sz @ ( produc1111759555eal_sz @ Prod ) @ ( produc1269210629eal_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_46_prod_Ocollapse,axiom,
! [Prod: produc1821101996_sc_sz] :
( ( produc789536734_sc_sz @ ( produc1503602930_sc_sz @ Prod ) @ ( produc974239792_sc_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_47_prod_Ocollapse,axiom,
! [Prod: produc190496183real_c] :
( ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_48_cr__good__interp__def,axiom,
( freche58918398_sc_sz
= ( ^ [X2: denota610675952t_unit,Y2: freche2095075217_sc_sz] :
( X2
= ( freche1421597129_sc_sz @ Y2 ) ) ) ) ).
% cr_good_interp_def
thf(fact_49_cr__strm__def,axiom,
( freche1244000341_sf_sz
= ( ^ [X2: trm_sf_sz,Y2: frechet_strm_sf_sz] :
( X2
= ( freche782854530_sf_sz @ Y2 ) ) ) ) ).
% cr_strm_def
thf(fact_50_cr__strm__def,axiom,
( frechet_cr_strm_a_c
= ( ^ [X2: trm_a_c,Y2: frechet_strm_a_c] :
( X2
= ( frechet_raw_term_a_c @ Y2 ) ) ) ) ).
% cr_strm_def
thf(fact_51_dfree__Times__simps,axiom,
! [A2: trm_a_c,B2: trm_a_c] :
( ( dfree_a_c @ ( times_a_c @ A2 @ B2 ) )
= ( ( dfree_a_c @ A2 )
& ( dfree_a_c @ B2 ) ) ) ).
% dfree_Times_simps
thf(fact_52_dfree__Times__simps,axiom,
! [A2: trm_sf_sz,B2: trm_sf_sz] :
( ( dfree_sf_sz @ ( times_sf_sz @ A2 @ B2 ) )
= ( ( dfree_sf_sz @ A2 )
& ( dfree_sf_sz @ B2 ) ) ) ).
% dfree_Times_simps
thf(fact_53_dfree__Plus__simps,axiom,
! [A2: trm_sf_sz,B2: trm_sf_sz] :
( ( dfree_sf_sz @ ( plus_sf_sz @ A2 @ B2 ) )
= ( ( dfree_sf_sz @ A2 )
& ( dfree_sf_sz @ B2 ) ) ) ).
% dfree_Plus_simps
thf(fact_54_dfree__Plus__simps,axiom,
! [A2: trm_a_c,B2: trm_a_c] :
( ( dfree_a_c @ ( plus_a_c @ A2 @ B2 ) )
= ( ( dfree_a_c @ A2 )
& ( dfree_a_c @ B2 ) ) ) ).
% dfree_Plus_simps
thf(fact_55_dfree__Fun__simps,axiom,
! [I: a,Args: c > trm_a_c] :
( ( dfree_a_c @ ( function_a_c @ I @ Args ) )
= ( ! [X2: c] : ( dfree_a_c @ ( Args @ X2 ) ) ) ) ).
% dfree_Fun_simps
thf(fact_56_dfree__Fun__simps,axiom,
! [I: sf,Args: sz > trm_sf_sz] :
( ( dfree_sf_sz @ ( function_sf_sz @ I @ Args ) )
= ( ! [X2: sz] : ( dfree_sf_sz @ ( Args @ X2 ) ) ) ) ).
% dfree_Fun_simps
thf(fact_57_simple__term__cases,axiom,
! [X: frechet_strm_a_c] :
~ ! [Y3: trm_a_c] :
( ( X
= ( freche665377492rm_a_c @ Y3 ) )
=> ~ ( member_trm_a_c @ Y3 @ ( collect_trm_a_c @ dfree_a_c ) ) ) ).
% simple_term_cases
thf(fact_58_simple__term__cases,axiom,
! [X: frechet_strm_sf_sz] :
~ ! [Y3: trm_sf_sz] :
( ( X
= ( freche1046279700_sf_sz @ Y3 ) )
=> ~ ( member_trm_sf_sz @ Y3 @ ( collect_trm_sf_sz @ dfree_sf_sz ) ) ) ).
% simple_term_cases
thf(fact_59_raw__term__inverse,axiom,
! [X: frechet_strm_a_c] :
( ( freche665377492rm_a_c @ ( frechet_raw_term_a_c @ X ) )
= X ) ).
% raw_term_inverse
thf(fact_60_raw__term__inverse,axiom,
! [X: frechet_strm_sf_sz] :
( ( freche1046279700_sf_sz @ ( freche782854530_sf_sz @ X ) )
= X ) ).
% raw_term_inverse
thf(fact_61_simple__term__inject,axiom,
! [X: trm_a_c,Y: trm_a_c] :
( ( member_trm_a_c @ X @ ( collect_trm_a_c @ dfree_a_c ) )
=> ( ( member_trm_a_c @ Y @ ( collect_trm_a_c @ dfree_a_c ) )
=> ( ( ( freche665377492rm_a_c @ X )
= ( freche665377492rm_a_c @ Y ) )
= ( X = Y ) ) ) ) ).
% simple_term_inject
thf(fact_62_simple__term__inject,axiom,
! [X: trm_sf_sz,Y: trm_sf_sz] :
( ( member_trm_sf_sz @ X @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ( member_trm_sf_sz @ Y @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( ( ( freche1046279700_sf_sz @ X )
= ( freche1046279700_sf_sz @ Y ) )
= ( X = Y ) ) ) ) ).
% simple_term_inject
thf(fact_63_simple__term__induct,axiom,
! [P: frechet_strm_a_c > $o,X: frechet_strm_a_c] :
( ! [Y3: trm_a_c] :
( ( member_trm_a_c @ Y3 @ ( collect_trm_a_c @ dfree_a_c ) )
=> ( P @ ( freche665377492rm_a_c @ Y3 ) ) )
=> ( P @ X ) ) ).
% simple_term_induct
thf(fact_64_simple__term__induct,axiom,
! [P: frechet_strm_sf_sz > $o,X: frechet_strm_sf_sz] :
( ! [Y3: trm_sf_sz] :
( ( member_trm_sf_sz @ Y3 @ ( collect_trm_sf_sz @ dfree_sf_sz ) )
=> ( P @ ( freche1046279700_sf_sz @ Y3 ) ) )
=> ( P @ X ) ) ).
% simple_term_induct
thf(fact_65_old_Oprod_Oinject,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A3: finite1398487019real_c,B3: finite1398487019real_c] :
( ( ( produc394644079real_c @ A2 @ B2 )
= ( produc394644079real_c @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_66_old_Oprod_Oinject,axiom,
! [A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz,A3: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ A2 @ B2 )
= ( produc1822718231_sc_sz @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_67_old_Oprod_Oinject,axiom,
! [A2: set_Fi291318197eal_sz,B2: set_Fi291318197eal_sz,A3: set_Fi291318197eal_sz,B3: set_Fi291318197eal_sz] :
( ( ( produc1235791063eal_sz @ A2 @ B2 )
= ( produc1235791063eal_sz @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_68_old_Oprod_Oinject,axiom,
! [A2: bounde472938360z_real,B2: bounde472938360z_real,A3: bounde472938360z_real,B3: bounde472938360z_real] :
( ( ( produc784663575z_real @ A2 @ B2 )
= ( produc784663575z_real @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_69_old_Oprod_Oinject,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A3: denota610675952t_unit,B3: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ A2 @ B2 )
= ( produc789536734_sc_sz @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_70_prod_Oinject,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c,Y1: finite1398487019real_c,Y22: finite1398487019real_c] :
( ( ( produc394644079real_c @ X1 @ X22 )
= ( produc394644079real_c @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_71_prod_Oinject,axiom,
! [X1: list_f1238882004_sc_sz,X22: list_f1238882004_sc_sz,Y1: list_f1238882004_sc_sz,Y22: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ X1 @ X22 )
= ( produc1822718231_sc_sz @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_72_prod_Oinject,axiom,
! [X1: set_Fi291318197eal_sz,X22: set_Fi291318197eal_sz,Y1: set_Fi291318197eal_sz,Y22: set_Fi291318197eal_sz] :
( ( ( produc1235791063eal_sz @ X1 @ X22 )
= ( produc1235791063eal_sz @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_73_prod_Oinject,axiom,
! [X1: bounde472938360z_real,X22: bounde472938360z_real,Y1: bounde472938360z_real,Y22: bounde472938360z_real] :
( ( ( produc784663575z_real @ X1 @ X22 )
= ( produc784663575z_real @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_74_prod_Oinject,axiom,
! [X1: denota610675952t_unit,X22: produc866628903_sc_sz,Y1: denota610675952t_unit,Y22: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ X1 @ X22 )
= ( produc789536734_sc_sz @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_75_trm_Oinject_I3_J,axiom,
! [X31: a,X32: c > trm_a_c,Y31: a,Y32: c > trm_a_c] :
( ( ( function_a_c @ X31 @ X32 )
= ( function_a_c @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% trm.inject(3)
thf(fact_76_trm_Oinject_I3_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,Y31: sf,Y32: sz > trm_sf_sz] :
( ( ( function_sf_sz @ X31 @ X32 )
= ( function_sf_sz @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% trm.inject(3)
thf(fact_77_trm_Oinject_I4_J,axiom,
! [X41: trm_sf_sz,X42: trm_sf_sz,Y41: trm_sf_sz,Y42: trm_sf_sz] :
( ( ( plus_sf_sz @ X41 @ X42 )
= ( plus_sf_sz @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_78_trm_Oinject_I4_J,axiom,
! [X41: trm_a_c,X42: trm_a_c,Y41: trm_a_c,Y42: trm_a_c] :
( ( ( plus_a_c @ X41 @ X42 )
= ( plus_a_c @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_79_trm_Oinject_I5_J,axiom,
! [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_80_trm_Oinject_I5_J,axiom,
! [X51: trm_sf_sz,X52: trm_sf_sz,Y51: trm_sf_sz,Y52: trm_sf_sz] :
( ( ( times_sf_sz @ X51 @ X52 )
= ( times_sf_sz @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_81_mem__Collect__eq,axiom,
! [A2: finite824932053eal_sz,P: finite824932053eal_sz > $o] :
( ( member1693951742eal_sz @ A2 @ ( collec1002602816eal_sz @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A2: produc1821101996_sc_sz,P: produc1821101996_sc_sz > $o] :
( ( member1057565379_sc_sz @ A2 @ ( collec1199592193_sc_sz @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A2: trm_sf_sz,P: trm_sf_sz > $o] :
( ( member_trm_sf_sz @ A2 @ ( collect_trm_sf_sz @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A2: trm_a_c,P: trm_a_c > $o] :
( ( member_trm_a_c @ A2 @ ( collect_trm_a_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
! [A2: sum_su737395349um_b_c,P: sum_su737395349um_b_c > $o] :
( ( member1893232190um_b_c @ A2 @ ( collec2086029184um_b_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A: set_Fi291318197eal_sz] :
( ( collec1002602816eal_sz
@ ^ [X2: finite824932053eal_sz] : ( member1693951742eal_sz @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_Pr1169339874_sc_sz] :
( ( collec1199592193_sc_sz
@ ^ [X2: produc1821101996_sc_sz] : ( member1057565379_sc_sz @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_trm_sf_sz] :
( ( collect_trm_sf_sz
@ ^ [X2: trm_sf_sz] : ( member_trm_sf_sz @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A: set_trm_a_c] :
( ( collect_trm_a_c
@ ^ [X2: trm_a_c] : ( member_trm_a_c @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A: set_Su1783761653um_b_c] :
( ( collec2086029184um_b_c
@ ^ [X2: sum_su737395349um_b_c] : ( member1893232190um_b_c @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_93_Collect__cong,axiom,
! [P: finite824932053eal_sz > $o,Q: finite824932053eal_sz > $o] :
( ! [X3: finite824932053eal_sz] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1002602816eal_sz @ P )
= ( collec1002602816eal_sz @ Q ) ) ) ).
