TPTP Problem File: ITP041^1.p
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%------------------------------------------------------------------------------
% File : ITP041^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_905__7236378_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_905__7236378_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 508 ( 181 unt; 155 typ; 0 def)
% Number of atoms : 1012 ( 266 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2981 ( 51 ~; 13 |; 80 &;2490 @)
% ( 0 <=>; 347 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 396 ( 396 >; 0 *; 0 +; 0 <<)
% Number of symbols : 133 ( 132 usr; 17 con; 0-4 aty)
% Number of variables : 1051 ( 158 ^; 881 !; 12 ?;1051 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:45:05.406
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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thf(sy_c_Set_OCollect_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
collec230941376real_c: ( finite1398487019real_c > $o ) > set_Fi1407883041real_c ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
collec1643251106real_c: ( produc190496183real_c > $o ) > set_Pr1389752855real_c ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
collec95329456um_c_c: ( set_Sum_sum_c_c > $o ) > set_set_Sum_sum_c_c ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
collect_Sum_sum_c_c: ( sum_sum_c_c > $o ) > set_Sum_sum_c_c ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
collect_Sum_sum_c_d: ( sum_sum_c_d > $o ) > set_Sum_sum_c_d ).
thf(sy_c_Set_OCollect_001tf__c,type,
collect_c: ( c > $o ) > set_c ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
image_1467045143um_c_c: ( produc190496183real_c > sum_sum_c_c ) > set_Pr1389752855real_c > set_Sum_sum_c_c ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
image_1475306648um_c_d: ( produc190496183real_c > sum_sum_c_d ) > set_Pr1389752855real_c > set_Sum_sum_c_d ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
image_327677598real_c: ( real > finite1398487019real_c ) > set_real > set_Fi1407883041real_c ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
image_1146069259real_c: ( sum_sum_c_c > produc190496183real_c ) > set_Sum_sum_c_c > set_Pr1389752855real_c ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
image_666880337um_c_c: ( sum_sum_c_c > sum_sum_c_c ) > set_Sum_sum_c_c > set_Sum_sum_c_c ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
image_675141842um_c_d: ( sum_sum_c_c > sum_sum_c_d ) > set_Sum_sum_c_c > set_Sum_sum_c_d ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__c_Mtf__d_J_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
image_1980705354real_c: ( sum_sum_c_d > produc190496183real_c ) > set_Sum_sum_c_d > set_Pr1389752855real_c ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__c_Mtf__d_J_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
image_1558941394um_c_c: ( sum_sum_c_d > sum_sum_c_c ) > set_Sum_sum_c_d > set_Sum_sum_c_c ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__c_Mtf__d_J_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
image_1567202899um_c_d: ( sum_sum_c_d > sum_sum_c_d ) > set_Sum_sum_c_d > set_Sum_sum_c_d ).
thf(sy_c_Set_Oimage_001tf__c_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
image_c_Sum_sum_c_c: ( c > sum_sum_c_c ) > set_c > set_Sum_sum_c_c ).
thf(sy_c_Set_Oimage_001tf__c_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
image_c_Sum_sum_c_d: ( c > sum_sum_c_d ) > set_c > set_Sum_sum_c_d ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
set_or385782508real_c: produc190496183real_c > produc190496183real_c > set_Pr1389752855real_c ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or656347191t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
set_or935154662um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
set_or2003963744real_c: finite1398487019real_c > set_Fi1407883041real_c ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
set_or10334466real_c: produc190496183real_c > set_Pr1389752855real_c ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
set_ord_atLeast_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
set_or2028906832um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
set_or1478397028real_c: finite1398487019real_c > set_Fi1407883041real_c ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
set_or248124158real_c: produc190496183real_c > set_Pr1389752855real_c ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
set_or326480468um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
thf(sy_c_Static__Semantics_OBVO_001tf__a_001tf__c,type,
static_BVO_a_c: oDE_a_c > set_Sum_sum_c_c ).
thf(sy_c_Static__Semantics_OFVF_001tf__a_001tf__b_001tf__c,type,
static_FVF_a_b_c: formula_a_b_c > set_Sum_sum_c_c ).
thf(sy_c_Static__Semantics_OFVO_001tf__a_001tf__c,type,
static_FVO_a_c: oDE_a_c > set_c ).
thf(sy_c_Static__Semantics_OSIGF_001tf__a_001tf__b_001tf__c,type,
static_SIGF_a_b_c: formula_a_b_c > set_Su1783761653um_b_c ).
thf(sy_c_Sum__Type_OInl_001tf__c_001tf__c,type,
sum_Inl_c_c: c > sum_sum_c_c ).
thf(sy_c_Sum__Type_OInl_001tf__c_001tf__d,type,
sum_Inl_c_d: c > sum_sum_c_d ).
thf(sy_c_Sum__Type_OInr_001tf__c_001tf__c,type,
sum_Inr_c_c: c > sum_sum_c_c ).
thf(sy_c_Syntax_OEquiv_001tf__a_001tf__b_001tf__c,type,
equiv_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
thf(sy_c_Syntax_OImplies_001tf__a_001tf__b_001tf__c,type,
implies_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
thf(sy_c_Syntax_OODE__dom_001tf__a_001tf__c,type,
oDE_dom_a_c: oDE_a_c > set_c ).
thf(sy_c_Syntax_Ofsafe_001tf__a_001tf__b_001tf__c,type,
fsafe_a_b_c: formula_a_b_c > $o ).
thf(sy_c_Syntax_Ohp_OEvolveODE_001tf__a_001tf__c_001tf__b,type,
evolveODE_a_c_b: oDE_a_c > formula_a_b_c > hp_a_b_c ).
thf(sy_c_Syntax_Ohpsafe_001tf__a_001tf__b_001tf__c,type,
hpsafe_a_b_c: hp_a_b_c > $o ).
thf(sy_c_Syntax_Oosafe_001tf__a_001tf__c,type,
osafe_a_c: oDE_a_c > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo1830106092n_real: real > set_real > filter_real ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
member1261661570real_c: finite1398487019real_c > set_Fi1407883041real_c > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
member1895684704real_c: produc190496183real_c > set_Pr1389752855real_c > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
member2124950898um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
member_Sum_sum_c_c: sum_sum_c_c > set_Sum_sum_c_c > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
member_Sum_sum_c_d: sum_sum_c_d > set_Sum_sum_c_d > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_v_I____,type,
i: denota231621370t_unit ).
thf(sy_v_J____,type,
j: denota231621370t_unit ).
thf(sy_v_ODE____,type,
ode: oDE_a_c ).
thf(sy_v_P____,type,
p: formula_a_b_c ).
thf(sy_v_V____,type,
v: set_Sum_sum_c_c ).
thf(sy_v_a____,type,
a: finite1398487019real_c ).
thf(sy_v_aa____,type,
aa: finite1398487019real_c ).
thf(sy_v_ab____,type,
ab: finite1398487019real_c ).
thf(sy_v_b____,type,
b: finite1398487019real_c ).
thf(sy_v_ba____,type,
ba: finite1398487019real_c ).
thf(sy_v_bb____,type,
bb: finite1398487019real_c ).
thf(sy_v_s____,type,
s: real ).
thf(sy_v_sol____,type,
sol: real > finite1398487019real_c ).
thf(sy_v_t____,type,
t: real ).
% Relevant facts (352)
thf(fact_0_osafe,axiom,
osafe_a_c @ ode ).
% osafe
thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062,axiom,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) )
& ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ).
% \<open>Vagree (mk_v J ODE (aa, ba) (sol s)) (aa, ba) (- semBV J ODE) \<and> Vagree (mk_v J ODE (aa, ba) (sol s)) (mk_xode J ODE (sol s)) (semBV J ODE)\<close>
thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062,axiom,
! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ S ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ).
% \<open>\<And>sa. Vagree (mk_v J ODE (aa, ba) (sol sa)) (aa, ba) (- semBV J ODE)\<close>
thf(fact_3_mk__v__agree,axiom,
! [I: denota231621370t_unit,ODE: oDE_a_c,Nu: produc190496183real_c,Sol: finite1398487019real_c] :
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu @ Sol ) @ Nu @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
& ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu @ Sol ) @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% mk_v_agree
thf(fact_4_prod_Oinject,axiom,
! [X1: finite1398487019real_c,X2: finite1398487019real_c,Y1: finite1398487019real_c,Y2: finite1398487019real_c] :
( ( ( produc394644079real_c @ X1 @ X2 )
= ( produc394644079real_c @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_old_Oprod_Oinject,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( ( produc394644079real_c @ A @ B )
= ( produc394644079real_c @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_6_mk__v__exists,axiom,
! [Nu: produc190496183real_c,I: denota231621370t_unit,ODE: oDE_a_c,Sol: finite1398487019real_c] :
? [Omega: produc190496183real_c] :
( ( denota1997846518gree_c @ Omega @ Nu @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
& ( denota1997846518gree_c @ Omega @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% mk_v_exists
thf(fact_7_VA,axiom,
denota1997846518gree_c @ ( produc394644079real_c @ a @ b ) @ ( produc394644079real_c @ aa @ ba ) @ v ).
% VA
thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ S ) ) @ ( denota1896987029_a_b_c @ i @ ode @ ( sol @ S ) ) @ ( denota1293057250_a_b_c @ i @ ode ) ) ).
