TPTP Problem File: ITP040^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP040^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_718__7224732_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_718__7224732_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 547 ( 170 unt; 178 typ; 0 def)
% Number of atoms : 1039 ( 297 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3951 ( 92 ~; 9 |; 68 &;3407 @)
% ( 0 <=>; 375 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 622 ( 622 >; 0 *; 0 +; 0 <<)
% Number of symbols : 148 ( 145 usr; 18 con; 0-4 aty)
% Number of variables : 1129 ( 213 ^; 905 !; 11 ?;1129 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:28.764
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
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% Relevant facts (352)
thf(fact_0_ode__to__fvo,axiom,
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( ( member_c @ X @ ( denota540094197_a_b_c @ I @ ODE ) )
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% ode_to_fvo
thf(fact_1_osafe,axiom,
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% osafe
thf(fact_2_eqP,axiom,
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% eqP
thf(fact_3_x,axiom,
ord_less_eq_real @ zero_zero_real @ x ).
% x
thf(fact_4_fsafe,axiom,
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% fsafe
thf(fact_5__092_060open_062_IInl_Ai_A_092_060in_062_AsemBV_AI_AODE_J_A_061_A_IInr_Ai_A_092_060in_062_AsemBV_AI_AODE_J_092_060close_062,axiom,
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= ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( denota1293057250_a_b_c @ i @ oDEa ) ) ) ).
% \<open>(Inl i \<in> semBV I ODE) = (Inr i \<in> semBV I ODE)\<close>
thf(fact_6_VA,axiom,
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( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ X ) )
@ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ X ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) )
@ ( static_FVF_a_b_c @ phi ) ) ).
% VA
thf(fact_7_prod_Ocollapse,axiom,
! [Prod: produc190496183real_c] :
( ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_8_bvo__to__fvo,axiom,
! [X: c,ODE: oDE_a_c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ X ) @ ( static_BVO_a_c @ ODE ) )
=> ( member_c @ X @ ( static_FVO_a_c @ ODE ) ) ) ).
% bvo_to_fvo
thf(fact_9_vec__lambda__eta,axiom,
! [G: finite1398487019real_c] :
( ( finite557796307c_real @ ( finite772340578real_c @ G ) )
= G ) ).
% vec_lambda_eta
thf(fact_10_Inl__Inr__False,axiom,
! [X: c,Y: d] :
( ( sum_Inl_c_d @ X )
!= ( sum_Inr_d_c @ Y ) ) ).
% Inl_Inr_False
thf(fact_11_Inl__Inr__False,axiom,
! [X: c,Y: c] :
( ( sum_Inl_c_c @ X )
!= ( sum_Inr_c_c @ Y ) ) ).
% Inl_Inr_False
thf(fact_12_Inr__Inl__False,axiom,
! [X: d,Y: c] :
( ( sum_Inr_d_c @ X )
!= ( sum_Inl_c_d @ Y ) ) ).
% Inr_Inl_False
thf(fact_13_Inr__Inl__False,axiom,
! [X: c,Y: c] :
( ( sum_Inr_c_c @ X )
!= ( sum_Inl_c_c @ Y ) ) ).
% Inr_Inl_False
thf(fact_14_vec__lambda__beta,axiom,
! [G: c > real,I3: c] :
( ( finite772340578real_c @ ( finite557796307c_real @ G ) @ I3 )
= ( G @ I3 ) ) ).
% vec_lambda_beta
thf(fact_15_alt__sem__lemma,axiom,
! [ODE: oDE_a_c,I: denota231621370t_unit,Sol: real > finite1398487019real_c,T: real,Ab: finite1398487019real_c] :
( ( osafe_a_c @ ODE )
=> ( ( denota1275485728_a_b_c @ I @ ODE @ ( Sol @ T ) )
= ( denota1275485728_a_b_c @ I @ ODE
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ ODE ) ) @ ( finite772340578real_c @ ( Sol @ T ) @ I2 ) @ ( finite772340578real_c @ Ab @ I2 ) ) ) ) ) ) ).