% Collect_cong
thf(fact_94_Collect__cong,axiom,
! [P: produc1821101996_sc_sz > $o,Q: produc1821101996_sc_sz > $o] :
( ! [X3: produc1821101996_sc_sz] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1199592193_sc_sz @ P )
= ( collec1199592193_sc_sz @ Q ) ) ) ).
% Collect_cong
thf(fact_95_Collect__cong,axiom,
! [P: trm_sf_sz > $o,Q: trm_sf_sz > $o] :
( ! [X3: trm_sf_sz] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_trm_sf_sz @ P )
= ( collect_trm_sf_sz @ Q ) ) ) ).
% Collect_cong
thf(fact_96_Collect__cong,axiom,
! [P: trm_a_c > $o,Q: trm_a_c > $o] :
( ! [X3: trm_a_c] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_trm_a_c @ P )
= ( collect_trm_a_c @ Q ) ) ) ).
% Collect_cong
thf(fact_97_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_98_Collect__cong,axiom,
! [P: sum_su737395349um_b_c > $o,Q: sum_su737395349um_b_c > $o] :
( ! [X3: sum_su737395349um_b_c] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec2086029184um_b_c @ P )
= ( collec2086029184um_b_c @ Q ) ) ) ).
% Collect_cong
thf(fact_99_agree__sub,axiom,
! [A: set_Sum_sum_sz_sz,B: set_Sum_sum_sz_sz,Nu: produc1149990247eal_sz,Omega: produc1149990247eal_sz] :
( ( ord_le1110131657_sz_sz @ A @ B )
=> ( ( denota102713844ree_sz @ Nu @ Omega @ B )
=> ( denota102713844ree_sz @ Nu @ Omega @ A ) ) ) ).
% agree_sub
thf(fact_100_agree__sub,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,Nu: produc190496183real_c,Omega: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( denota1997846518gree_c @ Nu @ Omega @ B )
=> ( denota1997846518gree_c @ Nu @ Omega @ A ) ) ) ).
% agree_sub
thf(fact_101_agree__supset,axiom,
! [B: set_Sum_sum_sz_sz,A: set_Sum_sum_sz_sz,Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz] :
( ( ord_le1110131657_sz_sz @ B @ A )
=> ( ( denota102713844ree_sz @ Nu @ Nu2 @ A )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ B ) ) ) ).
% agree_supset
thf(fact_102_agree__supset,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,Nu: produc190496183real_c,Nu2: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ A )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ B ) ) ) ).
% agree_supset
thf(fact_103_old_Oprod_Oinducts,axiom,
! [P: produc190496183real_c > $o,Prod: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_104_old_Oprod_Oinducts,axiom,
! [P: produc866628903_sc_sz > $o,Prod: produc866628903_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] : ( P @ ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_105_old_Oprod_Oinducts,axiom,
! [P: produc1241881959eal_sz > $o,Prod: produc1241881959eal_sz] :
( ! [A4: set_Fi291318197eal_sz,B4: set_Fi291318197eal_sz] : ( P @ ( produc1235791063eal_sz @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_106_old_Oprod_Oinducts,axiom,
! [P: produc1892666919z_real > $o,Prod: produc1892666919z_real] :
( ! [A4: bounde472938360z_real,B4: bounde472938360z_real] : ( P @ ( produc784663575z_real @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_107_old_Oprod_Oinducts,axiom,
! [P: produc1821101996_sc_sz > $o,Prod: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_108_old_Oprod_Oexhaust,axiom,
! [Y: produc190496183real_c] :
~ ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( Y
!= ( produc394644079real_c @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_109_old_Oprod_Oexhaust,axiom,
! [Y: produc866628903_sc_sz] :
~ ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] :
( Y
!= ( produc1822718231_sc_sz @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_110_old_Oprod_Oexhaust,axiom,
! [Y: produc1241881959eal_sz] :
~ ! [A4: set_Fi291318197eal_sz,B4: set_Fi291318197eal_sz] :
( Y
!= ( produc1235791063eal_sz @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_111_old_Oprod_Oexhaust,axiom,
! [Y: produc1892666919z_real] :
~ ! [A4: bounde472938360z_real,B4: bounde472938360z_real] :
( Y
!= ( produc784663575z_real @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_112_old_Oprod_Oexhaust,axiom,
! [Y: produc1821101996_sc_sz] :
~ ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] :
( Y
!= ( produc789536734_sc_sz @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_113_Pair__inject,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A3: finite1398487019real_c,B3: finite1398487019real_c] :
( ( ( produc394644079real_c @ A2 @ B2 )
= ( produc394644079real_c @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_114_Pair__inject,axiom,
! [A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz,A3: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ A2 @ B2 )
= ( produc1822718231_sc_sz @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_115_Pair__inject,axiom,
! [A2: set_Fi291318197eal_sz,B2: set_Fi291318197eal_sz,A3: set_Fi291318197eal_sz,B3: set_Fi291318197eal_sz] :
( ( ( produc1235791063eal_sz @ A2 @ B2 )
= ( produc1235791063eal_sz @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_116_Pair__inject,axiom,
! [A2: bounde472938360z_real,B2: bounde472938360z_real,A3: bounde472938360z_real,B3: bounde472938360z_real] :
( ( ( produc784663575z_real @ A2 @ B2 )
= ( produc784663575z_real @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_117_Pair__inject,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A3: denota610675952t_unit,B3: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ A2 @ B2 )
= ( produc789536734_sc_sz @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_118_prod__cases,axiom,
! [P: produc190496183real_c > $o,P2: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_119_prod__cases,axiom,
! [P: produc866628903_sc_sz > $o,P2: produc866628903_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] : ( P @ ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_120_prod__cases,axiom,
! [P: produc1241881959eal_sz > $o,P2: produc1241881959eal_sz] :
( ! [A4: set_Fi291318197eal_sz,B4: set_Fi291318197eal_sz] : ( P @ ( produc1235791063eal_sz @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_121_prod__cases,axiom,
! [P: produc1892666919z_real > $o,P2: produc1892666919z_real] :
( ! [A4: bounde472938360z_real,B4: bounde472938360z_real] : ( P @ ( produc784663575z_real @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_122_prod__cases,axiom,
! [P: produc1821101996_sc_sz > $o,P2: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_123_surj__pair,axiom,
! [P2: produc190496183real_c] :
? [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_124_surj__pair,axiom,
! [P2: produc866628903_sc_sz] :
? [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_125_surj__pair,axiom,
! [P2: produc1241881959eal_sz] :
? [X3: set_Fi291318197eal_sz,Y3: set_Fi291318197eal_sz] :
( P2
= ( produc1235791063eal_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_126_surj__pair,axiom,
! [P2: produc1892666919z_real] :
? [X3: bounde472938360z_real,Y3: bounde472938360z_real] :
( P2
= ( produc784663575z_real @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_127_surj__pair,axiom,
! [P2: produc1821101996_sc_sz] :
? [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( P2
= ( produc789536734_sc_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_128_prod__induct3,axiom,
! [P: produc1821101996_sc_sz > $o,X: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B4: list_f1238882004_sc_sz,C: list_f1238882004_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ ( produc1822718231_sc_sz @ B4 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_129_prod__cases3,axiom,
! [Y: produc1821101996_sc_sz] :
~ ! [A4: denota610675952t_unit,B4: list_f1238882004_sc_sz,C: list_f1238882004_sc_sz] :
( Y
!= ( produc789536734_sc_sz @ A4 @ ( produc1822718231_sc_sz @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_130_fst__conv,axiom,
! [X1: list_f1238882004_sc_sz,X22: list_f1238882004_sc_sz] :
( ( produc548504323_sc_sz @ ( produc1822718231_sc_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_131_fst__conv,axiom,
! [X1: set_Fi291318197eal_sz,X22: set_Fi291318197eal_sz] :
( ( produc286845635eal_sz @ ( produc1235791063eal_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_132_fst__conv,axiom,
! [X1: bounde472938360z_real,X22: bounde472938360z_real] :
( ( produc870616579z_real @ ( produc784663575z_real @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_133_fst__conv,axiom,
! [X1: finite824932053eal_sz,X22: finite824932053eal_sz] :
( ( produc1111759555eal_sz @ ( produc1308130519eal_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_134_fst__conv,axiom,
! [X1: denota610675952t_unit,X22: produc866628903_sc_sz] :
( ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_135_fst__conv,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c] :
( ( produc2010422875real_c @ ( produc394644079real_c @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_136_fst__eqD,axiom,
! [X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz,A2: list_f1238882004_sc_sz] :
( ( ( produc548504323_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_137_fst__eqD,axiom,
! [X: set_Fi291318197eal_sz,Y: set_Fi291318197eal_sz,A2: set_Fi291318197eal_sz] :
( ( ( produc286845635eal_sz @ ( produc1235791063eal_sz @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_138_fst__eqD,axiom,
! [X: bounde472938360z_real,Y: bounde472938360z_real,A2: bounde472938360z_real] :