% \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (mk_xode I ODE (sol sa)) (semBV I ODE)\<close>
thf(fact_9_VAOV,axiom,
denota1997846518gree_c @ ( produc394644079real_c @ a @ b ) @ ( produc394644079real_c @ aa @ ba ) @ ( static_BVO_a_c @ ode ) ).
% VAOV
thf(fact_10_agree__comm,axiom,
! [A3: produc190496183real_c,B3: produc190496183real_c,V: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ A3 @ B3 @ V )
=> ( denota1997846518gree_c @ B3 @ A3 @ V ) ) ).
% agree_comm
thf(fact_11_agree__refl,axiom,
! [Nu: produc190496183real_c,A3: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu @ Nu @ A3 ) ).
% agree_refl
thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062,axiom,
! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ S ) ) @ ( produc394644079real_c @ a @ b ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) ) ) ).
% \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (a, b) (- semBV I ODE)\<close>
thf(fact_13_surj__pair,axiom,
! [P: produc190496183real_c] :
? [X: finite1398487019real_c,Y: finite1398487019real_c] :
( P
= ( produc394644079real_c @ X @ Y ) ) ).
% surj_pair
thf(fact_14_prod__cases,axiom,
! [P2: produc190496183real_c > $o,P: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P2 @ ( produc394644079real_c @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_15_OVsub,axiom,
ord_le1772180283um_c_c @ ( static_BVO_a_c @ ode ) @ v ).
% OVsub
thf(fact_16_veq,axiom,
( ( produc394644079real_c @ ab @ bb )
= ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ t ) ) ) ).
% veq
thf(fact_17_old_Oprod_Oinducts,axiom,
! [P2: produc190496183real_c > $o,Prod: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P2 @ ( produc394644079real_c @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_18_old_Oprod_Oexhaust,axiom,
! [Y3: produc190496183real_c] :
~ ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( Y3
!= ( produc394644079real_c @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_19_Pair__inject,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( ( produc394644079real_c @ A @ B )
= ( produc394644079real_c @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_20_OVsub_H_H,axiom,
ord_le1772180283um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) @ v ).
% OVsub''
thf(fact_21_semBVsub,axiom,
ord_le1772180283um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) @ ( static_BVO_a_c @ ode ) ).
% semBVsub
thf(fact_22_uminus__Pair,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c] :
( ( uminus17964590real_c @ ( produc394644079real_c @ A @ B ) )
= ( produc394644079real_c @ ( uminus1737280820real_c @ A ) @ ( uminus1737280820real_c @ B ) ) ) ).
% uminus_Pair
thf(fact_23_uminus__Pair,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( uminus1225118734um_c_c @ ( produc1439061071um_c_c @ A @ B ) )
= ( produc1439061071um_c_c @ ( uminus1381786404um_c_c @ A ) @ ( uminus1381786404um_c_c @ B ) ) ) ).
% uminus_Pair
thf(fact_24_uminus__Pair,axiom,
! [A: set_Sum_sum_c_c,B: real] :
( ( uminus125610847c_real @ ( produc1528874528c_real @ A @ B ) )
= ( produc1528874528c_real @ ( uminus1381786404um_c_c @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% uminus_Pair
thf(fact_25_uminus__Pair,axiom,
! [A: real,B: set_Sum_sum_c_c] :
( ( uminus1629073887um_c_c @ ( produc1186709024um_c_c @ A @ B ) )
= ( produc1186709024um_c_c @ ( uminus_uminus_real @ A ) @ ( uminus1381786404um_c_c @ B ) ) ) ).
% uminus_Pair
thf(fact_26_uminus__Pair,axiom,
! [A: real,B: real] :
( ( uminus856758960l_real @ ( produc705216881l_real @ A @ B ) )
= ( produc705216881l_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% uminus_Pair
thf(fact_27_ComplI,axiom,
! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
( ~ ( member1895684704real_c @ C @ A3 )
=> ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) ) ) ).
% ComplI
thf(fact_28_ComplI,axiom,
! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
( ~ ( member_Sum_sum_c_d @ C @ A3 )
=> ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) ) ) ).
% ComplI
thf(fact_29_ComplI,axiom,
! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ~ ( member_Sum_sum_c_c @ C @ A3 )
=> ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) ) ) ).
% ComplI
thf(fact_30_Compl__iff,axiom,
! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) )
= ( ~ ( member1895684704real_c @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_31_Compl__iff,axiom,
! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) )
= ( ~ ( member_Sum_sum_c_d @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_32_Compl__iff,axiom,
! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) )
= ( ~ ( member_Sum_sum_c_c @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_33_Compl__eq__Compl__iff,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( ( uminus1381786404um_c_c @ A3 )
= ( uminus1381786404um_c_c @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_34_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_35_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_36_subset__antisym,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ( ord_le1772180283um_c_c @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_37_subsetI,axiom,
! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c] :
( ! [X: produc190496183real_c] :
( ( member1895684704real_c @ X @ A3 )
=> ( member1895684704real_c @ X @ B3 ) )
=> ( ord_le977353143real_c @ A3 @ B3 ) ) ).
% subsetI
thf(fact_38_subsetI,axiom,
! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d] :
( ! [X: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ A3 )
=> ( member_Sum_sum_c_d @ X @ B3 ) )
=> ( ord_le764260924um_c_d @ A3 @ B3 ) ) ).
% subsetI
thf(fact_39_subsetI,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ! [X: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ A3 )
=> ( member_Sum_sum_c_c @ X @ B3 ) )
=> ( ord_le1772180283um_c_c @ A3 @ B3 ) ) ).
% subsetI
thf(fact_40_compl__eq__compl__iff,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ( uminus1381786404um_c_c @ X3 )
= ( uminus1381786404um_c_c @ Y3 ) )
= ( X3 = Y3 ) ) ).
% compl_eq_compl_iff
thf(fact_41_double__compl,axiom,
! [X3: set_Sum_sum_c_c] :
( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ X3 ) )
= X3 ) ).
% double_compl
thf(fact_42_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_43_mem__Collect__eq,axiom,
! [A: produc190496183real_c,P2: produc190496183real_c > $o] :
( ( member1895684704real_c @ A @ ( collec1643251106real_c @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: sum_sum_c_c,P2: sum_sum_c_c > $o] :
( ( member_Sum_sum_c_c @ A @ ( collect_Sum_sum_c_c @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: sum_sum_c_d,P2: sum_sum_c_d > $o] :
( ( member_Sum_sum_c_d @ A @ ( collect_Sum_sum_c_d @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: finite1398487019real_c,P2: finite1398487019real_c > $o] :
( ( member1261661570real_c @ A @ ( collec230941376real_c @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: c,P2: c > $o] :
( ( member_c @ A @ ( collect_c @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_Pr1389752855real_c] :
( ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A3: set_Sum_sum_c_c] :
( ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A3: set_Sum_sum_c_d] :
( ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A3: set_Fi1407883041real_c] :
( ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A3: set_c] :
( ( collect_c
@ ^ [X4: c] : ( member_c @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_Collect__cong,axiom,
! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ! [X: finite1398487019real_c] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec230941376real_c @ P2 )
= ( collec230941376real_c @ Q ) ) ) ).
% Collect_cong
thf(fact_54_Collect__cong,axiom,
! [P2: c > $o,Q: c > $o] :
( ! [X: c] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect_c @ P2 )
= ( collect_c @ Q ) ) ) ).
% Collect_cong
thf(fact_55_compl__le__compl__iff,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X3 ) @ ( uminus1381786404um_c_c @ Y3 ) )
= ( ord_le1772180283um_c_c @ Y3 @ X3 ) ) ).
% compl_le_compl_iff
thf(fact_56_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_57_Compl__subset__Compl__iff,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ A3 ) @ ( uminus1381786404um_c_c @ B3 ) )
= ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_58_Compl__anti__mono,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ B3 ) @ ( uminus1381786404um_c_c @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_59_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_60_Collect__mono__iff,axiom,
! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ( ord_le1327118209real_c @ ( collec230941376real_c @ P2 ) @ ( collec230941376real_c @ Q ) )
= ( ! [X4: finite1398487019real_c] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_61_Collect__mono__iff,axiom,
! [P2: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ ( collect_c @ P2 ) @ ( collect_c @ Q ) )
= ( ! [X4: c] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_62_Collect__mono__iff,axiom,
! [P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
( ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P2 ) @ ( collect_Sum_sum_c_c @ Q ) )
= ( ! [X4: sum_sum_c_c] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_63_set__eq__subset,axiom,
( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : Y4 = Z )
= ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A5 @ B5 )
& ( ord_le1772180283um_c_c @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_64_subset__trans,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ( ord_le1772180283um_c_c @ B3 @ C2 )
=> ( ord_le1772180283um_c_c @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_65_Collect__mono,axiom,
! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ! [X: finite1398487019real_c] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le1327118209real_c @ ( collec230941376real_c @ P2 ) @ ( collec230941376real_c @ Q ) ) ) ).
% Collect_mono
thf(fact_66_Collect__mono,axiom,
! [P2: c > $o,Q: c > $o] :
( ! [X: c] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_c @ ( collect_c @ P2 ) @ ( collect_c @ Q ) ) ) ).
% Collect_mono
thf(fact_67_Collect__mono,axiom,
! [P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
( ! [X: sum_sum_c_c] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P2 ) @ ( collect_Sum_sum_c_c @ Q ) ) ) ).