% alt_sem_lemma
thf(fact_16_snd__zero,axiom,
( ( produc314122909real_c @ zero_z1506780526real_c )
= zero_z109254132real_c ) ).
% snd_zero
thf(fact_17_fst__zero,axiom,
( ( produc2010422875real_c @ zero_z1506780526real_c )
= zero_z109254132real_c ) ).
% fst_zero
thf(fact_18_zero__index,axiom,
! [I3: c] :
( ( finite772340578real_c @ zero_z109254132real_c @ I3 )
= zero_zero_real ) ).
% zero_index
thf(fact_19_hpsafe__Evolve_OIH,axiom,
coinci501627771_a_b_c @ p ).
% hpsafe_Evolve.IH
thf(fact_20_hpsafe__Evolve_Ohyps_I1_J,axiom,
osafe_a_c @ ode ).
% hpsafe_Evolve.hyps(1)
thf(fact_21_hpsafe__Evolve_Ohyps_I2_J,axiom,
fsafe_a_b_c @ p ).
% hpsafe_Evolve.hyps(2)
thf(fact_22_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_23_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_24_t,axiom,
ord_less_eq_real @ x @ t ).
% t
thf(fact_25_all,axiom,
! [I4: c] :
( ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ I4 ) @ ( static_BVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I4 )
= ( finite772340578real_c @ ab @ I4 ) ) )
& ( ( member_Sum_sum_c_d @ ( sum_Inl_c_d @ I4 ) @ ( image_c_Sum_sum_c_d @ sum_Inl_c_d @ ( static_FVO_a_c @ oDEa ) ) )
=> ( ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I4 )
= ( finite772340578real_c @ ab @ I4 ) ) )
& ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ I4 ) @ ( static_FVF_a_b_c @ phi ) )
=> ( ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I4 )
= ( finite772340578real_c @ ab @ I4 ) ) ) ) ).
% all
thf(fact_26_less__eq__vec__def,axiom,
( ord_le775706699real_c
= ( ^ [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
! [I2: c] : ( ord_less_eq_real @ ( finite772340578real_c @ X3 @ I2 ) @ ( finite772340578real_c @ Y3 @ I2 ) ) ) ) ).
% less_eq_vec_def
thf(fact_27_old_Oprod_Oinducts,axiom,
! [P: produc190496183real_c > $o,Prod: produc190496183real_c] :
( ! [A3: finite1398487019real_c,B3: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_28_old_Oprod_Oexhaust,axiom,
! [Y: produc190496183real_c] :
~ ! [A3: finite1398487019real_c,B3: finite1398487019real_c] :
( Y
!= ( produc394644079real_c @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_29_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_30_prod__cases,axiom,
! [P: produc190496183real_c > $o,P2: produc190496183real_c] :
( ! [A3: finite1398487019real_c,B3: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_31_surj__pair,axiom,
! [P2: produc190496183real_c] :
? [X4: finite1398487019real_c,Y4: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_32_vec__nth__inject,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( ( finite772340578real_c @ X )
= ( finite772340578real_c @ Y ) )
= ( X = Y ) ) ).
% vec_nth_inject
thf(fact_33_cond__component,axiom,
! [B: $o,X: finite1398487019real_c,Y: finite1398487019real_c,I3: c] :
( ( B
=> ( ( finite772340578real_c @ ( if_Fin36944805real_c @ B @ X @ Y ) @ I3 )
= ( finite772340578real_c @ X @ I3 ) ) )
& ( ~ B
=> ( ( finite772340578real_c @ ( if_Fin36944805real_c @ B @ X @ Y ) @ I3 )
= ( finite772340578real_c @ Y @ I3 ) ) ) ) ).
% cond_component
thf(fact_34_vec__eq__iff,axiom,
( ( ^ [Y5: finite1398487019real_c,Z: finite1398487019real_c] : Y5 = Z )
= ( ^ [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
! [I2: c] :
( ( finite772340578real_c @ X3 @ I2 )
= ( finite772340578real_c @ Y3 @ I2 ) ) ) ) ).
% vec_eq_iff
thf(fact_35_zero__prod__def,axiom,
( zero_z1506780526real_c
= ( produc394644079real_c @ zero_z109254132real_c @ zero_z109254132real_c ) ) ).