( ( ( produc870616579z_real @ ( produc784663575z_real @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_139_fst__eqD,axiom,
! [X: finite824932053eal_sz,Y: finite824932053eal_sz,A2: finite824932053eal_sz] :
( ( ( produc1111759555eal_sz @ ( produc1308130519eal_sz @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_140_fst__eqD,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz,A2: denota610675952t_unit] :
( ( ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_141_fst__eqD,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,A2: finite1398487019real_c] :
( ( ( produc2010422875real_c @ ( produc394644079real_c @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_142_snd__conv,axiom,
! [X1: list_f1238882004_sc_sz,X22: list_f1238882004_sc_sz] :
( ( produc4753477_sc_sz @ ( produc1822718231_sc_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_143_snd__conv,axiom,
! [X1: set_Fi291318197eal_sz,X22: set_Fi291318197eal_sz] :
( ( produc1746886149eal_sz @ ( produc1235791063eal_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_144_snd__conv,axiom,
! [X1: bounde472938360z_real,X22: bounde472938360z_real] :
( ( produc1422611269z_real @ ( produc784663575z_real @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_145_snd__conv,axiom,
! [X1: finite824932053eal_sz,X22: finite824932053eal_sz] :
( ( produc1269210629eal_sz @ ( produc1308130519eal_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_146_snd__conv,axiom,
! [X1: denota610675952t_unit,X22: produc866628903_sc_sz] :
( ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_147_snd__conv,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c] :
( ( produc314122909real_c @ ( produc394644079real_c @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_148_snd__eqD,axiom,
! [X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz,A2: list_f1238882004_sc_sz] :
( ( ( produc4753477_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_149_snd__eqD,axiom,
! [X: set_Fi291318197eal_sz,Y: set_Fi291318197eal_sz,A2: set_Fi291318197eal_sz] :
( ( ( produc1746886149eal_sz @ ( produc1235791063eal_sz @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_150_snd__eqD,axiom,
! [X: bounde472938360z_real,Y: bounde472938360z_real,A2: bounde472938360z_real] :
( ( ( produc1422611269z_real @ ( produc784663575z_real @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_151_snd__eqD,axiom,
! [X: finite824932053eal_sz,Y: finite824932053eal_sz,A2: finite824932053eal_sz] :
( ( ( produc1269210629eal_sz @ ( produc1308130519eal_sz @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_152_snd__eqD,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz,A2: produc866628903_sc_sz] :
( ( ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_153_snd__eqD,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,A2: finite1398487019real_c] :
( ( ( produc314122909real_c @ ( produc394644079real_c @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_154_prod__eq__iff,axiom,
( ( ^ [Y4: produc1149990247eal_sz,Z: produc1149990247eal_sz] : Y4 = Z )
= ( ^ [S2: produc1149990247eal_sz,T: produc1149990247eal_sz] :
( ( ( produc1111759555eal_sz @ S2 )
= ( produc1111759555eal_sz @ T ) )
& ( ( produc1269210629eal_sz @ S2 )
= ( produc1269210629eal_sz @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_155_prod__eq__iff,axiom,
( ( ^ [Y4: produc1821101996_sc_sz,Z: produc1821101996_sc_sz] : Y4 = Z )
= ( ^ [S2: produc1821101996_sc_sz,T: produc1821101996_sc_sz] :
( ( ( produc1503602930_sc_sz @ S2 )
= ( produc1503602930_sc_sz @ T ) )
& ( ( produc974239792_sc_sz @ S2 )
= ( produc974239792_sc_sz @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_156_prod__eq__iff,axiom,
( ( ^ [Y4: produc190496183real_c,Z: produc190496183real_c] : Y4 = Z )
= ( ^ [S2: produc190496183real_c,T: produc190496183real_c] :
( ( ( produc2010422875real_c @ S2 )
= ( produc2010422875real_c @ T ) )
& ( ( produc314122909real_c @ S2 )
= ( produc314122909real_c @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_157_prod_Oexpand,axiom,
! [Prod: produc1149990247eal_sz,Prod2: produc1149990247eal_sz] :
( ( ( ( produc1111759555eal_sz @ Prod )
= ( produc1111759555eal_sz @ Prod2 ) )
& ( ( produc1269210629eal_sz @ Prod )
= ( produc1269210629eal_sz @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_158_prod_Oexpand,axiom,
! [Prod: produc1821101996_sc_sz,Prod2: produc1821101996_sc_sz] :
( ( ( ( produc1503602930_sc_sz @ Prod )
= ( produc1503602930_sc_sz @ Prod2 ) )
& ( ( produc974239792_sc_sz @ Prod )
= ( produc974239792_sc_sz @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_159_prod_Oexpand,axiom,
! [Prod: produc190496183real_c,Prod2: produc190496183real_c] :
( ( ( ( produc2010422875real_c @ Prod )
= ( produc2010422875real_c @ Prod2 ) )
& ( ( produc314122909real_c @ Prod )
= ( produc314122909real_c @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_160_prod__eqI,axiom,
! [P2: produc1149990247eal_sz,Q2: produc1149990247eal_sz] :
( ( ( produc1111759555eal_sz @ P2 )
= ( produc1111759555eal_sz @ Q2 ) )
=> ( ( ( produc1269210629eal_sz @ P2 )
= ( produc1269210629eal_sz @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_161_prod__eqI,axiom,
! [P2: produc1821101996_sc_sz,Q2: produc1821101996_sc_sz] :
( ( ( produc1503602930_sc_sz @ P2 )
= ( produc1503602930_sc_sz @ Q2 ) )
=> ( ( ( produc974239792_sc_sz @ P2 )
= ( produc974239792_sc_sz @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_162_prod__eqI,axiom,
! [P2: produc190496183real_c,Q2: produc190496183real_c] :
( ( ( produc2010422875real_c @ P2 )
= ( produc2010422875real_c @ Q2 ) )
=> ( ( ( produc314122909real_c @ P2 )
= ( produc314122909real_c @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_163_dfree__Fun,axiom,
! [Args: c > trm_a_c,I: a] :
( ! [I4: c] : ( dfree_a_c @ ( Args @ I4 ) )
=> ( dfree_a_c @ ( function_a_c @ I @ Args ) ) ) ).
% dfree_Fun
thf(fact_164_dfree__Fun,axiom,
! [Args: sz > trm_sf_sz,I: sf] :
( ! [I4: sz] : ( dfree_sf_sz @ ( Args @ I4 ) )
=> ( dfree_sf_sz @ ( function_sf_sz @ I @ Args ) ) ) ).
% dfree_Fun
thf(fact_165_dfree_Odfree__Plus,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_1 )
=> ( ( dfree_sf_sz @ Theta_2 )
=> ( dfree_sf_sz @ ( plus_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree.dfree_Plus
thf(fact_166_dfree_Odfree__Plus,axiom,
! [Theta_1: trm_a_c,Theta_2: trm_a_c] :
( ( dfree_a_c @ Theta_1 )
=> ( ( dfree_a_c @ Theta_2 )
=> ( dfree_a_c @ ( plus_a_c @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree.dfree_Plus
thf(fact_167_dfree__Times,axiom,
! [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_Times
thf(fact_168_dfree__Times,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_1 )
=> ( ( dfree_sf_sz @ Theta_2 )
=> ( dfree_sf_sz @ ( times_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree_Times
thf(fact_169_trm_Odistinct_I23_J,axiom,
! [X31: a,X32: c > trm_a_c,X41: trm_a_c,X42: trm_a_c] :
( ( function_a_c @ X31 @ X32 )
!= ( plus_a_c @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_170_trm_Odistinct_I23_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,X41: trm_sf_sz,X42: trm_sf_sz] :
( ( function_sf_sz @ X31 @ X32 )
!= ( plus_sf_sz @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_171_trm_Odistinct_I25_J,axiom,
! [X31: a,X32: c > trm_a_c,X51: trm_a_c,X52: trm_a_c] :
( ( function_a_c @ X31 @ X32 )
!= ( times_a_c @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_172_trm_Odistinct_I25_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,X51: trm_sf_sz,X52: trm_sf_sz] :
( ( function_sf_sz @ X31 @ X32 )
!= ( times_sf_sz @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_173_trm_Odistinct_I31_J,axiom,
! [X41: trm_sf_sz,X42: trm_sf_sz,X51: trm_sf_sz,X52: trm_sf_sz] :
( ( plus_sf_sz @ X41 @ X42 )
!= ( times_sf_sz @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_174_trm_Odistinct_I31_J,axiom,
! [X41: trm_a_c,X42: trm_a_c,X51: trm_a_c,X52: trm_a_c] :
( ( plus_a_c @ X41 @ X42 )
!= ( times_a_c @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_175_surjective__pairing,axiom,
! [T3: produc866628903_sc_sz] :
( T3
= ( produc1822718231_sc_sz @ ( produc548504323_sc_sz @ T3 ) @ ( produc4753477_sc_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_176_surjective__pairing,axiom,
! [T3: produc1241881959eal_sz] :
( T3
= ( produc1235791063eal_sz @ ( produc286845635eal_sz @ T3 ) @ ( produc1746886149eal_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_177_surjective__pairing,axiom,
! [T3: produc1892666919z_real] :
( T3
= ( produc784663575z_real @ ( produc870616579z_real @ T3 ) @ ( produc1422611269z_real @ T3 ) ) ) ).
% surjective_pairing
thf(fact_178_surjective__pairing,axiom,
! [T3: produc1149990247eal_sz] :
( T3
= ( produc1308130519eal_sz @ ( produc1111759555eal_sz @ T3 ) @ ( produc1269210629eal_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_179_surjective__pairing,axiom,
! [T3: produc1821101996_sc_sz] :
( T3
= ( produc789536734_sc_sz @ ( produc1503602930_sc_sz @ T3 ) @ ( produc974239792_sc_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_180_surjective__pairing,axiom,
! [T3: produc190496183real_c] :
( T3
= ( produc394644079real_c @ ( produc2010422875real_c @ T3 ) @ ( produc314122909real_c @ T3 ) ) ) ).