% Collect_mono
thf(fact_68_subset__refl,axiom,
! [A3: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A3 @ A3 ) ).
% subset_refl
thf(fact_69_subset__iff,axiom,
( ord_le977353143real_c
= ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
! [T: produc190496183real_c] :
( ( member1895684704real_c @ T @ A5 )
=> ( member1895684704real_c @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_70_subset__iff,axiom,
( ord_le764260924um_c_d
= ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
! [T: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ T @ A5 )
=> ( member_Sum_sum_c_d @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_71_subset__iff,axiom,
( ord_le1772180283um_c_c
= ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
! [T: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ T @ A5 )
=> ( member_Sum_sum_c_c @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_72_equalityD2,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( A3 = B3 )
=> ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_73_equalityD1,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( A3 = B3 )
=> ( ord_le1772180283um_c_c @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_74_subset__eq,axiom,
( ord_le977353143real_c
= ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
! [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A5 )
=> ( member1895684704real_c @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_75_subset__eq,axiom,
( ord_le764260924um_c_d
= ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A5 )
=> ( member_Sum_sum_c_d @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_76_subset__eq,axiom,
( ord_le1772180283um_c_c
= ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A5 )
=> ( member_Sum_sum_c_c @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_77_equalityE,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( A3 = B3 )
=> ~ ( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ~ ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_78_subsetD,axiom,
! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c,C: produc190496183real_c] :
( ( ord_le977353143real_c @ A3 @ B3 )
=> ( ( member1895684704real_c @ C @ A3 )
=> ( member1895684704real_c @ C @ B3 ) ) ) ).
% subsetD
thf(fact_79_subsetD,axiom,
! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d,C: sum_sum_c_d] :
( ( ord_le764260924um_c_d @ A3 @ B3 )
=> ( ( member_Sum_sum_c_d @ C @ A3 )
=> ( member_Sum_sum_c_d @ C @ B3 ) ) ) ).
% subsetD
thf(fact_80_subsetD,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,C: sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ( member_Sum_sum_c_c @ C @ A3 )
=> ( member_Sum_sum_c_c @ C @ B3 ) ) ) ).
% subsetD
thf(fact_81_in__mono,axiom,
! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c,X3: produc190496183real_c] :
( ( ord_le977353143real_c @ A3 @ B3 )
=> ( ( member1895684704real_c @ X3 @ A3 )
=> ( member1895684704real_c @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_82_in__mono,axiom,
! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d,X3: sum_sum_c_d] :
( ( ord_le764260924um_c_d @ A3 @ B3 )
=> ( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_83_in__mono,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,X3: sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_84_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_85_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_86_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_87_compl__le__swap2,axiom,
! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y3 ) @ X3 )
=> ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X3 ) @ Y3 ) ) ).
% compl_le_swap2
thf(fact_88_compl__le__swap1,axiom,
! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y3 @ ( uminus1381786404um_c_c @ X3 ) )
=> ( ord_le1772180283um_c_c @ X3 @ ( uminus1381786404um_c_c @ Y3 ) ) ) ).
% compl_le_swap1
thf(fact_89_compl__mono,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X3 @ Y3 )
=> ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y3 ) @ ( uminus1381786404um_c_c @ X3 ) ) ) ).
% compl_mono
thf(fact_90_agree__sub,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,Nu: produc190496183real_c,Omega2: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ( denota1997846518gree_c @ Nu @ Omega2 @ B3 )
=> ( denota1997846518gree_c @ Nu @ Omega2 @ A3 ) ) ) ).
% agree_sub
thf(fact_91_agree__supset,axiom,
! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,Nu: produc190496183real_c,Nu2: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ B3 @ A3 )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ A3 )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ B3 ) ) ) ).
% agree_supset
thf(fact_92_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_93_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_94_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_95_double__complement,axiom,
! [A3: set_Sum_sum_c_c] :
( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_96_ComplD,axiom,
! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) )
=> ~ ( member1895684704real_c @ C @ A3 ) ) ).
% ComplD
thf(fact_97_ComplD,axiom,
! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) )
=> ~ ( member_Sum_sum_c_d @ C @ A3 ) ) ).
% ComplD
thf(fact_98_ComplD,axiom,
! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) )
=> ~ ( member_Sum_sum_c_c @ C @ A3 ) ) ).
% ComplD
thf(fact_99_Pair__le,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,C: finite1398487019real_c,D: finite1398487019real_c] :
( ( ord_le850691415real_c @ ( produc394644079real_c @ A @ B ) @ ( produc394644079real_c @ C @ D ) )
= ( ( ord_le775706699real_c @ A @ C )
& ( ord_le775706699real_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_100_Pair__le,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
( ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ A @ B ) @ ( produc1439061071um_c_c @ C @ D ) )
= ( ( ord_le1772180283um_c_c @ A @ C )
& ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_101_Pair__le,axiom,
! [A: set_Sum_sum_c_c,B: real,C: set_Sum_sum_c_c,D: real] :
( ( ord_le835231240c_real @ ( produc1528874528c_real @ A @ B ) @ ( produc1528874528c_real @ C @ D ) )
= ( ( ord_le1772180283um_c_c @ A @ C )
& ( ord_less_eq_real @ B @ D ) ) ) ).
% Pair_le
thf(fact_102_Pair__le,axiom,
! [A: real,B: set_Sum_sum_c_c,C: real,D: set_Sum_sum_c_c] :
( ( ord_le191210632um_c_c @ ( produc1186709024um_c_c @ A @ B ) @ ( produc1186709024um_c_c @ C @ D ) )
= ( ( ord_less_eq_real @ A @ C )
& ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_103_Pair__le,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
= ( ( ord_less_eq_real @ A @ C )
& ( ord_less_eq_real @ B @ D ) ) ) ).
% Pair_le
thf(fact_104_order__refl,axiom,
! [X3: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X3 @ X3 ) ).
% order_refl
thf(fact_105_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_106_Pair__mono,axiom,
! [X3: finite1398487019real_c,X5: finite1398487019real_c,Y3: finite1398487019real_c,Y5: finite1398487019real_c] :
( ( ord_le775706699real_c @ X3 @ X5 )
=> ( ( ord_le775706699real_c @ Y3 @ Y5 )
=> ( ord_le850691415real_c @ ( produc394644079real_c @ X3 @ Y3 ) @ ( produc394644079real_c @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_107_Pair__mono,axiom,
! [X3: set_Sum_sum_c_c,X5: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X3 @ X5 )
=> ( ( ord_le1772180283um_c_c @ Y3 @ Y5 )
=> ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ X3 @ Y3 ) @ ( produc1439061071um_c_c @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_108_Pair__mono,axiom,
! [X3: set_Sum_sum_c_c,X5: set_Sum_sum_c_c,Y3: real,Y5: real] :
( ( ord_le1772180283um_c_c @ X3 @ X5 )
=> ( ( ord_less_eq_real @ Y3 @ Y5 )
=> ( ord_le835231240c_real @ ( produc1528874528c_real @ X3 @ Y3 ) @ ( produc1528874528c_real @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_109_Pair__mono,axiom,
! [X3: real,X5: real,Y3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
( ( ord_less_eq_real @ X3 @ X5 )
=> ( ( ord_le1772180283um_c_c @ Y3 @ Y5 )
=> ( ord_le191210632um_c_c @ ( produc1186709024um_c_c @ X3 @ Y3 ) @ ( produc1186709024um_c_c @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_110_Pair__mono,axiom,
! [X3: real,X5: real,Y3: real,Y5: real] :
( ( ord_less_eq_real @ X3 @ X5 )
=> ( ( ord_less_eq_real @ Y3 @ Y5 )
=> ( ord_le1342644953l_real @ ( produc705216881l_real @ X3 @ Y3 ) @ ( produc705216881l_real @ X5 @ Y5 ) ) ) ) ).
% Pair_mono
thf(fact_111_Fsub,axiom,
ord_le1772180283um_c_c @ ( static_FVF_a_b_c @ p ) @ v ).
% Fsub
thf(fact_112_solSem,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ t )
=> ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ X3 ) ) @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% solSem
thf(fact_113_relChain__def,axiom,
( bNF_Ca1693016812um_c_c
= ( ^ [R: set_Pr1389752855real_c,As: finite1398487019real_c > set_Sum_sum_c_c] :
! [I2: finite1398487019real_c,J: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ I2 @ J ) @ R )
=> ( ord_le1772180283um_c_c @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_114_relChain__def,axiom,
( bNF_Ca449034301c_real
= ( ^ [R: set_Pr1389752855real_c,As: finite1398487019real_c > real] :
! [I2: finite1398487019real_c,J: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ I2 @ J ) @ R )
=> ( ord_less_eq_real @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_115_hpsafe__Evolve_OIH,axiom,
coinci501627771_a_b_c @ p ).
% hpsafe_Evolve.IH
thf(fact_116_t,axiom,
ord_less_eq_real @ zero_zero_real @ t ).
% t
thf(fact_117_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_118_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_119_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_120_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_121_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_122_IH_H,axiom,
! [A6: finite1398487019real_c,B6: finite1398487019real_c,Aa: finite1398487019real_c,Ba: finite1398487019real_c] :
( ( denota1997846518gree_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( produc394644079real_c @ Aa @ Ba ) @ ( static_FVF_a_b_c @ p ) )
=> ( ( member1895684704real_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( denota968303861_a_b_c @ i @ p ) )
= ( member1895684704real_c @ ( produc394644079real_c @ Aa @ Ba ) @ ( denota968303861_a_b_c @ j @ p ) ) ) ) ).