% zero_prod_def
thf(fact_36_zero__prod__def,axiom,
( zero_z659284464l_real
= ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_37_mem__Collect__eq,axiom,
! [A: sum_sum_c_c,P: sum_sum_c_c > $o] :
( ( member_Sum_sum_c_c @ A @ ( collect_Sum_sum_c_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_38_mem__Collect__eq,axiom,
! [A: sum_sum_c_d,P: sum_sum_c_d > $o] :
( ( member_Sum_sum_c_d @ A @ ( collect_Sum_sum_c_d @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_39_mem__Collect__eq,axiom,
! [A: produc190496183real_c,P: produc190496183real_c > $o] :
( ( member1895684704real_c @ A @ ( collec1643251106real_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
! [A: real > finite1398487019real_c,P: ( real > finite1398487019real_c ) > $o] :
( ( member1321351885real_c @ A @ ( collec913357835real_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: finite1398487019real_c,P: finite1398487019real_c > $o] :
( ( member1261661570real_c @ A @ ( collec230941376real_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: c,P: c > $o] :
( ( member_c @ A @ ( collect_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A4: set_Sum_sum_c_c] :
( ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] : ( member_Sum_sum_c_c @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A4: set_Sum_sum_c_d] :
( ( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] : ( member_Sum_sum_c_d @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A4: set_Pr1389752855real_c] :
( ( collec1643251106real_c
@ ^ [X3: produc190496183real_c] : ( member1895684704real_c @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A4: set_re2066790124real_c] :
( ( collec913357835real_c
@ ^ [X3: real > finite1398487019real_c] : ( member1321351885real_c @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A4: set_Fi1407883041real_c] :
( ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] : ( member1261661570real_c @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A4: set_c] :
( ( collect_c
@ ^ [X3: c] : ( member_c @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ! [X4: finite1398487019real_c] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec230941376real_c @ P )
= ( collec230941376real_c @ Q ) ) ) ).
% Collect_cong
thf(fact_50_Collect__cong,axiom,
! [P: c > $o,Q: c > $o] :
( ! [X4: c] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_c @ P )
= ( collect_c @ Q ) ) ) ).
% Collect_cong
thf(fact_51_fst__conv,axiom,
! [X1: finite1398487019real_c,X2: finite1398487019real_c] :
( ( produc2010422875real_c @ ( produc394644079real_c @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_52_fst__eqD,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,A: finite1398487019real_c] :
( ( ( produc2010422875real_c @ ( produc394644079real_c @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_53_snd__conv,axiom,
! [X1: finite1398487019real_c,X2: finite1398487019real_c] :
( ( produc314122909real_c @ ( produc394644079real_c @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_54_snd__eqD,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,A: finite1398487019real_c] :
( ( ( produc314122909real_c @ ( produc394644079real_c @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_55_prod__eq__iff,axiom,
( ( ^ [Y5: produc190496183real_c,Z: produc190496183real_c] : Y5 = Z )
= ( ^ [S: produc190496183real_c,T2: produc190496183real_c] :
( ( ( produc2010422875real_c @ S )
= ( produc2010422875real_c @ T2 ) )
& ( ( produc314122909real_c @ S )
= ( produc314122909real_c @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_56_prod_Oexpand,axiom,
! [Prod: produc190496183real_c,Prod2: produc190496183real_c] :
( ( ( ( produc2010422875real_c @ Prod )
= ( produc2010422875real_c @ Prod2 ) )
& ( ( produc314122909real_c @ Prod )
= ( produc314122909real_c @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_57_prod__eqI,axiom,
! [P2: produc190496183real_c,Q2: produc190496183real_c] :
( ( ( produc2010422875real_c @ P2 )
= ( produc2010422875real_c @ Q2 ) )
=> ( ( ( produc314122909real_c @ P2 )
= ( produc314122909real_c @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_58_vec__nth__inverse,axiom,
! [X: finite1398487019real_c] :
( ( finite557796307c_real @ ( finite772340578real_c @ X ) )
= X ) ).