% surjective_pairing
thf(fact_181_prod_Oexhaust__sel,axiom,
! [Prod: produc866628903_sc_sz] :
( Prod
= ( produc1822718231_sc_sz @ ( produc548504323_sc_sz @ Prod ) @ ( produc4753477_sc_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_182_prod_Oexhaust__sel,axiom,
! [Prod: produc1241881959eal_sz] :
( Prod
= ( produc1235791063eal_sz @ ( produc286845635eal_sz @ Prod ) @ ( produc1746886149eal_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_183_prod_Oexhaust__sel,axiom,
! [Prod: produc1892666919z_real] :
( Prod
= ( produc784663575z_real @ ( produc870616579z_real @ Prod ) @ ( produc1422611269z_real @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_184_prod_Oexhaust__sel,axiom,
! [Prod: produc1149990247eal_sz] :
( Prod
= ( produc1308130519eal_sz @ ( produc1111759555eal_sz @ Prod ) @ ( produc1269210629eal_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_185_prod_Oexhaust__sel,axiom,
! [Prod: produc1821101996_sc_sz] :
( Prod
= ( produc789536734_sc_sz @ ( produc1503602930_sc_sz @ Prod ) @ ( produc974239792_sc_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_186_prod_Oexhaust__sel,axiom,
! [Prod: produc190496183real_c] :
( Prod
= ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_187_Pair__le,axiom,
! [A2: set_a,B2: set_a,C2: set_a,D: set_a] :
( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A2 @ B2 ) @ ( produc1928581911_set_a @ C2 @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_188_Pair__le,axiom,
! [A2: set_a,B2: a > $o,C2: set_a,D: a > $o] :
( ( ord_le501349868_a_a_o @ ( produc634809990_a_a_o @ A2 @ B2 ) @ ( produc634809990_a_a_o @ C2 @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_a_o @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_189_Pair__le,axiom,
! [A2: a > $o,B2: set_a,C2: a > $o,D: set_a] :
( ( ord_le72828020_set_a @ ( produc1916559302_set_a @ A2 @ B2 ) @ ( produc1916559302_set_a @ C2 @ D ) )
= ( ( ord_less_eq_a_o @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_190_Pair__le,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,C2: finite1398487019real_c,D: finite1398487019real_c] :
( ( ord_le850691415real_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc394644079real_c @ C2 @ D ) )
= ( ( ord_le775706699real_c @ A2 @ C2 )
& ( ord_le775706699real_c @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_191_Pair__le,axiom,
! [A2: a > $o,B2: a > $o,C2: a > $o,D: a > $o] :
( ( ord_le1883211903_o_a_o @ ( product_Pair_a_o_a_o @ A2 @ B2 ) @ ( product_Pair_a_o_a_o @ C2 @ D ) )
= ( ( ord_less_eq_a_o @ A2 @ C2 )
& ( ord_less_eq_a_o @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_192_Pair__le,axiom,
! [A2: set_Fi291318197eal_sz,B2: set_Fi291318197eal_sz,C2: set_Fi291318197eal_sz,D: set_Fi291318197eal_sz] :
( ( ord_le978883847eal_sz @ ( produc1235791063eal_sz @ A2 @ B2 ) @ ( produc1235791063eal_sz @ C2 @ D ) )
= ( ( ord_le41263445eal_sz @ A2 @ C2 )
& ( ord_le41263445eal_sz @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_193_Pair__le,axiom,
! [A2: set_a,B2: trm_sf_sz > trm_sf_sz > $o,C2: set_a,D: trm_sf_sz > trm_sf_sz > $o] :
( ( ord_le1884788375f_sz_o @ ( produc1174558183f_sz_o @ A2 @ B2 ) @ ( produc1174558183f_sz_o @ C2 @ D ) )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_le8515534f_sz_o @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_194_Pair__le,axiom,
! [A2: trm_sf_sz > trm_sf_sz > $o,B2: set_a,C2: trm_sf_sz > trm_sf_sz > $o,D: set_a] :
( ( ord_le1643089911_set_a @ ( produc1543226183_set_a @ A2 @ B2 ) @ ( produc1543226183_set_a @ C2 @ D ) )
= ( ( ord_le8515534f_sz_o @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_195_Pair__le,axiom,
! [A2: trm_sf_sz > trm_sf_sz > $o,B2: a > $o,C2: trm_sf_sz > trm_sf_sz > $o,D: a > $o] :
( ( ord_le2132572860_o_a_o @ ( produc1152932438_o_a_o @ A2 @ B2 ) @ ( produc1152932438_o_a_o @ C2 @ D ) )
= ( ( ord_le8515534f_sz_o @ A2 @ C2 )
& ( ord_less_eq_a_o @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_196_Pair__le,axiom,
! [A2: a > $o,B2: trm_sf_sz > trm_sf_sz > $o,C2: a > $o,D: trm_sf_sz > trm_sf_sz > $o] :
( ( ord_le472894788f_sz_o @ ( produc593159062f_sz_o @ A2 @ B2 ) @ ( produc593159062f_sz_o @ C2 @ D ) )
= ( ( ord_less_eq_a_o @ A2 @ C2 )
& ( ord_le8515534f_sz_o @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_197_directional__derivative__def,axiom,
( denota2112424896_a_b_c
= ( ^ [I5: denota231621370t_unit,T: trm_a_c,V2: produc190496183real_c] : ( denota229585092_a_b_c @ I5 @ T @ ( produc2010422875real_c @ V2 ) @ ( produc314122909real_c @ V2 ) ) ) ) ).
% directional_derivative_def
thf(fact_198_directional__derivative__def,axiom,
( denota453121236_sc_sz
= ( ^ [I5: denota610675952t_unit,T: trm_sf_sz,V2: produc1149990247eal_sz] : ( denota1979904720_sc_sz @ I5 @ T @ ( produc1111759555eal_sz @ V2 ) @ ( produc1269210629eal_sz @ V2 ) ) ) ) ).
% directional_derivative_def
thf(fact_199_raw__interp__inverse,axiom,
! [X: freche2095075217_sc_sz] :
( ( freche1784963216_sc_sz @ ( freche1421597129_sc_sz @ X ) )
= X ) ).
% raw_interp_inverse
thf(fact_200_type__definition__strm,axiom,
type_d1238563341rm_a_c @ frechet_raw_term_a_c @ freche665377492rm_a_c @ ( collect_trm_a_c @ dfree_a_c ) ).
% type_definition_strm
thf(fact_201_type__definition__strm,axiom,
type_d46389773_sf_sz @ freche782854530_sf_sz @ freche1046279700_sf_sz @ ( collect_trm_sf_sz @ dfree_sf_sz ) ).
% type_definition_strm
thf(fact_202_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_f1238882004_sc_sz > list_f1238882004_sc_sz > $o,X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz,A2: produc866628903_sc_sz] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc1822718231_sc_sz @ X @ Y ) )
=> ( P @ ( produc548504323_sc_sz @ A2 ) @ ( produc4753477_sc_sz @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_203_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: set_Fi291318197eal_sz > set_Fi291318197eal_sz > $o,X: set_Fi291318197eal_sz,Y: set_Fi291318197eal_sz,A2: produc1241881959eal_sz] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc1235791063eal_sz @ X @ Y ) )
=> ( P @ ( produc286845635eal_sz @ A2 ) @ ( produc1746886149eal_sz @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_204_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: bounde472938360z_real > bounde472938360z_real > $o,X: bounde472938360z_real,Y: bounde472938360z_real,A2: produc1892666919z_real] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc784663575z_real @ X @ Y ) )
=> ( P @ ( produc870616579z_real @ A2 ) @ ( produc1422611269z_real @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_205_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: finite824932053eal_sz > finite824932053eal_sz > $o,X: finite824932053eal_sz,Y: finite824932053eal_sz,A2: produc1149990247eal_sz] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc1308130519eal_sz @ X @ Y ) )
=> ( P @ ( produc1111759555eal_sz @ A2 ) @ ( produc1269210629eal_sz @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_206_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: denota610675952t_unit > produc866628903_sc_sz > $o,X: denota610675952t_unit,Y: produc866628903_sc_sz,A2: produc1821101996_sc_sz] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc789536734_sc_sz @ X @ Y ) )
=> ( P @ ( produc1503602930_sc_sz @ A2 ) @ ( produc974239792_sc_sz @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_207_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,X: finite1398487019real_c,Y: finite1398487019real_c,A2: produc190496183real_c] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc394644079real_c @ X @ Y ) )
=> ( P @ ( produc2010422875real_c @ A2 ) @ ( produc314122909real_c @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_208_conjI__realizer,axiom,
! [P: list_f1238882004_sc_sz > $o,P2: list_f1238882004_sc_sz,Q: list_f1238882004_sc_sz > $o,Q2: list_f1238882004_sc_sz] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc548504323_sc_sz @ ( produc1822718231_sc_sz @ P2 @ Q2 ) ) )
& ( Q @ ( produc4753477_sc_sz @ ( produc1822718231_sc_sz @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_209_conjI__realizer,axiom,
! [P: set_Fi291318197eal_sz > $o,P2: set_Fi291318197eal_sz,Q: set_Fi291318197eal_sz > $o,Q2: set_Fi291318197eal_sz] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc286845635eal_sz @ ( produc1235791063eal_sz @ P2 @ Q2 ) ) )
& ( Q @ ( produc1746886149eal_sz @ ( produc1235791063eal_sz @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_210_conjI__realizer,axiom,
! [P: bounde472938360z_real > $o,P2: bounde472938360z_real,Q: bounde472938360z_real > $o,Q2: bounde472938360z_real] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc870616579z_real @ ( produc784663575z_real @ P2 @ Q2 ) ) )
& ( Q @ ( produc1422611269z_real @ ( produc784663575z_real @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_211_conjI__realizer,axiom,
! [P: finite824932053eal_sz > $o,P2: finite824932053eal_sz,Q: finite824932053eal_sz > $o,Q2: finite824932053eal_sz] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc1111759555eal_sz @ ( produc1308130519eal_sz @ P2 @ Q2 ) ) )
& ( Q @ ( produc1269210629eal_sz @ ( produc1308130519eal_sz @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_212_conjI__realizer,axiom,
! [P: denota610675952t_unit > $o,P2: denota610675952t_unit,Q: produc866628903_sc_sz > $o,Q2: produc866628903_sc_sz] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ P2 @ Q2 ) ) )
& ( Q @ ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_213_conjI__realizer,axiom,
! [P: finite1398487019real_c > $o,P2: finite1398487019real_c,Q: finite1398487019real_c > $o,Q2: finite1398487019real_c] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc2010422875real_c @ ( produc394644079real_c @ P2 @ Q2 ) ) )
& ( Q @ ( produc314122909real_c @ ( produc394644079real_c @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_214_exI__realizer,axiom,
! [P: list_f1238882004_sc_sz > list_f1238882004_sc_sz > $o,Y: list_f1238882004_sc_sz,X: list_f1238882004_sc_sz] :
( ( P @ Y @ X )
=> ( P @ ( produc4753477_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) ) @ ( produc548504323_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_215_exI__realizer,axiom,
! [P: set_Fi291318197eal_sz > set_Fi291318197eal_sz > $o,Y: set_Fi291318197eal_sz,X: set_Fi291318197eal_sz] :
( ( P @ Y @ X )
=> ( P @ ( produc1746886149eal_sz @ ( produc1235791063eal_sz @ X @ Y ) ) @ ( produc286845635eal_sz @ ( produc1235791063eal_sz @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_216_exI__realizer,axiom,
! [P: bounde472938360z_real > bounde472938360z_real > $o,Y: bounde472938360z_real,X: bounde472938360z_real] :
( ( P @ Y @ X )
=> ( P @ ( produc1422611269z_real @ ( produc784663575z_real @ X @ Y ) ) @ ( produc870616579z_real @ ( produc784663575z_real @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_217_exI__realizer,axiom,
! [P: finite824932053eal_sz > finite824932053eal_sz > $o,Y: finite824932053eal_sz,X: finite824932053eal_sz] :
( ( P @ Y @ X )
=> ( P @ ( produc1269210629eal_sz @ ( produc1308130519eal_sz @ X @ Y ) ) @ ( produc1111759555eal_sz @ ( produc1308130519eal_sz @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_218_exI__realizer,axiom,
! [P: produc866628903_sc_sz > denota610675952t_unit > $o,Y: produc866628903_sc_sz,X: denota610675952t_unit] :
( ( P @ Y @ X )
=> ( P @ ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) ) @ ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_219_exI__realizer,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,Y: finite1398487019real_c,X: finite1398487019real_c] :
( ( P @ Y @ X )