% IH'
thf(fact_123_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_124_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_125_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_126_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_127_fsafe,axiom,
fsafe_a_b_c @ p ).
% fsafe
thf(fact_128_zero__prod__def,axiom,
( zero_z1506780526real_c
= ( produc394644079real_c @ zero_z109254132real_c @ zero_z109254132real_c ) ) ).
% zero_prod_def
thf(fact_129_zero__prod__def,axiom,
( zero_z659284464l_real
= ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_130_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_131_iff__to__impl,axiom,
! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
( ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
= ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) )
= ( ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
=> ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) )
& ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) )
=> ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) ) ) ) ) ).
% iff_to_impl
thf(fact_132_order__subst1,axiom,
! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst1,axiom,
! [A: set_Sum_sum_c_c,F: real > set_Sum_sum_c_c,B: real,C: real] :
( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_134_order__subst1,axiom,
! [A: real,F: set_Sum_sum_c_c > real,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_135_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_136_order__subst2,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > real,C: real] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: real,B: real,F: real > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_140_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__eq__le__subst,axiom,
! [A: real,F: set_Sum_sum_c_c > real,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_142_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_c_c,F: real > set_Sum_sum_c_c,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_143_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_144_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_145_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > real,C: real] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_146_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_147_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_eq__iff,axiom,
( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : Y4 = Z )
= ( ^ [X4: set_Sum_sum_c_c,Y6: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X4 @ Y6 )
& ( ord_le1772180283um_c_c @ Y6 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_149_eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
& ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_150_antisym,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X3 @ Y3 )
=> ( ( ord_le1772180283um_c_c @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_151_antisym,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_152_linear,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
| ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% linear
thf(fact_153_eq__refl,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( X3 = Y3 )
=> ( ord_le1772180283um_c_c @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_154_eq__refl,axiom,
! [X3: real,Y3: real] :
( ( X3 = Y3 )
=> ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_155_le__cases,axiom,
! [X3: real,Y3: real] :
( ~ ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% le_cases
thf(fact_156_order_Otrans,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% order.trans
thf(fact_157_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_158_le__cases3,axiom,
! [X3: real,Y3: real,Z2: real] :
( ( ( ord_less_eq_real @ X3 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_159_antisym__conv,axiom,
! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y3 @ X3 )
=> ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_160_antisym__conv,axiom,
! [Y3: real,X3: real] :
( ( ord_less_eq_real @ Y3 @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_161_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : Y4 = Z )
= ( ^ [A7: set_Sum_sum_c_c,B7: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A7 @ B7 )
& ( ord_le1772180283um_c_c @ B7 @ A7 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_162_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [A7: real,B7: real] :
( ( ord_less_eq_real @ A7 @ B7 )
& ( ord_less_eq_real @ B7 @ A7 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_163_ord__eq__le__trans,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( A = B )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_164_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_165_ord__le__eq__trans,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( B = C )
=> ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_166_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_167_order__class_Oorder_Oantisym,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_168_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_169_order__trans,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X3 @ Y3 )
=> ( ( ord_le1772180283um_c_c @ Y3 @ Z2 )
=> ( ord_le1772180283um_c_c @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_170_order__trans,axiom,
! [X3: real,Y3: real,Z2: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( ord_less_eq_real @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_171_dual__order_Orefl,axiom,
! [A: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A @ A ) ).
% dual_order.refl
thf(fact_172_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_173_linorder__wlog,axiom,
! [P2: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_174_dual__order_Otrans,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( ord_le1772180283um_c_c @ C @ B )
=> ( ord_le1772180283um_c_c @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_175_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_176_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : Y4 = Z )
= ( ^ [A7: set_Sum_sum_c_c,B7: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B7 @ A7 )
& ( ord_le1772180283um_c_c @ A7 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_177_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [A7: real,B7: real] :
( ( ord_less_eq_real @ B7 @ A7 )
& ( ord_less_eq_real @ A7 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_178_dual__order_Oantisym,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( ord_le1772180283um_c_c @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_179_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_180_iff__sem,axiom,
! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ ( equiv_a_b_c @ A3 @ B3 ) ) )
= ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
= ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) ) ) ).
% iff_sem
thf(fact_181_IHF,axiom,
! [A6: finite1398487019real_c,B6: finite1398487019real_c,Aa: finite1398487019real_c,Ba: finite1398487019real_c] :
( ( denota69106024_a_b_c @ i @ j @ ( static_SIGF_a_b_c @ p ) )
=> ( ( denota1997846518gree_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( produc394644079real_c @ Aa @ Ba ) @ ( static_FVF_a_b_c @ p ) )
=> ( ( member1895684704real_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( denota968303861_a_b_c @ i @ p ) )
= ( member1895684704real_c @ ( produc394644079real_c @ Aa @ Ba ) @ ( denota968303861_a_b_c @ j @ p ) ) ) ) ) ).
% IHF
thf(fact_182_minus__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% minus_le_self_iff
thf(fact_183_le__minus__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% le_minus_self_iff
thf(fact_184_real__eq__0__iff__le__ge__0,axiom,
! [X3: real] :
( ( X3 = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_185_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_186_coincide__fml__def,axiom,
( coinci501627771_a_b_c
= ( ^ [Phi: formula_a_b_c] :
! [Nu3: produc190496183real_c,Nu4: produc190496183real_c,I3: denota231621370t_unit,J2: denota231621370t_unit] :
( ( denota69106024_a_b_c @ I3 @ J2 @ ( static_SIGF_a_b_c @ Phi ) )
=> ( ( denota1997846518gree_c @ Nu3 @ Nu4 @ ( static_FVF_a_b_c @ Phi ) )
=> ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I3 @ Phi ) )
= ( member1895684704real_c @ Nu4 @ ( denota968303861_a_b_c @ J2 @ Phi ) ) ) ) ) ) ) ).
% coincide_fml_def
thf(fact_187_IAP,axiom,
denota69106024_a_b_c @ i @ j @ ( static_SIGF_a_b_c @ p ) ).
% IAP
thf(fact_188_Iagree__sub,axiom,
! [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c,I: denota231621370t_unit,J3: denota231621370t_unit] :
( ( ord_le929608341um_b_c @ A3 @ B3 )
=> ( ( denota69106024_a_b_c @ I @ J3 @ B3 )
=> ( denota69106024_a_b_c @ I @ J3 @ A3 ) ) ) ).
% Iagree_sub
thf(fact_189_Iagree__comm,axiom,
! [A3: denota231621370t_unit,B3: denota231621370t_unit,V: set_Su1783761653um_b_c] :
( ( denota69106024_a_b_c @ A3 @ B3 @ V )
=> ( denota69106024_a_b_c @ B3 @ A3 @ V ) ) ).
% Iagree_comm
thf(fact_190_Iagree__refl,axiom,
! [I: denota231621370t_unit,A3: set_Su1783761653um_b_c] : ( denota69106024_a_b_c @ I @ I @ A3 ) ).
% Iagree_refl
thf(fact_191__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062,axiom,
coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ j ).
% \<open>ode_sem_equiv (EvolveODE ODE P) J\<close>
thf(fact_192__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062,axiom,
coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ i ).
% \<open>ode_sem_equiv (EvolveODE ODE P) I\<close>
thf(fact_193_impl__sem,axiom,
! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ ( implies_a_b_c @ A3 @ B3 ) ) )
= ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
=> ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) ) ) ).
% impl_sem
thf(fact_194_equiv,axiom,
! [I: denota231621370t_unit] : ( coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ I ) ).
% equiv
thf(fact_195_coincide__hp_H__def,axiom,
( coinci1852866155_a_b_c
= ( ^ [Alpha: hp_a_b_c] :
! [I3: denota231621370t_unit,J2: denota231621370t_unit] :
( ( coinci354700372_a_b_c @ Alpha @ I3 @ J2 )
& ( coinci772602909_a_b_c @ Alpha @ I3 ) ) ) ) ).