% vec_nth_inverse
thf(fact_59_vec__lambda__unique,axiom,
! [F: finite1398487019real_c,G: c > real] :
( ( ! [I2: c] :
( ( finite772340578real_c @ F @ I2 )
= ( G @ I2 ) ) )
= ( ( finite557796307c_real @ G )
= F ) ) ).
% vec_lambda_unique
thf(fact_60_zero__vec__def,axiom,
( zero_z109254132real_c
= ( finite557796307c_real
@ ^ [I2: c] : zero_zero_real ) ) ).
% zero_vec_def
thf(fact_61_surjective__pairing,axiom,
! [T: produc190496183real_c] :
( T
= ( produc394644079real_c @ ( produc2010422875real_c @ T ) @ ( produc314122909real_c @ T ) ) ) ).
% surjective_pairing
thf(fact_62_prod_Oexhaust__sel,axiom,
! [Prod: produc190496183real_c] :
( Prod
= ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_63_mk__v__concrete,axiom,
( denota161327353_a_b_c
= ( ^ [I5: denota231621370t_unit,ODE2: oDE_a_c,Nu: produc190496183real_c,Sol2: finite1398487019real_c] :
( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( finite772340578real_c @ ( if_Fin36944805real_c @ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ I2 ) @ ( denota1293057250_a_b_c @ I5 @ ODE2 ) ) @ Sol2 @ ( produc2010422875real_c @ Nu ) ) @ I2 ) )
@ ( finite557796307c_real
@ ^ [I2: c] : ( finite772340578real_c @ ( if_Fin36944805real_c @ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ I2 ) @ ( denota1293057250_a_b_c @ I5 @ ODE2 ) ) @ ( denota1275485728_a_b_c @ I5 @ ODE2 @ Sol2 ) @ ( produc314122909real_c @ Nu ) ) @ I2 ) ) ) ) ) ).
% mk_v_concrete
thf(fact_64__092_060open_062Vagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_Imk__xode_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
( ( denota1997846518gree_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) )
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ i @ oDEa ) ) )
& ( denota1997846518gree_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) )
@ ( denota1896987029_a_b_c @ i @ oDEa
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) )
@ ( denota1293057250_a_b_c @ i @ oDEa ) ) ) ).
% \<open>Vagree (mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (- semBV I ODE) \<and> Vagree (mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (mk_xode I ODE (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (semBV I ODE)\<close>
thf(fact_65_Vagree__def,axiom,
( denota1997846518gree_c
= ( ^ [Nu: produc190496183real_c,Nu2: produc190496183real_c,V: set_Sum_sum_c_c] :
( ! [I2: c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ I2 ) @ V )
=> ( ( finite772340578real_c @ ( produc2010422875real_c @ Nu ) @ I2 )
= ( finite772340578real_c @ ( produc2010422875real_c @ Nu2 ) @ I2 ) ) )
& ! [I2: c] :
( ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ I2 ) @ V )
=> ( ( finite772340578real_c @ ( produc314122909real_c @ Nu ) @ I2 )
= ( finite772340578real_c @ ( produc314122909real_c @ Nu2 ) @ I2 ) ) ) ) ) ) ).
% Vagree_def
thf(fact_66__092_060open_062Vagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Iab_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Imk__xode_AI_AODE_A_Isol_Ax_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) @ ( produc394644079real_c @ ab @ bb ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ i @ oDEa ) ) )
& ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) @ ( denota1896987029_a_b_c @ i @ oDEa @ ( sol @ x ) ) @ ( denota1293057250_a_b_c @ i @ oDEa ) ) ) ).