=> ( P @ ( produc314122909real_c @ ( produc394644079real_c @ X @ Y ) ) @ ( produc2010422875real_c @ ( produc394644079real_c @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_220_less__eq__prod__def,axiom,
( ord_le850691415real_c
= ( ^ [X2: produc190496183real_c,Y2: produc190496183real_c] :
( ( ord_le775706699real_c @ ( produc2010422875real_c @ X2 ) @ ( produc2010422875real_c @ Y2 ) )
& ( ord_le775706699real_c @ ( produc314122909real_c @ X2 ) @ ( produc314122909real_c @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_221_less__eq__prod__def,axiom,
( ord_le486764743_set_a
= ( ^ [X2: produc1691597095_set_a,Y2: produc1691597095_set_a] :
( ( ord_less_eq_set_a @ ( produc1926412547_set_a @ X2 ) @ ( produc1926412547_set_a @ Y2 ) )
& ( ord_less_eq_set_a @ ( produc918972485_set_a @ X2 ) @ ( produc918972485_set_a @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_222_less__eq__prod__def,axiom,
( ord_le501349868_a_a_o
= ( ^ [X2: produc1989635468_a_a_o,Y2: produc1989635468_a_a_o] :
( ( ord_less_eq_set_a @ ( produc1911461786_a_a_o @ X2 ) @ ( produc1911461786_a_a_o @ Y2 ) )
& ( ord_less_eq_a_o @ ( produc23639768_a_a_o @ X2 ) @ ( produc23639768_a_a_o @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_223_less__eq__prod__def,axiom,
( ord_le72828020_set_a
= ( ^ [X2: produc1561113620_set_a,Y2: produc1561113620_set_a] :
( ( ord_less_eq_a_o @ ( produc1045727450_set_a @ X2 ) @ ( produc1045727450_set_a @ Y2 ) )
& ( ord_less_eq_set_a @ ( produc1305389080_set_a @ X2 ) @ ( produc1305389080_set_a @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_224_less__eq__prod__def,axiom,
( ord_le473656583eal_sz
= ( ^ [X2: produc1149990247eal_sz,Y2: produc1149990247eal_sz] :
( ( ord_le787639925eal_sz @ ( produc1111759555eal_sz @ X2 ) @ ( produc1111759555eal_sz @ Y2 ) )
& ( ord_le787639925eal_sz @ ( produc1269210629eal_sz @ X2 ) @ ( produc1269210629eal_sz @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_225_less__eq__prod__def,axiom,
( ord_le1883211903_o_a_o
= ( ^ [X2: product_prod_a_o_a_o,Y2: product_prod_a_o_a_o] :
( ( ord_less_eq_a_o @ ( product_fst_a_o_a_o @ X2 ) @ ( product_fst_a_o_a_o @ Y2 ) )
& ( ord_less_eq_a_o @ ( product_snd_a_o_a_o @ X2 ) @ ( product_snd_a_o_a_o @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_226_less__eq__prod__def,axiom,
( ord_le1884788375f_sz_o
= ( ^ [X2: produc1144880887f_sz_o,Y2: produc1144880887f_sz_o] :
( ( ord_less_eq_set_a @ ( produc38604755f_sz_o @ X2 ) @ ( produc38604755f_sz_o @ Y2 ) )
& ( ord_le8515534f_sz_o @ ( produc407510293f_sz_o @ X2 ) @ ( produc407510293f_sz_o @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_227_less__eq__prod__def,axiom,
( ord_le1643089911_set_a
= ( ^ [X2: produc903182423_set_a,Y2: produc903182423_set_a] :
( ( ord_le8515534f_sz_o @ ( produc407272755_set_a @ X2 ) @ ( produc407272755_set_a @ Y2 ) )
& ( ord_less_eq_set_a @ ( produc776178293_set_a @ X2 ) @ ( produc776178293_set_a @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_228_less__eq__prod__def,axiom,
( ord_le2132572860_o_a_o
= ( ^ [X2: produc1065634396_o_a_o,Y2: produc1065634396_o_a_o] :
( ( ord_le8515534f_sz_o @ ( produc2013638506_o_a_o @ X2 ) @ ( produc2013638506_o_a_o @ Y2 ) )
& ( ord_less_eq_a_o @ ( produc1630531752_o_a_o @ X2 ) @ ( produc1630531752_o_a_o @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_229_less__eq__prod__def,axiom,
( ord_le472894788f_sz_o
= ( ^ [X2: produc1553439972f_sz_o,Y2: produc1553439972f_sz_o] :
( ( ord_less_eq_a_o @ ( produc1453865130f_sz_o @ X2 ) @ ( produc1453865130f_sz_o @ Y2 ) )
& ( ord_le8515534f_sz_o @ ( produc1070758376f_sz_o @ X2 ) @ ( produc1070758376f_sz_o @ Y2 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_230_Iagree__sub,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,I3: denota231621370t_unit,J: denota231621370t_unit] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( denota69106024_a_b_c @ I3 @ J @ B )
=> ( denota69106024_a_b_c @ I3 @ J @ A ) ) ) ).
% Iagree_sub
thf(fact_231_subset__Collect__iff,axiom,
! [B: set_Fi291318197eal_sz,A: set_Fi291318197eal_sz,P: finite824932053eal_sz > $o] :
( ( ord_le41263445eal_sz @ B @ A )
=> ( ( ord_le41263445eal_sz @ B
@ ( collec1002602816eal_sz
@ ^ [X2: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_232_subset__Collect__iff,axiom,
! [B: set_Pr1169339874_sc_sz,A: set_Pr1169339874_sc_sz,P: produc1821101996_sc_sz > $o] :
( ( ord_le1230423618_sc_sz @ B @ A )
=> ( ( ord_le1230423618_sc_sz @ B
@ ( collec1199592193_sc_sz
@ ^ [X2: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_233_subset__Collect__iff,axiom,
! [B: set_trm_sf_sz,A: set_trm_sf_sz,P: trm_sf_sz > $o] :
( ( ord_le1697056455_sf_sz @ B @ A )
=> ( ( ord_le1697056455_sf_sz @ B
@ ( collect_trm_sf_sz
@ ^ [X2: trm_sf_sz] :
( ( member_trm_sf_sz @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: trm_sf_sz] :
( ( member_trm_sf_sz @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_234_subset__Collect__iff,axiom,
! [B: set_trm_a_c,A: set_trm_a_c,P: trm_a_c > $o] :
( ( ord_le1369644495rm_a_c @ B @ A )
=> ( ( ord_le1369644495rm_a_c @ B
@ ( collect_trm_a_c
@ ^ [X2: trm_a_c] :
( ( member_trm_a_c @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: trm_a_c] :
( ( member_trm_a_c @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_235_subset__Collect__iff,axiom,
! [B: set_a,A: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_236_subset__Collect__iff,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,P: sum_su737395349um_b_c > $o] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( ( ord_le929608341um_b_c @ B
@ ( collec2086029184um_b_c
@ ^ [X2: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X2 @ A )
& ( P @ X2 ) ) ) )
= ( ! [X2: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X2 @ B )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_237_subset__CollectI,axiom,
! [B: set_Fi291318197eal_sz,A: set_Fi291318197eal_sz,Q: finite824932053eal_sz > $o,P: finite824932053eal_sz > $o] :
( ( ord_le41263445eal_sz @ B @ A )
=> ( ! [X3: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le41263445eal_sz
@ ( collec1002602816eal_sz
@ ^ [X2: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec1002602816eal_sz
@ ^ [X2: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_238_subset__CollectI,axiom,
! [B: set_Pr1169339874_sc_sz,A: set_Pr1169339874_sc_sz,Q: produc1821101996_sc_sz > $o,P: produc1821101996_sc_sz > $o] :
( ( ord_le1230423618_sc_sz @ B @ A )
=> ( ! [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1230423618_sc_sz
@ ( collec1199592193_sc_sz
@ ^ [X2: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec1199592193_sc_sz
@ ^ [X2: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_239_subset__CollectI,axiom,
! [B: set_trm_sf_sz,A: set_trm_sf_sz,Q: trm_sf_sz > $o,P: trm_sf_sz > $o] :
( ( ord_le1697056455_sf_sz @ B @ A )
=> ( ! [X3: trm_sf_sz] :
( ( member_trm_sf_sz @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1697056455_sf_sz
@ ( collect_trm_sf_sz
@ ^ [X2: trm_sf_sz] :
( ( member_trm_sf_sz @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_trm_sf_sz
@ ^ [X2: trm_sf_sz] :
( ( member_trm_sf_sz @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_240_subset__CollectI,axiom,
! [B: set_trm_a_c,A: set_trm_a_c,Q: trm_a_c > $o,P: trm_a_c > $o] :
( ( ord_le1369644495rm_a_c @ B @ A )
=> ( ! [X3: trm_a_c] :
( ( member_trm_a_c @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le1369644495rm_a_c
@ ( collect_trm_a_c
@ ^ [X2: trm_a_c] :
( ( member_trm_a_c @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_trm_a_c
@ ^ [X2: trm_a_c] :
( ( member_trm_a_c @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_241_subset__CollectI,axiom,
! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_242_subset__CollectI,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,Q: sum_su737395349um_b_c > $o,P: sum_su737395349um_b_c > $o] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( ! [X3: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X3 @ B )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le929608341um_b_c
@ ( collec2086029184um_b_c
@ ^ [X2: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X2 @ B )
& ( Q @ X2 ) ) )
@ ( collec2086029184um_b_c
@ ^ [X2: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X2 @ A )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_243_Pair__mono,axiom,
! [X: set_a,X4: set_a,Y: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X @ X4 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X @ Y ) @ ( produc1928581911_set_a @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_244_Pair__mono,axiom,
! [X: set_a,X4: set_a,Y: a > $o,Y5: a > $o] :
( ( ord_less_eq_set_a @ X @ X4 )
=> ( ( ord_less_eq_a_o @ Y @ Y5 )
=> ( ord_le501349868_a_a_o @ ( produc634809990_a_a_o @ X @ Y ) @ ( produc634809990_a_a_o @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_245_Pair__mono,axiom,
! [X: a > $o,X4: a > $o,Y: set_a,Y5: set_a] :
( ( ord_less_eq_a_o @ X @ X4 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le72828020_set_a @ ( produc1916559302_set_a @ X @ Y ) @ ( produc1916559302_set_a @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_246_Pair__mono,axiom,
! [X: finite1398487019real_c,X4: finite1398487019real_c,Y: finite1398487019real_c,Y5: finite1398487019real_c] :
( ( ord_le775706699real_c @ X @ X4 )
=> ( ( ord_le775706699real_c @ Y @ Y5 )
=> ( ord_le850691415real_c @ ( produc394644079real_c @ X @ Y ) @ ( produc394644079real_c @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_247_Pair__mono,axiom,
! [X: a > $o,X4: a > $o,Y: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X @ X4 )
=> ( ( ord_less_eq_a_o @ Y @ Y5 )
=> ( ord_le1883211903_o_a_o @ ( product_Pair_a_o_a_o @ X @ Y ) @ ( product_Pair_a_o_a_o @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_248_Pair__mono,axiom,
! [X: set_Fi291318197eal_sz,X4: set_Fi291318197eal_sz,Y: set_Fi291318197eal_sz,Y5: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ X @ X4 )
=> ( ( ord_le41263445eal_sz @ Y @ Y5 )
=> ( ord_le978883847eal_sz @ ( produc1235791063eal_sz @ X @ Y ) @ ( produc1235791063eal_sz @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_249_Pair__mono,axiom,
! [X: set_a,X4: set_a,Y: trm_sf_sz > trm_sf_sz > $o,Y5: trm_sf_sz > trm_sf_sz > $o] :
( ( ord_less_eq_set_a @ X @ X4 )
=> ( ( ord_le8515534f_sz_o @ Y @ Y5 )
=> ( ord_le1884788375f_sz_o @ ( produc1174558183f_sz_o @ X @ Y ) @ ( produc1174558183f_sz_o @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_250_Pair__mono,axiom,
! [X: trm_sf_sz > trm_sf_sz > $o,X4: trm_sf_sz > trm_sf_sz > $o,Y: set_a,Y5: set_a] :
( ( ord_le8515534f_sz_o @ X @ X4 )
=> ( ( ord_less_eq_set_a @ Y @ Y5 )
=> ( ord_le1643089911_set_a @ ( produc1543226183_set_a @ X @ Y ) @ ( produc1543226183_set_a @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_251_Pair__mono,axiom,
! [X: trm_sf_sz > trm_sf_sz > $o,X4: trm_sf_sz > trm_sf_sz > $o,Y: a > $o,Y5: a > $o] :
( ( ord_le8515534f_sz_o @ X @ X4 )
=> ( ( ord_less_eq_a_o @ Y @ Y5 )
=> ( ord_le2132572860_o_a_o @ ( produc1152932438_o_a_o @ X @ Y ) @ ( produc1152932438_o_a_o @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_252_Pair__mono,axiom,
! [X: a > $o,X4: a > $o,Y: trm_sf_sz > trm_sf_sz > $o,Y5: trm_sf_sz > trm_sf_sz > $o] :
( ( ord_less_eq_a_o @ X @ X4 )
=> ( ( ord_le8515534f_sz_o @ Y @ Y5 )
=> ( ord_le472894788f_sz_o @ ( produc593159062f_sz_o @ X @ Y ) @ ( produc593159062f_sz_o @ X4 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_253_fst__mono,axiom,
! [X: produc1149990247eal_sz,Y: produc1149990247eal_sz] :
( ( ord_le473656583eal_sz @ X @ Y )
=> ( ord_le787639925eal_sz @ ( produc1111759555eal_sz @ X ) @ ( produc1111759555eal_sz @ Y ) ) ) ).