% coincide_hp'_def
thf(fact_196_hp_Oinject_I5_J,axiom,
! [X51: oDE_a_c,X52: formula_a_b_c,Y51: oDE_a_c,Y52: formula_a_b_c] :
( ( ( evolveODE_a_c_b @ X51 @ X52 )
= ( evolveODE_a_c_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% hp.inject(5)
thf(fact_197_sol,axiom,
( initia1631504802real_c @ sol
@ ^ [A7: real] : ( denota1275485728_a_b_c @ i @ ode )
@ ( set_or656347191t_real @ zero_zero_real @ t )
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ X4 ) @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% sol
thf(fact_198_less__eq__set__def,axiom,
( ord_le977353143real_c
= ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
( ord_le1132241894al_c_o
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A5 )
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_199_less__eq__set__def,axiom,
( ord_le764260924um_c_d
= ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
( ord_le2143422793_c_d_o
@ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A5 )
@ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_200_less__eq__set__def,axiom,
( ord_le1772180283um_c_c
= ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
( ord_le1315653386_c_c_o
@ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A5 )
@ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_201_Collect__subset,axiom,
! [A3: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
( ord_le977353143real_c
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A3 )
& ( P2 @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_202_Collect__subset,axiom,
! [A3: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
( ord_le764260924um_c_d
@ ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A3 )
& ( P2 @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_203_Collect__subset,axiom,
! [A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
( ord_le1327118209real_c
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ A3 )
& ( P2 @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_204_Collect__subset,axiom,
! [A3: set_c,P2: c > $o] :
( ord_less_eq_set_c
@ ( collect_c
@ ^ [X4: c] :
( ( member_c @ X4 @ A3 )
& ( P2 @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_205_Collect__subset,axiom,
! [A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
( ord_le1772180283um_c_c
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A3 )
& ( P2 @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_206_Compl__eq,axiom,
( uminus232257166real_c
= ( ^ [A5: set_Pr1389752855real_c] :
( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
~ ( member1895684704real_c @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_207_Compl__eq,axiom,
( uminus373867045um_c_d
= ( ^ [A5: set_Sum_sum_c_d] :
( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
~ ( member_Sum_sum_c_d @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_208_Compl__eq,axiom,
( uminus102402154real_c
= ( ^ [A5: set_Fi1407883041real_c] :
( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
~ ( member1261661570real_c @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_209_Compl__eq,axiom,
( uminus_uminus_set_c
= ( ^ [A5: set_c] :
( collect_c
@ ^ [X4: c] :
~ ( member_c @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_210_Compl__eq,axiom,
( uminus1381786404um_c_c
= ( ^ [A5: set_Sum_sum_c_c] :
( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
~ ( member_Sum_sum_c_c @ X4 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_211_Collect__neg__eq,axiom,
! [P2: finite1398487019real_c > $o] :
( ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
~ ( P2 @ X4 ) )
= ( uminus102402154real_c @ ( collec230941376real_c @ P2 ) ) ) ).
% Collect_neg_eq
thf(fact_212_Collect__neg__eq,axiom,
! [P2: c > $o] :
( ( collect_c
@ ^ [X4: c] :
~ ( P2 @ X4 ) )
= ( uminus_uminus_set_c @ ( collect_c @ P2 ) ) ) ).
% Collect_neg_eq
thf(fact_213_Collect__neg__eq,axiom,
! [P2: sum_sum_c_c > $o] :
( ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
~ ( P2 @ X4 ) )
= ( uminus1381786404um_c_c @ ( collect_Sum_sum_c_c @ P2 ) ) ) ).
% Collect_neg_eq
thf(fact_214_uminus__set__def,axiom,
( uminus232257166real_c
= ( ^ [A5: set_Pr1389752855real_c] :
( collec1643251106real_c
@ ( uminus32155087al_c_o
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_215_uminus__set__def,axiom,
( uminus373867045um_c_d
= ( ^ [A5: set_Sum_sum_c_d] :
( collect_Sum_sum_c_d
@ ( uminus916499104_c_d_o
@ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_216_uminus__set__def,axiom,
( uminus102402154real_c
= ( ^ [A5: set_Fi1407883041real_c] :
( collec230941376real_c
@ ( uminus23580315al_c_o
@ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_217_uminus__set__def,axiom,
( uminus_uminus_set_c
= ( ^ [A5: set_c] :
( collect_c
@ ( uminus_uminus_c_o
@ ^ [X4: c] : ( member_c @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_218_uminus__set__def,axiom,
( uminus1381786404um_c_c
= ( ^ [A5: set_Sum_sum_c_c] :
( collect_Sum_sum_c_c
@ ( uminus88729697_c_c_o
@ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_219_mk__xode_Oelims,axiom,
! [X3: denota231621370t_unit,Xa: oDE_a_c,Xb: finite1398487019real_c,Y3: produc190496183real_c] :
( ( ( denota1896987029_a_b_c @ X3 @ Xa @ Xb )
= Y3 )
=> ( Y3
= ( produc394644079real_c @ Xb @ ( denota1275485728_a_b_c @ X3 @ Xa @ Xb ) ) ) ) ).
% mk_xode.elims
thf(fact_220_mk__xode_Osimps,axiom,
( denota1896987029_a_b_c
= ( ^ [I3: denota231621370t_unit,ODE2: oDE_a_c,Sol2: finite1398487019real_c] : ( produc394644079real_c @ Sol2 @ ( denota1275485728_a_b_c @ I3 @ ODE2 @ Sol2 ) ) ) ) ).
% mk_xode.simps
thf(fact_221_atLeastatMost__subset__iff,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ A @ B ) @ ( set_or935154662um_c_c @ C @ D ) )
= ( ~ ( ord_le1772180283um_c_c @ A @ B )
| ( ( ord_le1772180283um_c_c @ C @ A )
& ( ord_le1772180283um_c_c @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_222_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_223_atLeastAtMost__iff,axiom,
! [I4: produc190496183real_c,L: produc190496183real_c,U: produc190496183real_c] :
( ( member1895684704real_c @ I4 @ ( set_or385782508real_c @ L @ U ) )
= ( ( ord_le850691415real_c @ L @ I4 )
& ( ord_le850691415real_c @ I4 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_224_atLeastAtMost__iff,axiom,
! [I4: set_Sum_sum_c_c,L: set_Sum_sum_c_c,U: set_Sum_sum_c_c] :
( ( member2124950898um_c_c @ I4 @ ( set_or935154662um_c_c @ L @ U ) )
= ( ( ord_le1772180283um_c_c @ L @ I4 )
& ( ord_le1772180283um_c_c @ I4 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_225_atLeastAtMost__iff,axiom,
! [I4: real,L: real,U: real] :
( ( member_real @ I4 @ ( set_or656347191t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I4 )
& ( ord_less_eq_real @ I4 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_226_Icc__eq__Icc,axiom,
! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,L2: set_Sum_sum_c_c,H2: set_Sum_sum_c_c] :
( ( ( set_or935154662um_c_c @ L @ H )
= ( set_or935154662um_c_c @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le1772180283um_c_c @ L @ H )
& ~ ( ord_le1772180283um_c_c @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_227_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or656347191t_real @ L @ H )
= ( set_or656347191t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_228_pred__subset__eq,axiom,
! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
( ( ord_le1132241894al_c_o
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ R2 )
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ S2 ) )
= ( ord_le977353143real_c @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_229_pred__subset__eq,axiom,
! [R2: set_Sum_sum_c_d,S2: set_Sum_sum_c_d] :
( ( ord_le2143422793_c_d_o
@ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ R2 )
@ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ S2 ) )
= ( ord_le764260924um_c_d @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_230_pred__subset__eq,axiom,
! [R2: set_Sum_sum_c_c,S2: set_Sum_sum_c_c] :
( ( ord_le1315653386_c_c_o
@ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ R2 )
@ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ S2 ) )
= ( ord_le1772180283um_c_c @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_231_solDeriv,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ t )
=> ( has_ve1904308670real_c @ sol @ ( denota1275485728_a_b_c @ i @ ode @ ( sol @ X3 ) ) @ ( topolo1830106092n_real @ X3 @ ( set_or656347191t_real @ zero_zero_real @ t ) ) ) ) ) ).
% solDeriv
thf(fact_232_mk__v__def,axiom,
( denota161327353_a_b_c
= ( ^ [I3: denota231621370t_unit,ODE2: oDE_a_c,Nu3: produc190496183real_c,Sol2: finite1398487019real_c] :
( the_Pr1290640446real_c
@ ^ [Omega3: produc190496183real_c] :
( ( denota1997846518gree_c @ Omega3 @ Nu3 @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I3 @ ODE2 ) ) )
& ( denota1997846518gree_c @ Omega3 @ ( denota1896987029_a_b_c @ I3 @ ODE2 @ Sol2 ) @ ( denota1293057250_a_b_c @ I3 @ ODE2 ) ) ) ) ) ) ).
% mk_v_def
thf(fact_233_subrelI,axiom,
! [R3: set_Pr1389752855real_c,S: set_Pr1389752855real_c] :
( ! [X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ R3 )
=> ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ S ) )
=> ( ord_le977353143real_c @ R3 @ S ) ) ).
% subrelI
thf(fact_234_pred__subset__eq2,axiom,
! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
( ( ord_le1382916304al_c_o
@ ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ R2 )
@ ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ S2 ) )
= ( ord_le977353143real_c @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_235_pred__equals__eq2,axiom,
! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
( ( ( ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ R2 ) )
= ( ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_236_has__vector__derivative__Pair,axiom,
! [F: real > finite1398487019real_c,F2: finite1398487019real_c,X3: real,S: set_real,G: real > finite1398487019real_c,G2: finite1398487019real_c] :
( ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ S ) )
=> ( ( has_ve1904308670real_c @ G @ G2 @ ( topolo1830106092n_real @ X3 @ S ) )
=> ( has_ve373335844real_c
@ ^ [X4: real] : ( produc394644079real_c @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( produc394644079real_c @ F2 @ G2 )
@ ( topolo1830106092n_real @ X3 @ S ) ) ) ) ).
% has_vector_derivative_Pair
thf(fact_237_has__vector__derivative__minus,axiom,
! [F: real > real,F2: real,Net: filter_real] :
( ( has_ve217091583e_real @ F @ F2 @ Net )
=> ( has_ve217091583e_real
@ ^ [X4: real] : ( uminus_uminus_real @ ( F @ X4 ) )
@ ( uminus_uminus_real @ F2 )
@ Net ) ) ).
% has_vector_derivative_minus
thf(fact_238_has__vector__derivative__minus,axiom,
! [F: real > finite1398487019real_c,F2: finite1398487019real_c,Net: filter_real] :
( ( has_ve1904308670real_c @ F @ F2 @ Net )
=> ( has_ve1904308670real_c
@ ^ [X4: real] : ( uminus1737280820real_c @ ( F @ X4 ) )
@ ( uminus1737280820real_c @ F2 )
@ Net ) ) ).