% \<open>Vagree (mk_v I ODE (ab, bb) (sol x)) (ab, bb) (- semBV I ODE) \<and> Vagree (mk_v I ODE (ab, bb) (sol x)) (mk_xode I ODE (sol x)) (semBV I ODE)\<close>
thf(fact_67_concrete__v_Osimps,axiom,
( denota310462944_a_b_c
= ( ^ [I5: denota231621370t_unit,ODE2: oDE_a_c,Nu: produc190496183real_c,Sol2: finite1398487019real_c] :
( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( finite772340578real_c @ ( if_Fin36944805real_c @ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ I2 ) @ ( denota1293057250_a_b_c @ I5 @ ODE2 ) ) @ Sol2 @ ( produc2010422875real_c @ Nu ) ) @ I2 ) )
@ ( finite557796307c_real
@ ^ [I2: c] : ( finite772340578real_c @ ( if_Fin36944805real_c @ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ I2 ) @ ( denota1293057250_a_b_c @ I5 @ ODE2 ) ) @ ( denota1275485728_a_b_c @ I5 @ ODE2 @ Sol2 ) @ ( produc314122909real_c @ Nu ) ) @ I2 ) ) ) ) ) ).
% concrete_v.simps
thf(fact_68_concrete__v_Oelims,axiom,
! [X: denota231621370t_unit,Xa: oDE_a_c,Xb: produc190496183real_c,Xc: finite1398487019real_c,Y: produc190496183real_c] :
( ( ( denota310462944_a_b_c @ X @ Xa @ Xb @ Xc )
= Y )
=> ( Y
= ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( finite772340578real_c @ ( if_Fin36944805real_c @ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ I2 ) @ ( denota1293057250_a_b_c @ X @ Xa ) ) @ Xc @ ( produc2010422875real_c @ Xb ) ) @ I2 ) )
@ ( finite557796307c_real
@ ^ [I2: c] : ( finite772340578real_c @ ( if_Fin36944805real_c @ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ I2 ) @ ( denota1293057250_a_b_c @ X @ Xa ) ) @ ( denota1275485728_a_b_c @ X @ Xa @ Xc ) @ ( produc314122909real_c @ Xb ) ) @ I2 ) ) ) ) ) ).
% concrete_v.elims
thf(fact_69_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_70_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_71_sem,axiom,
! [Nu3: produc190496183real_c,Nu4: produc190496183real_c] :
( ( denota1997846518gree_c @ Nu3 @ Nu4 @ ( static_FVF_a_b_c @ p ) )
=> ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ i @ p ) )
= ( member1895684704real_c @ Nu4 @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% sem
thf(fact_72_aaba,axiom,
( ( produc394644079real_c @ aa @ ba )
= ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ t ) ) ) ).
% aaba
thf(fact_73_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_74_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_75_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_76_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_77_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_78_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_79_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_80_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_81_vector__uminus__component,axiom,
! [X: finite1398487019real_c,I3: c] :
( ( finite772340578real_c @ ( uminus1737280820real_c @ X ) @ I3 )
= ( uminus_uminus_real @ ( finite772340578real_c @ X @ I3 ) ) ) ).
% vector_uminus_component
thf(fact_82_fst__uminus,axiom,
! [X: produc190496183real_c] :
( ( produc2010422875real_c @ ( uminus17964590real_c @ X ) )
= ( uminus1737280820real_c @ ( produc2010422875real_c @ X ) ) ) ).
% fst_uminus
thf(fact_83_snd__uminus,axiom,
! [X: produc190496183real_c] :
( ( produc314122909real_c @ ( uminus17964590real_c @ X ) )
= ( uminus1737280820real_c @ ( produc314122909real_c @ X ) ) ) ).
% snd_uminus
thf(fact_84_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_85_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_86_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_87_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_88_allT,axiom,
! [S2: real] :
( ( ord_less_eq_real @ zero_zero_real @ S2 )
=> ( ( ord_less_eq_real @ S2 @ t )
=> ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ S2 ) ) @ ( denota968303861_a_b_c @ i @ phi ) ) ) ) ).
% allT
thf(fact_89_iff__to__impl,axiom,
! [Nu3: produc190496183real_c,I: denota231621370t_unit,A4: formula_a_b_c,B4: formula_a_b_c] :
( ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ A4 ) )
= ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ B4 ) ) )
= ( ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ A4 ) )
=> ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ B4 ) ) )
& ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ B4 ) )
=> ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ A4 ) ) ) ) ) ).