% fst_mono
thf(fact_254_fst__mono,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( ord_le850691415real_c @ X @ Y )
=> ( ord_le775706699real_c @ ( produc2010422875real_c @ X ) @ ( produc2010422875real_c @ Y ) ) ) ).
% fst_mono
thf(fact_255_snd__mono,axiom,
! [X: produc1149990247eal_sz,Y: produc1149990247eal_sz] :
( ( ord_le473656583eal_sz @ X @ Y )
=> ( ord_le787639925eal_sz @ ( produc1269210629eal_sz @ X ) @ ( produc1269210629eal_sz @ Y ) ) ) ).
% snd_mono
thf(fact_256_snd__mono,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( ord_le850691415real_c @ X @ Y )
=> ( ord_le775706699real_c @ ( produc314122909real_c @ X ) @ ( produc314122909real_c @ Y ) ) ) ).
% snd_mono
thf(fact_257_exE__realizer_H,axiom,
! [P: finite824932053eal_sz > finite824932053eal_sz > $o,P2: produc1149990247eal_sz] :
( ( P @ ( produc1269210629eal_sz @ P2 ) @ ( produc1111759555eal_sz @ P2 ) )
=> ~ ! [X3: finite824932053eal_sz,Y3: finite824932053eal_sz] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_258_exE__realizer_H,axiom,
! [P: produc866628903_sc_sz > denota610675952t_unit > $o,P2: produc1821101996_sc_sz] :
( ( P @ ( produc974239792_sc_sz @ P2 ) @ ( produc1503602930_sc_sz @ P2 ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_259_exE__realizer_H,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,P2: produc190496183real_c] :
( ( P @ ( produc314122909real_c @ P2 ) @ ( produc2010422875real_c @ P2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_260_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_In2139678582um_b_c @ X1 )
= ( sum_In2139678582um_b_c @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_261_old_Osum_Oinject_I1_J,axiom,
! [A2: a,A3: a] :
( ( ( sum_In2139678582um_b_c @ A2 )
= ( sum_In2139678582um_b_c @ A3 ) )
= ( A2 = A3 ) ) ).
% old.sum.inject(1)
thf(fact_262_good__interp__inverse,axiom,
! [Y: denota231621370t_unit] :
( ( member1777622435t_unit @ Y @ ( collec451781861t_unit @ denota258389581_a_b_c ) )
=> ( ( freche1608036107_a_b_c @ ( freche1473475332_a_b_c @ Y ) )
= Y ) ) ).
% good_interp_inverse
thf(fact_263_good__interp__inverse,axiom,
! [Y: denota610675952t_unit] :
( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ( freche1421597129_sc_sz @ ( freche1784963216_sc_sz @ Y ) )
= Y ) ) ).
% good_interp_inverse
thf(fact_264_f0__def,axiom,
( f0_sf_sz
= ( ^ [F: sf] : ( function_sf_sz @ F @ empty_sz_sf ) ) ) ).
% f0_def
thf(fact_265_subsetI,axiom,
! [A: set_Fi291318197eal_sz,B: set_Fi291318197eal_sz] :
( ! [X3: finite824932053eal_sz] :
( ( member1693951742eal_sz @ X3 @ A )
=> ( member1693951742eal_sz @ X3 @ B ) )
=> ( ord_le41263445eal_sz @ A @ B ) ) ).
% subsetI
thf(fact_266_subsetI,axiom,
! [A: set_Pr1169339874_sc_sz,B: set_Pr1169339874_sc_sz] :
( ! [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ A )
=> ( member1057565379_sc_sz @ X3 @ B ) )
=> ( ord_le1230423618_sc_sz @ A @ B ) ) ).
% subsetI
thf(fact_267_subsetI,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ! [X3: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X3 @ A )
=> ( member1893232190um_b_c @ X3 @ B ) )
=> ( ord_le929608341um_b_c @ A @ B ) ) ).
% subsetI
thf(fact_268_subsetI,axiom,
! [A: set_trm_sf_sz,B: set_trm_sf_sz] :
( ! [X3: trm_sf_sz] :
( ( member_trm_sf_sz @ X3 @ A )
=> ( member_trm_sf_sz @ X3 @ B ) )
=> ( ord_le1697056455_sf_sz @ A @ B ) ) ).
% subsetI
thf(fact_269_subsetI,axiom,
! [A: set_trm_a_c,B: set_trm_a_c] :
( ! [X3: trm_a_c] :
( ( member_trm_a_c @ X3 @ A )
=> ( member_trm_a_c @ X3 @ B ) )
=> ( ord_le1369644495rm_a_c @ A @ B ) ) ).
% subsetI
thf(fact_270_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_271_raw__interp__induct,axiom,
! [Y: denota231621370t_unit,P: denota231621370t_unit > $o] :
( ( member1777622435t_unit @ Y @ ( collec451781861t_unit @ denota258389581_a_b_c ) )
=> ( ! [X3: freche914690577_a_b_c] : ( P @ ( freche1608036107_a_b_c @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_interp_induct
thf(fact_272_raw__interp__induct,axiom,
! [Y: denota610675952t_unit,P: denota610675952t_unit > $o] :
( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ! [X3: freche2095075217_sc_sz] : ( P @ ( freche1421597129_sc_sz @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_interp_induct
thf(fact_273_raw__interp__cases,axiom,
! [Y: denota231621370t_unit] :
( ( member1777622435t_unit @ Y @ ( collec451781861t_unit @ denota258389581_a_b_c ) )
=> ~ ! [X3: freche914690577_a_b_c] :
( Y
!= ( freche1608036107_a_b_c @ X3 ) ) ) ).
% raw_interp_cases
thf(fact_274_raw__interp__cases,axiom,
! [Y: denota610675952t_unit] :
( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ~ ! [X3: freche2095075217_sc_sz] :
( Y
!= ( freche1421597129_sc_sz @ X3 ) ) ) ).
% raw_interp_cases
thf(fact_275_raw__interp,axiom,
! [X: freche914690577_a_b_c] : ( member1777622435t_unit @ ( freche1608036107_a_b_c @ X ) @ ( collec451781861t_unit @ denota258389581_a_b_c ) ) ).
% raw_interp
thf(fact_276_raw__interp,axiom,
! [X: freche2095075217_sc_sz] : ( member754147591t_unit @ ( freche1421597129_sc_sz @ X ) @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) ) ).
% raw_interp
thf(fact_277_good__interp__cases,axiom,
! [X: freche914690577_a_b_c] :
~ ! [Y3: denota231621370t_unit] :
( ( X
= ( freche1473475332_a_b_c @ Y3 ) )
=> ~ ( member1777622435t_unit @ Y3 @ ( collec451781861t_unit @ denota258389581_a_b_c ) ) ) ).
% good_interp_cases
thf(fact_278_good__interp__cases,axiom,
! [X: freche2095075217_sc_sz] :
~ ! [Y3: denota610675952t_unit] :
( ( X
= ( freche1784963216_sc_sz @ Y3 ) )
=> ~ ( member754147591t_unit @ Y3 @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) ) ) ).
% good_interp_cases
thf(fact_279_good__interp__induct,axiom,
! [P: freche914690577_a_b_c > $o,X: freche914690577_a_b_c] :
( ! [Y3: denota231621370t_unit] :
( ( member1777622435t_unit @ Y3 @ ( collec451781861t_unit @ denota258389581_a_b_c ) )
=> ( P @ ( freche1473475332_a_b_c @ Y3 ) ) )
=> ( P @ X ) ) ).
% good_interp_induct
thf(fact_280_good__interp__induct,axiom,
! [P: freche2095075217_sc_sz > $o,X: freche2095075217_sc_sz] :
( ! [Y3: denota610675952t_unit] :
( ( member754147591t_unit @ Y3 @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( P @ ( freche1784963216_sc_sz @ Y3 ) ) )
=> ( P @ X ) ) ).
% good_interp_induct
thf(fact_281_good__interp__inject,axiom,
! [X: denota231621370t_unit,Y: denota231621370t_unit] :
( ( member1777622435t_unit @ X @ ( collec451781861t_unit @ denota258389581_a_b_c ) )
=> ( ( member1777622435t_unit @ Y @ ( collec451781861t_unit @ denota258389581_a_b_c ) )
=> ( ( ( freche1473475332_a_b_c @ X )
= ( freche1473475332_a_b_c @ Y ) )
= ( X = Y ) ) ) ) ).
% good_interp_inject
thf(fact_282_good__interp__inject,axiom,
! [X: denota610675952t_unit,Y: denota610675952t_unit] :
( ( member754147591t_unit @ X @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ( member754147591t_unit @ Y @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) )
=> ( ( ( freche1784963216_sc_sz @ X )
= ( freche1784963216_sc_sz @ Y ) )
= ( X = Y ) ) ) ) ).