% has_vector_derivative_minus
thf(fact_239_has__vector__derivative__const,axiom,
! [C: real,Net: filter_real] :
( has_ve217091583e_real
@ ^ [X4: real] : C
@ zero_zero_real
@ Net ) ).
% has_vector_derivative_const
thf(fact_240_has__vector__derivative__const,axiom,
! [C: finite1398487019real_c,Net: filter_real] :
( has_ve1904308670real_c
@ ^ [X4: real] : C
@ zero_z109254132real_c
@ Net ) ).
% has_vector_derivative_const
thf(fact_241_has__vector__derivative__within__subset,axiom,
! [F: real > finite1398487019real_c,F2: finite1398487019real_c,X3: real,S2: set_real,T2: set_real] :
( ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ S2 ) )
=> ( ( ord_less_eq_set_real @ T2 @ S2 )
=> ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ T2 ) ) ) ) ).
% has_vector_derivative_within_subset
thf(fact_242_has__vector__derivative__weaken,axiom,
! [F: real > finite1398487019real_c,D2: finite1398487019real_c,X3: real,T2: set_real,S2: set_real,G: real > finite1398487019real_c] :
( ( has_ve1904308670real_c @ F @ D2 @ ( topolo1830106092n_real @ X3 @ T2 ) )
=> ( ( member_real @ X3 @ S2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X: real] :
( ( member_real @ X @ S2 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( has_ve1904308670real_c @ G @ D2 @ ( topolo1830106092n_real @ X3 @ S2 ) ) ) ) ) ) ).
% has_vector_derivative_weaken
thf(fact_243_solves__ode__supset__range,axiom,
! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,Y7: set_Fi1407883041real_c] :
( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
=> ( ( ord_le1327118209real_c @ X6 @ Y7 )
=> ( initia1631504802real_c @ X3 @ F @ T2 @ Y7 ) ) ) ).
% solves_ode_supset_range
thf(fact_244_solves__ode__subset,axiom,
! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,S2: set_real] :
( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( initia1631504802real_c @ X3 @ F @ S2 @ X6 ) ) ) ).
% solves_ode_subset
thf(fact_245_solves__ode__on__subset,axiom,
! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,S2: set_real,Y7: set_Fi1407883041real_c,T2: set_real,X6: set_Fi1407883041real_c] :
( ( initia1631504802real_c @ X3 @ F @ S2 @ Y7 )
=> ( ( ord_less_eq_set_real @ T2 @ S2 )
=> ( ( ord_le1327118209real_c @ Y7 @ X6 )
=> ( initia1631504802real_c @ X3 @ F @ T2 @ X6 ) ) ) ) ).
% solves_ode_on_subset
thf(fact_246_at__le,axiom,
! [S: set_real,T3: set_real,X3: real] :
( ( ord_less_eq_set_real @ S @ T3 )
=> ( ord_le132810396r_real @ ( topolo1830106092n_real @ X3 @ S ) @ ( topolo1830106092n_real @ X3 @ T3 ) ) ) ).
% at_le
thf(fact_247_intervalE,axiom,
! [A: produc190496183real_c,X3: produc190496183real_c,B: produc190496183real_c] :
( ( ( ord_le850691415real_c @ A @ X3 )
& ( ord_le850691415real_c @ X3 @ B ) )
=> ( member1895684704real_c @ X3 @ ( set_or385782508real_c @ A @ B ) ) ) ).
% intervalE
thf(fact_248_intervalE,axiom,
! [A: set_Sum_sum_c_c,X3: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ( ord_le1772180283um_c_c @ A @ X3 )
& ( ord_le1772180283um_c_c @ X3 @ B ) )
=> ( member2124950898um_c_c @ X3 @ ( set_or935154662um_c_c @ A @ B ) ) ) ).
% intervalE
thf(fact_249_intervalE,axiom,
! [A: real,X3: real,B: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( member_real @ X3 @ ( set_or656347191t_real @ A @ B ) ) ) ).
% intervalE
thf(fact_250_conj__subset__def,axiom,
! [A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ( ord_le1327118209real_c @ A3
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( P2 @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_le1327118209real_c @ A3 @ ( collec230941376real_c @ P2 ) )
& ( ord_le1327118209real_c @ A3 @ ( collec230941376real_c @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_251_conj__subset__def,axiom,
! [A3: set_c,P2: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ A3
@ ( collect_c
@ ^ [X4: c] :
( ( P2 @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_c @ A3 @ ( collect_c @ P2 ) )
& ( ord_less_eq_set_c @ A3 @ ( collect_c @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_252_conj__subset__def,axiom,
! [A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
( ( ord_le1772180283um_c_c @ A3
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( P2 @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_le1772180283um_c_c @ A3 @ ( collect_Sum_sum_c_c @ P2 ) )
& ( ord_le1772180283um_c_c @ A3 @ ( collect_Sum_sum_c_c @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_253_subset__Collect__iff,axiom,
! [B3: set_Pr1389752855real_c,A3: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
( ( ord_le977353143real_c @ B3 @ A3 )
=> ( ( ord_le977353143real_c @ B3
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ B3 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_254_subset__Collect__iff,axiom,
! [B3: set_Sum_sum_c_d,A3: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
( ( ord_le764260924um_c_d @ B3 @ A3 )
=> ( ( ord_le764260924um_c_d @ B3
@ ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A3 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ B3 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_255_subset__Collect__iff,axiom,
! [B3: set_Fi1407883041real_c,A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
( ( ord_le1327118209real_c @ B3 @ A3 )
=> ( ( ord_le1327118209real_c @ B3
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ B3 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_256_subset__Collect__iff,axiom,
! [B3: set_c,A3: set_c,P2: c > $o] :
( ( ord_less_eq_set_c @ B3 @ A3 )
=> ( ( ord_less_eq_set_c @ B3
@ ( collect_c
@ ^ [X4: c] :
( ( member_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: c] :
( ( member_c @ X4 @ B3 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_257_subset__Collect__iff,axiom,
! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
( ( ord_le1772180283um_c_c @ B3 @ A3 )
=> ( ( ord_le1772180283um_c_c @ B3
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ B3 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_258_subset__CollectI,axiom,
! [B3: set_Pr1389752855real_c,A3: set_Pr1389752855real_c,Q: produc190496183real_c > $o,P2: produc190496183real_c > $o] :
( ( ord_le977353143real_c @ B3 @ A3 )
=> ( ! [X: produc190496183real_c] :
( ( member1895684704real_c @ X @ B3 )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le977353143real_c
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_259_subset__CollectI,axiom,
! [B3: set_Sum_sum_c_d,A3: set_Sum_sum_c_d,Q: sum_sum_c_d > $o,P2: sum_sum_c_d > $o] :
( ( ord_le764260924um_c_d @ B3 @ A3 )
=> ( ! [X: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ B3 )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le764260924um_c_d
@ ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A3 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_260_subset__CollectI,axiom,
! [B3: set_Fi1407883041real_c,A3: set_Fi1407883041real_c,Q: finite1398487019real_c > $o,P2: finite1398487019real_c > $o] :
( ( ord_le1327118209real_c @ B3 @ A3 )
=> ( ! [X: finite1398487019real_c] :
( ( member1261661570real_c @ X @ B3 )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le1327118209real_c
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_261_subset__CollectI,axiom,
! [B3: set_c,A3: set_c,Q: c > $o,P2: c > $o] :
( ( ord_less_eq_set_c @ B3 @ A3 )
=> ( ! [X: c] :
( ( member_c @ X @ B3 )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq_set_c
@ ( collect_c
@ ^ [X4: c] :
( ( member_c @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_c
@ ^ [X4: c] :
( ( member_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_262_subset__CollectI,axiom,
! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,Q: sum_sum_c_c > $o,P2: sum_sum_c_c > $o] :
( ( ord_le1772180283um_c_c @ B3 @ A3 )
=> ( ! [X: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ B3 )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_le1772180283um_c_c
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A3 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_263_prop__restrict,axiom,
! [X3: produc190496183real_c,Z3: set_Pr1389752855real_c,X6: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
( ( member1895684704real_c @ X3 @ Z3 )
=> ( ( ord_le977353143real_c @ Z3
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_264_prop__restrict,axiom,
! [X3: sum_sum_c_d,Z3: set_Sum_sum_c_d,X6: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
( ( member_Sum_sum_c_d @ X3 @ Z3 )
=> ( ( ord_le764260924um_c_d @ Z3
@ ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_265_prop__restrict,axiom,
! [X3: finite1398487019real_c,Z3: set_Fi1407883041real_c,X6: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
( ( member1261661570real_c @ X3 @ Z3 )
=> ( ( ord_le1327118209real_c @ Z3
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_266_prop__restrict,axiom,
! [X3: c,Z3: set_c,X6: set_c,P2: c > $o] :
( ( member_c @ X3 @ Z3 )
=> ( ( ord_less_eq_set_c @ Z3
@ ( collect_c
@ ^ [X4: c] :
( ( member_c @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_267_prop__restrict,axiom,
! [X3: sum_sum_c_c,Z3: set_Sum_sum_c_c,X6: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
( ( member_Sum_sum_c_c @ X3 @ Z3 )
=> ( ( ord_le1772180283um_c_c @ Z3
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_268_Collect__restrict,axiom,
! [X6: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
( ord_le977353143real_c
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_269_Collect__restrict,axiom,
! [X6: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
( ord_le764260924um_c_d
@ ( collect_Sum_sum_c_d
@ ^ [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_270_Collect__restrict,axiom,
! [X6: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
( ord_le1327118209real_c
@ ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] :
( ( member1261661570real_c @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_271_Collect__restrict,axiom,
! [X6: set_c,P2: c > $o] :
( ord_less_eq_set_c
@ ( collect_c
@ ^ [X4: c] :
( ( member_c @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_272_Collect__restrict,axiom,
! [X6: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
( ord_le1772180283um_c_c
@ ( collect_Sum_sum_c_c
@ ^ [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_273_ssubst__Pair__rhs,axiom,
! [R3: finite1398487019real_c,S: finite1398487019real_c,R2: set_Pr1389752855real_c,S3: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ R3 @ S ) @ R2 )
=> ( ( S3 = S )
=> ( member1895684704real_c @ ( produc394644079real_c @ R3 @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_274_hpsafe__ODE__simps,axiom,
! [ODE: oDE_a_c,P: formula_a_b_c] :
( ( hpsafe_a_b_c @ ( evolveODE_a_c_b @ ODE @ P ) )
= ( ( osafe_a_c @ ODE )
& ( fsafe_a_b_c @ P ) ) ) ).
% hpsafe_ODE_simps
thf(fact_275_hpsafe__fsafe_Ohpsafe__Evolve,axiom,
! [ODE: oDE_a_c,P2: formula_a_b_c] :
( ( osafe_a_c @ ODE )
=> ( ( fsafe_a_b_c @ P2 )
=> ( hpsafe_a_b_c @ ( evolveODE_a_c_b @ ODE @ P2 ) ) ) ) ).
% hpsafe_fsafe.hpsafe_Evolve
thf(fact_276_Icc__subset__Ici__iff,axiom,
! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,L2: set_Sum_sum_c_c] :
( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ L @ H ) @ ( set_or2028906832um_c_c @ L2 ) )
= ( ~ ( ord_le1772180283um_c_c @ L @ H )
| ( ord_le1772180283um_c_c @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_277_Icc__subset__Ici__iff,axiom,
! [L: real,H: real,L2: real] :
( ( ord_less_eq_set_real @ ( set_or656347191t_real @ L @ H ) @ ( set_ord_atLeast_real @ L2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_278_Icc__subset__Iic__iff,axiom,
! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,H2: set_Sum_sum_c_c] :
( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ L @ H ) @ ( set_or326480468um_c_c @ H2 ) )
= ( ~ ( ord_le1772180283um_c_c @ L @ H )
| ( ord_le1772180283um_c_c @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_279_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H2: real] :
( ( ord_less_eq_set_real @ ( set_or656347191t_real @ L @ H ) @ ( set_ord_atMost_real @ H2 ) )
= ( ~ ( ord_less_eq_real @ L @ H )
| ( ord_less_eq_real @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_280_atMost__iff,axiom,
! [I4: produc190496183real_c,K: produc190496183real_c] :
( ( member1895684704real_c @ I4 @ ( set_or248124158real_c @ K ) )
= ( ord_le850691415real_c @ I4 @ K ) ) ).
% atMost_iff
thf(fact_281_atMost__iff,axiom,
! [I4: set_Sum_sum_c_c,K: set_Sum_sum_c_c] :
( ( member2124950898um_c_c @ I4 @ ( set_or326480468um_c_c @ K ) )
= ( ord_le1772180283um_c_c @ I4 @ K ) ) ).
% atMost_iff
thf(fact_282_atMost__iff,axiom,
! [I4: real,K: real] :
( ( member_real @ I4 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I4 @ K ) ) ).
% atMost_iff
thf(fact_283_atLeast__iff,axiom,
! [I4: produc190496183real_c,K: produc190496183real_c] :
( ( member1895684704real_c @ I4 @ ( set_or10334466real_c @ K ) )
= ( ord_le850691415real_c @ K @ I4 ) ) ).
% atLeast_iff
thf(fact_284_atLeast__iff,axiom,
! [I4: set_Sum_sum_c_c,K: set_Sum_sum_c_c] :
( ( member2124950898um_c_c @ I4 @ ( set_or2028906832um_c_c @ K ) )
= ( ord_le1772180283um_c_c @ K @ I4 ) ) ).
% atLeast_iff
thf(fact_285_atLeast__iff,axiom,
! [I4: real,K: real] :
( ( member_real @ I4 @ ( set_ord_atLeast_real @ K ) )
= ( ord_less_eq_real @ K @ I4 ) ) ).
% atLeast_iff
thf(fact_286_atMost__subset__iff,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le657554289um_c_c @ ( set_or326480468um_c_c @ X3 ) @ ( set_or326480468um_c_c @ Y3 ) )
= ( ord_le1772180283um_c_c @ X3 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_287_atMost__subset__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X3 ) @ ( set_ord_atMost_real @ Y3 ) )
= ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% atMost_subset_iff
thf(fact_288_atLeast__subset__iff,axiom,
! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le657554289um_c_c @ ( set_or2028906832um_c_c @ X3 ) @ ( set_or2028906832um_c_c @ Y3 ) )
= ( ord_le1772180283um_c_c @ Y3 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_289_atLeast__subset__iff,axiom,
! [X3: real,Y3: real] :
( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X3 ) @ ( set_ord_atLeast_real @ Y3 ) )
= ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_290_not__Ici__le__Icc,axiom,
! [L: real,L2: real,H2: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ L ) @ ( set_or656347191t_real @ L2 @ H2 ) ) ).
% not_Ici_le_Icc
thf(fact_291_not__Iic__le__Icc,axiom,
! [H: real,L2: real,H2: real] :
~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or656347191t_real @ L2 @ H2 ) ) ).
% not_Iic_le_Icc
thf(fact_292_atLeast__def,axiom,
( set_or2003963744real_c
= ( ^ [L3: finite1398487019real_c] : ( collec230941376real_c @ ( ord_le775706699real_c @ L3 ) ) ) ) ).
% atLeast_def
thf(fact_293_atLeast__def,axiom,
( set_or2028906832um_c_c
= ( ^ [L3: set_Sum_sum_c_c] : ( collec95329456um_c_c @ ( ord_le1772180283um_c_c @ L3 ) ) ) ) ).
% atLeast_def
thf(fact_294_atLeast__def,axiom,
( set_ord_atLeast_real
= ( ^ [L3: real] : ( collect_real @ ( ord_less_eq_real @ L3 ) ) ) ) ).
% atLeast_def
thf(fact_295_atMost__def,axiom,
( set_or1478397028real_c
= ( ^ [U2: finite1398487019real_c] :
( collec230941376real_c
@ ^ [X4: finite1398487019real_c] : ( ord_le775706699real_c @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_296_atMost__def,axiom,
( set_or326480468um_c_c
= ( ^ [U2: set_Sum_sum_c_c] :
( collec95329456um_c_c
@ ^ [X4: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_297_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_298_image__uminus__atMost,axiom,
! [X3: real] :
( ( image_real_real @ uminus_uminus_real @ ( set_ord_atMost_real @ X3 ) )
= ( set_ord_atLeast_real @ ( uminus_uminus_real @ X3 ) ) ) ).
% image_uminus_atMost
thf(fact_299_image__uminus__atLeast,axiom,
! [X3: real] :
( ( image_real_real @ uminus_uminus_real @ ( set_ord_atLeast_real @ X3 ) )
= ( set_ord_atMost_real @ ( uminus_uminus_real @ X3 ) ) ) ).