% good_interp_inject
thf(fact_283_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_284_Collect__mono__iff,axiom,
! [P: finite824932053eal_sz > $o,Q: finite824932053eal_sz > $o] :
( ( ord_le41263445eal_sz @ ( collec1002602816eal_sz @ P ) @ ( collec1002602816eal_sz @ Q ) )
= ( ! [X2: finite824932053eal_sz] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_285_Collect__mono__iff,axiom,
! [P: produc1821101996_sc_sz > $o,Q: produc1821101996_sc_sz > $o] :
( ( ord_le1230423618_sc_sz @ ( collec1199592193_sc_sz @ P ) @ ( collec1199592193_sc_sz @ Q ) )
= ( ! [X2: produc1821101996_sc_sz] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_286_Collect__mono__iff,axiom,
! [P: trm_sf_sz > $o,Q: trm_sf_sz > $o] :
( ( ord_le1697056455_sf_sz @ ( collect_trm_sf_sz @ P ) @ ( collect_trm_sf_sz @ Q ) )
= ( ! [X2: trm_sf_sz] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_287_Collect__mono__iff,axiom,
! [P: trm_a_c > $o,Q: trm_a_c > $o] :
( ( ord_le1369644495rm_a_c @ ( collect_trm_a_c @ P ) @ ( collect_trm_a_c @ Q ) )
= ( ! [X2: trm_a_c] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_288_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_289_Collect__mono__iff,axiom,
! [P: sum_su737395349um_b_c > $o,Q: sum_su737395349um_b_c > $o] :
( ( ord_le929608341um_b_c @ ( collec2086029184um_b_c @ P ) @ ( collec2086029184um_b_c @ Q ) )
= ( ! [X2: sum_su737395349um_b_c] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_290_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_291_subset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_292_Collect__mono,axiom,
! [P: produc1821101996_sc_sz > $o,Q: produc1821101996_sc_sz > $o] :
( ! [X3: produc1821101996_sc_sz] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1230423618_sc_sz @ ( collec1199592193_sc_sz @ P ) @ ( collec1199592193_sc_sz @ Q ) ) ) ).
% Collect_mono
thf(fact_293_Collect__mono,axiom,
! [P: trm_sf_sz > $o,Q: trm_sf_sz > $o] :
( ! [X3: trm_sf_sz] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1697056455_sf_sz @ ( collect_trm_sf_sz @ P ) @ ( collect_trm_sf_sz @ Q ) ) ) ).
% Collect_mono
thf(fact_294_Collect__mono,axiom,
! [P: trm_a_c > $o,Q: trm_a_c > $o] :
( ! [X3: trm_a_c] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1369644495rm_a_c @ ( collect_trm_a_c @ P ) @ ( collect_trm_a_c @ Q ) ) ) ).
% Collect_mono
thf(fact_295_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_296_Collect__mono,axiom,
! [P: sum_su737395349um_b_c > $o,Q: sum_su737395349um_b_c > $o] :
( ! [X3: sum_su737395349um_b_c] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le929608341um_b_c @ ( collec2086029184um_b_c @ P ) @ ( collec2086029184um_b_c @ Q ) ) ) ).
% Collect_mono
thf(fact_297_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_298_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_299_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_300_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_301_Inl__inject,axiom,
! [X: a,Y: a] :
( ( ( sum_In2139678582um_b_c @ X )
= ( sum_In2139678582um_b_c @ Y ) )
=> ( X = Y ) ) ).
% Inl_inject
thf(fact_302_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_303_Collect__subset,axiom,
! [A: set_Su1783761653um_b_c,P: sum_su737395349um_b_c > $o] :
( ord_le929608341um_b_c
@ ( collec2086029184um_b_c
@ ^ [X2: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_304_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_305_sterm__determines__frechet,axiom,
! [I3: denota231621370t_unit,J: denota231621370t_unit,Theta_12: trm_a_c,Theta_22: trm_a_c,Nu: produc190496183real_c] :
( ( denota258389581_a_b_c @ I3 )
=> ( ( denota258389581_a_b_c @ J )
=> ( ( dfree_a_c @ Theta_12 )
=> ( ( dfree_a_c @ Theta_22 )
=> ( ( ( denota2042094639_a_b_c @ I3 @ Theta_12 )
= ( denota2042094639_a_b_c @ J @ Theta_22 ) )
=> ( ( denota229585092_a_b_c @ I3 @ Theta_12 @ ( produc2010422875real_c @ Nu ) @ ( produc314122909real_c @ Nu ) )
= ( denota229585092_a_b_c @ J @ Theta_22 @ ( produc2010422875real_c @ Nu ) @ ( produc314122909real_c @ Nu ) ) ) ) ) ) ) ) ).
% sterm_determines_frechet
thf(fact_306_sterm__determines__frechet,axiom,
! [I3: denota610675952t_unit,J: denota610675952t_unit,Theta_12: trm_sf_sz,Theta_22: trm_sf_sz,Nu: produc1149990247eal_sz] :
( ( denota1579475975_sc_sz @ I3 )
=> ( ( denota1579475975_sc_sz @ J )
=> ( ( dfree_sf_sz @ Theta_12 )
=> ( ( dfree_sf_sz @ Theta_22 )
=> ( ( ( denota1179238309_sc_sz @ I3 @ Theta_12 )
= ( denota1179238309_sc_sz @ J @ Theta_22 ) )
=> ( ( denota1979904720_sc_sz @ I3 @ Theta_12 @ ( produc1111759555eal_sz @ Nu ) @ ( produc1269210629eal_sz @ Nu ) )
= ( denota1979904720_sc_sz @ J @ Theta_22 @ ( produc1111759555eal_sz @ Nu ) @ ( produc1269210629eal_sz @ Nu ) ) ) ) ) ) ) ) ).
% sterm_determines_frechet
thf(fact_307_type__definition__good__interp,axiom,
type_d1092523951t_unit @ freche1421597129_sc_sz @ freche1784963216_sc_sz @ ( collec1767975749t_unit @ denota1579475975_sc_sz ) ).
% type_definition_good_interp
thf(fact_308_image2__def,axiom,
( bNF_Gr1775200401_sc_sz
= ( ^ [A5: set_a,F: a > denota610675952t_unit,G: a > produc866628903_sc_sz] :
( collec1199592193_sc_sz
@ ^ [Uu: produc1821101996_sc_sz] :
? [A6: a] :
( ( Uu
= ( produc789536734_sc_sz @ ( F @ A6 ) @ ( G @ A6 ) ) )
& ( member_a @ A6 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_309_image2__eqI,axiom,
! [B2: denota610675952t_unit,F2: a > denota610675952t_unit,X: a,C2: produc866628903_sc_sz,G2: a > produc866628903_sc_sz,A: set_a] :
( ( B2
= ( F2 @ X ) )
=> ( ( C2
= ( G2 @ X ) )
=> ( ( member_a @ X @ A )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ B2 @ C2 ) @ ( bNF_Gr1775200401_sc_sz @ A @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_310_coincidence__sterm_H,axiom,
! [Theta: trm_a_c,Nu: produc190496183real_c,Nu2: produc190496183real_c,I3: denota231621370t_unit,J: denota231621370t_unit] :
( ( dfree_a_c @ Theta )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVT_a_c @ Theta ) )
=> ( ( denota69106024_a_b_c @ I3 @ J
@ ( collec2086029184um_b_c
@ ^ [Uu: sum_su737395349um_b_c] :
? [X2: a] :
( ( Uu
= ( sum_In2139678582um_b_c @ X2 ) )
& ( member_a @ X2 @ ( static_SIGT_a_c @ Theta ) ) ) ) )
=> ( ( denota2042094639_a_b_c @ I3 @ Theta @ ( produc2010422875real_c @ Nu ) )
= ( denota2042094639_a_b_c @ J @ Theta @ ( produc2010422875real_c @ Nu2 ) ) ) ) ) ) ).
% coincidence_sterm'
thf(fact_311_pred__subset__eq,axiom,
! [R: set_a,S3: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S3 ) )
= ( ord_less_eq_set_a @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_312_sterm__continuous_H,axiom,
! [I3: denota610675952t_unit,Theta: trm_sf_sz,S3: set_Fi291318197eal_sz] :
( ( denota1579475975_sc_sz @ I3 )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo1348430467z_real @ S3 @ ( denota1179238309_sc_sz @ I3 @ Theta ) ) ) ) ).
% sterm_continuous'
thf(fact_313_agree__func__fvt,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,F2: sf,Args: sz > trm_sf_sz,I: sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVT_sf_sz @ ( function_sf_sz @ F2 @ Args ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVT_sf_sz @ ( Args @ I ) ) ) ) ).
% agree_func_fvt
thf(fact_314_subrelI,axiom,
! [R2: set_Pr1169339874_sc_sz,S4: set_Pr1169339874_sc_sz] :
( ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X3 @ Y3 ) @ R2 )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X3 @ Y3 ) @ S4 ) )
=> ( ord_le1230423618_sc_sz @ R2 @ S4 ) ) ).
% subrelI
thf(fact_315_FVDiff__sub,axiom,
! [F2: trm_a_c] : ( ord_le1772180283um_c_c @ ( static_FVT_a_c @ F2 ) @ ( static_FVDiff_a_c @ F2 ) ) ).
% FVDiff_sub
thf(fact_316_pred__equals__eq2,axiom,
! [R: set_Pr1169339874_sc_sz,S3: set_Pr1169339874_sc_sz] :
( ( ( ^ [X2: denota610675952t_unit,Y2: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X2 @ Y2 ) @ R ) )
= ( ^ [X2: denota610675952t_unit,Y2: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X2 @ Y2 ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_317_pred__subset__eq2,axiom,
! [R: set_Pr1169339874_sc_sz,S3: set_Pr1169339874_sc_sz] :
( ( ord_le108471557c_sz_o
@ ^ [X2: denota610675952t_unit,Y2: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X2 @ Y2 ) @ R )
@ ^ [X2: denota610675952t_unit,Y2: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X2 @ Y2 ) @ S3 ) )
= ( ord_le1230423618_sc_sz @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_318_sterm__continuous,axiom,
! [I3: denota610675952t_unit,Theta: trm_sf_sz] :
( ( denota1579475975_sc_sz @ I3 )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo1348430467z_real @ top_to1873962757eal_sz @ ( denota1179238309_sc_sz @ I3 @ Theta ) ) ) ) ).
% sterm_continuous
thf(fact_319_continuous__on__cases__le,axiom,
! [S4: set_Fi291318197eal_sz,H: finite824932053eal_sz > real,A2: real,F2: finite824932053eal_sz > bounde472938360z_real,G2: finite824932053eal_sz > bounde472938360z_real] :
( ( topolo885751137z_real
@ ( collec1002602816eal_sz
@ ^ [T: finite824932053eal_sz] :
( ( member1693951742eal_sz @ T @ S4 )
& ( ord_less_eq_real @ ( H @ T ) @ A2 ) ) )
@ F2 )
=> ( ( topolo885751137z_real
@ ( collec1002602816eal_sz
@ ^ [T: finite824932053eal_sz] :
( ( member1693951742eal_sz @ T @ S4 )
& ( ord_less_eq_real @ A2 @ ( H @ T ) ) ) )
@ G2 )
=> ( ( topolo1348430467z_real @ S4 @ H )
=> ( ! [T4: finite824932053eal_sz] :
( ( member1693951742eal_sz @ T4 @ S4 )
=> ( ( ( H @ T4 )
= A2 )
=> ( ( F2 @ T4 )
= ( G2 @ T4 ) ) ) )
=> ( topolo885751137z_real @ S4
@ ^ [T: finite824932053eal_sz] : ( if_Bou1785791806z_real @ ( ord_less_eq_real @ ( H @ T ) @ A2 ) @ ( F2 @ T ) @ ( G2 @ T ) ) ) ) ) ) ) ).