% image_uminus_atLeast
thf(fact_300_image__eqI,axiom,
! [B: sum_sum_c_c,F: c > sum_sum_c_c,X3: c,A3: set_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ B @ ( image_c_Sum_sum_c_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_301_image__eqI,axiom,
! [B: sum_sum_c_d,F: c > sum_sum_c_d,X3: c,A3: set_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member_c @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ B @ ( image_c_Sum_sum_c_d @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_302_image__eqI,axiom,
! [B: sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ B @ ( image_666880337um_c_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_303_image__eqI,axiom,
! [B: sum_sum_c_d,F: sum_sum_c_c > sum_sum_c_d,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ B @ ( image_675141842um_c_d @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_304_image__eqI,axiom,
! [B: sum_sum_c_c,F: sum_sum_c_d > sum_sum_c_c,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
( ( B
= ( F @ X3 ) )
=> ( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ B @ ( image_1558941394um_c_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_305_image__eqI,axiom,
! [B: sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
( ( B
= ( F @ X3 ) )
=> ( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ B @ ( image_1567202899um_c_d @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_306_image__eqI,axiom,
! [B: sum_sum_c_c,F: produc190496183real_c > sum_sum_c_c,X3: produc190496183real_c,A3: set_Pr1389752855real_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member1895684704real_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ B @ ( image_1467045143um_c_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_307_image__eqI,axiom,
! [B: sum_sum_c_d,F: produc190496183real_c > sum_sum_c_d,X3: produc190496183real_c,A3: set_Pr1389752855real_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member1895684704real_c @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ B @ ( image_1475306648um_c_d @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_308_image__eqI,axiom,
! [B: produc190496183real_c,F: sum_sum_c_c > produc190496183real_c,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( B
= ( F @ X3 ) )
=> ( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member1895684704real_c @ B @ ( image_1146069259real_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_309_image__eqI,axiom,
! [B: produc190496183real_c,F: sum_sum_c_d > produc190496183real_c,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
( ( B
= ( F @ X3 ) )
=> ( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member1895684704real_c @ B @ ( image_1980705354real_c @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_310_image__uminus__atLeastAtMost,axiom,
! [X3: real,Y3: real] :
( ( image_real_real @ uminus_uminus_real @ ( set_or656347191t_real @ X3 @ Y3 ) )
= ( set_or656347191t_real @ ( uminus_uminus_real @ Y3 ) @ ( uminus_uminus_real @ X3 ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_311_subset__image__iff,axiom,
! [B3: set_Sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
( ( ord_le764260924um_c_d @ B3 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A3 )
& ( B3
= ( image_c_Sum_sum_c_d @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_312_subset__image__iff,axiom,
! [B3: set_Sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
( ( ord_le1772180283um_c_c @ B3 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A3 )
& ( B3
= ( image_c_Sum_sum_c_c @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_313_subset__image__iff,axiom,
! [B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B3 @ ( image_666880337um_c_c @ F @ A3 ) )
= ( ? [AA: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ AA @ A3 )
& ( B3
= ( image_666880337um_c_c @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_314_image__subset__iff,axiom,
! [F: c > sum_sum_c_d,A3: set_c,B3: set_Sum_sum_c_d] :
( ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ B3 )
= ( ! [X4: c] :
( ( member_c @ X4 @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_315_image__subset__iff,axiom,
! [F: c > sum_sum_c_c,A3: set_c,B3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ B3 )
= ( ! [X4: c] :
( ( member_c @ X4 @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_316_subset__imageE,axiom,
! [B3: set_Sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
( ( ord_le764260924um_c_d @ B3 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A3 )
=> ( B3
!= ( image_c_Sum_sum_c_d @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_317_subset__imageE,axiom,
! [B3: set_Sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
( ( ord_le1772180283um_c_c @ B3 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A3 )
=> ( B3
!= ( image_c_Sum_sum_c_c @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_318_subset__imageE,axiom,
! [B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B3 @ ( image_666880337um_c_c @ F @ A3 ) )
=> ~ ! [C3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ C3 @ A3 )
=> ( B3
!= ( image_666880337um_c_c @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_319_image__subsetI,axiom,
! [A3: set_c,F: c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
( ! [X: c] :
( ( member_c @ X @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
=> ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_320_image__subsetI,axiom,
! [A3: set_c,F: c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ! [X: c] :
( ( member_c @ X @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
=> ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_321_image__subsetI,axiom,
! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
( ! [X: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
=> ( ord_le764260924um_c_d @ ( image_675141842um_c_d @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_322_image__subsetI,axiom,
! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,B3: set_Sum_sum_c_d] :
( ! [X: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
=> ( ord_le764260924um_c_d @ ( image_1567202899um_c_d @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_323_image__subsetI,axiom,
! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ! [X: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
=> ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_324_image__subsetI,axiom,
! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ! [X: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
=> ( ord_le1772180283um_c_c @ ( image_1558941394um_c_c @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_325_image__subsetI,axiom,
! [A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
( ! [X: produc190496183real_c] :
( ( member1895684704real_c @ X @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
=> ( ord_le764260924um_c_d @ ( image_1475306648um_c_d @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_326_image__subsetI,axiom,
! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > produc190496183real_c,B3: set_Pr1389752855real_c] :
( ! [X: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ A3 )
=> ( member1895684704real_c @ ( F @ X ) @ B3 ) )
=> ( ord_le977353143real_c @ ( image_1146069259real_c @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_327_image__subsetI,axiom,
! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > produc190496183real_c,B3: set_Pr1389752855real_c] :
( ! [X: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ A3 )
=> ( member1895684704real_c @ ( F @ X ) @ B3 ) )
=> ( ord_le977353143real_c @ ( image_1980705354real_c @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_328_image__subsetI,axiom,
! [A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ! [X: produc190496183real_c] :
( ( member1895684704real_c @ X @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
=> ( ord_le1772180283um_c_c @ ( image_1467045143um_c_c @ F @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_329_image__mono,axiom,
! [A3: set_c,B3: set_c,F: c > sum_sum_c_d] :
( ( ord_less_eq_set_c @ A3 @ B3 )
=> ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ ( image_c_Sum_sum_c_d @ F @ B3 ) ) ) ).
% image_mono
thf(fact_330_image__mono,axiom,
! [A3: set_c,B3: set_c,F: c > sum_sum_c_c] :
( ( ord_less_eq_set_c @ A3 @ B3 )
=> ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ ( image_c_Sum_sum_c_c @ F @ B3 ) ) ) ).
% image_mono
thf(fact_331_image__mono,axiom,
! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
=> ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A3 ) @ ( image_666880337um_c_c @ F @ B3 ) ) ) ).
% image_mono
thf(fact_332_solves__ode__subset__range,axiom,
! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,Y7: set_Fi1407883041real_c] :
( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
=> ( ( ord_le1327118209real_c @ ( image_327677598real_c @ X3 @ T2 ) @ Y7 )
=> ( initia1631504802real_c @ X3 @ F @ T2 @ Y7 ) ) ) ).
% solves_ode_subset_range
thf(fact_333_imageI,axiom,
! [X3: c,A3: set_c,F: c > sum_sum_c_c] :
( ( member_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_c_Sum_sum_c_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_334_imageI,axiom,
! [X3: c,A3: set_c,F: c > sum_sum_c_d] :
( ( member_c @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_c_Sum_sum_c_d @ F @ A3 ) ) ) ).
% imageI
thf(fact_335_imageI,axiom,
! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_666880337um_c_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_336_imageI,axiom,
! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d] :
( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_675141842um_c_d @ F @ A3 ) ) ) ).
% imageI
thf(fact_337_imageI,axiom,
! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c] :
( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_1558941394um_c_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_338_imageI,axiom,
! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_1567202899um_c_d @ F @ A3 ) ) ) ).
% imageI
thf(fact_339_imageI,axiom,
! [X3: produc190496183real_c,A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_c] :
( ( member1895684704real_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_1467045143um_c_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_340_imageI,axiom,
! [X3: produc190496183real_c,A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_d] :
( ( member1895684704real_c @ X3 @ A3 )
=> ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_1475306648um_c_d @ F @ A3 ) ) ) ).
% imageI
thf(fact_341_imageI,axiom,
! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > produc190496183real_c] :
( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member1895684704real_c @ ( F @ X3 ) @ ( image_1146069259real_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_342_imageI,axiom,
! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > produc190496183real_c] :
( ( member_Sum_sum_c_d @ X3 @ A3 )
=> ( member1895684704real_c @ ( F @ X3 ) @ ( image_1980705354real_c @ F @ A3 ) ) ) ).
% imageI
thf(fact_343_image__iff,axiom,
! [Z2: sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
( ( member_Sum_sum_c_c @ Z2 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
= ( ? [X4: c] :
( ( member_c @ X4 @ A3 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_344_image__iff,axiom,
! [Z2: sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
( ( member_Sum_sum_c_d @ Z2 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
= ( ? [X4: c] :
( ( member_c @ X4 @ A3 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_345_bex__imageD,axiom,
! [F: c > sum_sum_c_c,A3: set_c,P2: sum_sum_c_c > $o] :
( ? [X7: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X7 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
& ( P2 @ X7 ) )
=> ? [X: c] :
( ( member_c @ X @ A3 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_346_bex__imageD,axiom,
! [F: c > sum_sum_c_d,A3: set_c,P2: sum_sum_c_d > $o] :
( ? [X7: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X7 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
& ( P2 @ X7 ) )
=> ? [X: c] :
( ( member_c @ X @ A3 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_347_image__cong,axiom,
! [M: set_c,N: set_c,F: c > sum_sum_c_c,G: c > sum_sum_c_c] :
( ( M = N )
=> ( ! [X: c] :
( ( member_c @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_c_Sum_sum_c_c @ F @ M )
= ( image_c_Sum_sum_c_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_348_image__cong,axiom,
! [M: set_c,N: set_c,F: c > sum_sum_c_d,G: c > sum_sum_c_d] :
( ( M = N )
=> ( ! [X: c] :
( ( member_c @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_c_Sum_sum_c_d @ F @ M )
= ( image_c_Sum_sum_c_d @ G @ N ) ) ) ) ).
% image_cong
thf(fact_349_Osub,axiom,
ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( static_FVO_a_c @ ode ) ) @ v ).
% Osub
thf(fact_350_VSA,axiom,
( denota256060419gree_c @ ( sol @ zero_zero_real ) @ a
@ ( collect_c
@ ^ [Uu: c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_BVO_a_c @ ode ) )
| ( member_Sum_sum_c_d @ ( sum_Inl_c_d @ Uu ) @ ( image_c_Sum_sum_c_d @ sum_Inl_c_d @ ( static_FVO_a_c @ ode ) ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_FVF_a_b_c @ p ) ) ) ) ) ).
% VSA
thf(fact_351_OVsub_H,axiom,
ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( oDE_dom_a_c @ ode ) ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( oDE_dom_a_c @ ode ) ) ) @ v ).
% OVsub'
% Conjectures (1)
thf(conj_0,conjecture,
denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ).
%------------------------------------------------------------------------------