% continuous_on_cases_le
thf(fact_320_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_321_UNIV__I,axiom,
! [X: finite824932053eal_sz] : ( member1693951742eal_sz @ X @ top_to1873962757eal_sz ) ).
% UNIV_I
thf(fact_322_top_Oextremum__uniqueI,axiom,
! [A2: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ top_to1873962757eal_sz @ A2 )
=> ( A2 = top_to1873962757eal_sz ) ) ).
% top.extremum_uniqueI
thf(fact_323_top_Oextremum__unique,axiom,
! [A2: set_Fi291318197eal_sz] :
( ( ord_le41263445eal_sz @ top_to1873962757eal_sz @ A2 )
= ( A2 = top_to1873962757eal_sz ) ) ).
% top.extremum_unique
thf(fact_324_top__greatest,axiom,
! [A2: set_Fi291318197eal_sz] : ( ord_le41263445eal_sz @ A2 @ top_to1873962757eal_sz ) ).
% top_greatest
thf(fact_325_subset__UNIV,axiom,
! [A: set_Fi291318197eal_sz] : ( ord_le41263445eal_sz @ A @ top_to1873962757eal_sz ) ).
% subset_UNIV
thf(fact_326_ids_Oseq__sem_Ocases,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,X: produc1821101996_sc_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I2: denota610675952t_unit,S: produc866628903_sc_sz] :
( X
!= ( produc789536734_sc_sz @ I2 @ S ) ) ) ).
% ids.seq_sem.cases
thf(fact_327_UNIV__def,axiom,
( top_to524132421um_b_c
= ( collec2086029184um_b_c
@ ^ [X2: sum_su737395349um_b_c] : $true ) ) ).
% UNIV_def
thf(fact_328_UNIV__def,axiom,
( top_to1873962757eal_sz
= ( collec1002602816eal_sz
@ ^ [X2: finite824932053eal_sz] : $true ) ) ).
% UNIV_def
thf(fact_329_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_330_UNIV__witness,axiom,
? [X3: finite824932053eal_sz] : ( member1693951742eal_sz @ X3 @ top_to1873962757eal_sz ) ).
% UNIV_witness
thf(fact_331_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_332_UNIV__eq__I,axiom,
! [A: set_Fi291318197eal_sz] :
( ! [X3: finite824932053eal_sz] : ( member1693951742eal_sz @ X3 @ A )
=> ( top_to1873962757eal_sz = A ) ) ).
% UNIV_eq_I
thf(fact_333_top__prod__def,axiom,
( top_to1787697847eal_sz
= ( produc1235791063eal_sz @ top_to1873962757eal_sz @ top_to1873962757eal_sz ) ) ).
% top_prod_def
thf(fact_334_ids_OP__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,P2: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p_sc_sf_sz @ P2 )
= ( predic913887884_sf_sz @ P2 ) ) ) ).
% ids.P_def
thf(fact_335_frechet__continuous,axiom,
! [I3: denota610675952t_unit,Theta: trm_sf_sz] :
( ( denota1579475975_sc_sz @ I3 )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo885751137z_real @ top_to1873962757eal_sz @ ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ I3 ) @ ( freche1046279700_sf_sz @ Theta ) ) ) ) ) ).
% frechet_continuous
thf(fact_336_top__empty__eq2,axiom,
( top_to1399053141c_sz_o
= ( ^ [X2: denota610675952t_unit,Y2: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X2 @ Y2 ) @ top_to1499569298_sc_sz ) ) ) ).
% top_empty_eq2
thf(fact_337_top__set__def,axiom,
( top_to524132421um_b_c
= ( collec2086029184um_b_c @ top_to1201357528_b_c_o ) ) ).
% top_set_def
thf(fact_338_top__set__def,axiom,
( top_to1873962757eal_sz
= ( collec1002602816eal_sz @ top_to646513176l_sz_o ) ) ).
% top_set_def
thf(fact_339_Vagree__univ,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,C2: finite1398487019real_c,D: finite1398487019real_c] :
( ( denota1997846518gree_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc394644079real_c @ C2 @ D ) @ top_to1660572043um_c_c )
=> ( ( A2 = C2 )
& ( B2 = D ) ) ) ).
% Vagree_univ
thf(fact_340_agree__UNIV__eq,axiom,
! [Nu: produc190496183real_c,Omega: produc190496183real_c] :
( ( denota1997846518gree_c @ Nu @ Omega @ top_to1660572043um_c_c )
=> ( Nu = Omega ) ) ).
% agree_UNIV_eq
thf(fact_341_ids_Ofrechet__continuous,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,I3: denota610675952t_unit,Theta: trm_sf_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota1579475975_sc_sz @ I3 )
=> ( ( dfree_sf_sz @ Theta )
=> ( topolo885751137z_real @ top_to1873962757eal_sz @ ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ I3 ) @ ( freche1046279700_sf_sz @ Theta ) ) ) ) ) ) ).
% ids.frechet_continuous
thf(fact_342_frechet__blin,axiom,
! [I3: denota610675952t_unit,Theta: trm_sf_sz] :
( ( denota1579475975_sc_sz @ I3 )
=> ( ( dfree_sf_sz @ Theta )
=> ( ( ^ [V2: finite824932053eal_sz] : ( bounde397081123z_real @ ( denota1979904720_sc_sz @ I3 @ Theta @ V2 ) ) )
= ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ I3 ) @ ( freche1046279700_sf_sz @ Theta ) ) ) ) ) ).
% frechet_blin
thf(fact_343_continuous__on__const,axiom,
! [S4: set_Fi291318197eal_sz,C2: bounde472938360z_real] :
( topolo885751137z_real @ S4
@ ^ [X2: finite824932053eal_sz] : C2 ) ).
% continuous_on_const
thf(fact_344_continuous__on__subset,axiom,
! [S4: set_Fi291318197eal_sz,F2: finite824932053eal_sz > bounde472938360z_real,T3: set_Fi291318197eal_sz] :
( ( topolo885751137z_real @ S4 @ F2 )
=> ( ( ord_le41263445eal_sz @ T3 @ S4 )
=> ( topolo885751137z_real @ T3 @ F2 ) ) ) ).
% continuous_on_subset
thf(fact_345_continuous__on__Pair,axiom,
! [S4: set_Fi291318197eal_sz,F2: finite824932053eal_sz > bounde472938360z_real,G2: finite824932053eal_sz > bounde472938360z_real] :
( ( topolo885751137z_real @ S4 @ F2 )
=> ( ( topolo885751137z_real @ S4 @ G2 )
=> ( topolo1027477648z_real @ S4
@ ^ [X2: finite824932053eal_sz] : ( produc784663575z_real @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_346_ids_Ofrechet__blin,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,I3: denota610675952t_unit,Theta: trm_sf_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota1579475975_sc_sz @ I3 )
=> ( ( dfree_sf_sz @ Theta )
=> ( ( ^ [V2: finite824932053eal_sz] : ( bounde397081123z_real @ ( denota1979904720_sc_sz @ I3 @ Theta @ V2 ) ) )
= ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ I3 ) @ ( freche1046279700_sf_sz @ Theta ) ) ) ) ) ) ).
% ids.frechet_blin
thf(fact_347_blin__frechet_Oabs__eq,axiom,
! [Xb: denota610675952t_unit,Xa: trm_sf_sz,X: finite824932053eal_sz] :
( ( bNF_eq1660582207t_unit @ denota1579475975_sc_sz @ Xb @ Xb )
=> ( ( bNF_eq_onp_trm_sf_sz @ dfree_sf_sz @ Xa @ Xa )
=> ( ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ Xb ) @ ( freche1046279700_sf_sz @ Xa ) @ X )
= ( bounde397081123z_real @ ( denota1979904720_sc_sz @ Xb @ Xa @ X ) ) ) ) ) ).
% blin_frechet.abs_eq
thf(fact_348_ids_Oblin__frechet_Oabs__eq,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,Xb: denota610675952t_unit,Xa: trm_sf_sz,X: finite824932053eal_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( bNF_eq1660582207t_unit @ denota1579475975_sc_sz @ Xb @ Xb )
=> ( ( bNF_eq_onp_trm_sf_sz @ dfree_sf_sz @ Xa @ Xa )
=> ( ( freche585307148_sc_sz @ ( freche1784963216_sc_sz @ Xb ) @ ( freche1046279700_sf_sz @ Xa ) @ X )
= ( bounde397081123z_real @ ( denota1979904720_sc_sz @ Xb @ Xa @ X ) ) ) ) ) ) ).
% ids.blin_frechet.abs_eq
thf(fact_349_Quotient__good__interp,axiom,
quotie842200089_sc_sz @ ( bNF_eq1660582207t_unit @ denota1579475975_sc_sz ) @ freche1784963216_sc_sz @ freche1421597129_sc_sz @ freche58918398_sc_sz ).
% Quotient_good_interp
thf(fact_350_Quotient__strm,axiom,
quotie1467858509rm_a_c @ ( bNF_eq_onp_trm_a_c @ dfree_a_c ) @ freche665377492rm_a_c @ frechet_raw_term_a_c @ frechet_cr_strm_a_c ).
% Quotient_strm
thf(fact_351_Quotient__strm,axiom,
quotie1549861309_sf_sz @ ( bNF_eq_onp_trm_sf_sz @ dfree_sf_sz ) @ freche1046279700_sf_sz @ freche782854530_sf_sz @ freche1244000341_sf_sz ).
% Quotient_strm
thf(fact_352_eq__onp__le__eq,axiom,
! [P: denota610675952t_unit > $o] :
( ord_le2003534918unit_o @ ( bNF_eq1660582207t_unit @ P )
@ ^ [Y4: denota610675952t_unit,Z: denota610675952t_unit] : Y4 = Z ) ).
% eq_onp_le_eq
thf(fact_353_eq__onp__le__eq,axiom,
! [P: trm_sf_sz > $o] :
( ord_le8515534f_sz_o @ ( bNF_eq_onp_trm_sf_sz @ P )
@ ^ [Y4: trm_sf_sz,Z: trm_sf_sz] : Y4 = Z ) ).
% eq_onp_le_eq
% Helper facts (3)
thf(help_If_3_1_If_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_T,axiom,
! [X: bounde472938360z_real,Y: bounde472938360z_real] :
( ( if_Bou1785791806z_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Bounded____Linear____Function__Oblinfun_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Real__Oreal_J_T,axiom,
! [X: bounde472938360z_real,Y: bounde472938360z_real] :
( ( if_Bou1785791806z_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ t2 ) ).
%------------------------------------------------------------------------------