% iff_to_impl
thf(fact_90_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_91_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_92_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_93_agree__supset,axiom,
! [B4: set_Sum_sum_c_c,A4: set_Sum_sum_c_c,Nu3: produc190496183real_c,Nu4: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ B4 @ A4 )
=> ( ( denota1997846518gree_c @ Nu3 @ Nu4 @ A4 )
=> ( denota1997846518gree_c @ Nu3 @ Nu4 @ B4 ) ) ) ).
% agree_supset
thf(fact_94_agree__sub,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c,Nu3: produc190496183real_c,Omega: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ A4 @ B4 )
=> ( ( denota1997846518gree_c @ Nu3 @ Omega @ B4 )
=> ( denota1997846518gree_c @ Nu3 @ Omega @ A4 ) ) ) ).
% agree_sub
thf(fact_95_mk__v__exists,axiom,
! [Nu3: produc190496183real_c,I: denota231621370t_unit,ODE: oDE_a_c,Sol: finite1398487019real_c] :
? [Omega2: produc190496183real_c] :
( ( denota1997846518gree_c @ Omega2 @ Nu3 @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
& ( denota1997846518gree_c @ Omega2 @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% mk_v_exists
thf(fact_96_uminus__vec__def,axiom,
( uminus1737280820real_c
= ( ^ [X3: finite1398487019real_c] :
( finite557796307c_real
@ ^ [I2: c] : ( uminus_uminus_real @ ( finite772340578real_c @ X3 @ I2 ) ) ) ) ) ).
% uminus_vec_def
thf(fact_97_uminus__prod__def,axiom,
( uminus17964590real_c
= ( ^ [X3: produc190496183real_c] : ( produc394644079real_c @ ( uminus1737280820real_c @ ( produc2010422875real_c @ X3 ) ) @ ( uminus1737280820real_c @ ( produc314122909real_c @ X3 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_98_uminus__prod__def,axiom,
( uminus1225118734um_c_c
= ( ^ [X3: produc214722967um_c_c] : ( produc1439061071um_c_c @ ( uminus1381786404um_c_c @ ( produc1983230011um_c_c @ X3 ) ) @ ( uminus1381786404um_c_c @ ( produc1132019837um_c_c @ X3 ) ) ) ) ) ).
% uminus_prod_def
thf(fact_99_mk__v__agree,axiom,
! [I: denota231621370t_unit,ODE: oDE_a_c,Nu3: produc190496183real_c,Sol: finite1398487019real_c] :
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu3 @ Sol ) @ Nu3 @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
& ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu3 @ Sol ) @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% mk_v_agree
thf(fact_100_mk__xode_Oelims,axiom,
! [X: denota231621370t_unit,Xa: oDE_a_c,Xb: finite1398487019real_c,Y: produc190496183real_c] :
( ( ( denota1896987029_a_b_c @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( produc394644079real_c @ Xb @ ( denota1275485728_a_b_c @ X @ Xa @ Xb ) ) ) ) ).
% mk_xode.elims
thf(fact_101_mk__xode_Osimps,axiom,
( denota1896987029_a_b_c
= ( ^ [I5: denota231621370t_unit,ODE2: oDE_a_c,Sol2: finite1398487019real_c] : ( produc394644079real_c @ Sol2 @ ( denota1275485728_a_b_c @ I5 @ ODE2 @ Sol2 ) ) ) ) ).
% mk_xode.simps
thf(fact_102_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_103_agree__refl,axiom,
! [Nu3: produc190496183real_c,A4: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu3 @ Nu3 @ A4 ) ).
% agree_refl
thf(fact_104_agree__comm,axiom,
! [A4: produc190496183real_c,B4: produc190496183real_c,V2: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ A4 @ B4 @ V2 )
=> ( denota1997846518gree_c @ B4 @ A4 @ V2 ) ) ).
% agree_comm
thf(fact_105_Pair__mono,axiom,
! [X: finite1398487019real_c,X5: finite1398487019real_c,Y: finite1398487019real_c,Y6: finite1398487019real_c] :
( ( ord_le775706699real_c @ X @ X5 )
=> ( ( ord_le775706699real_c @ Y @ Y6 )
=> ( ord_le850691415real_c @ ( produc394644079real_c @ X @ Y ) @ ( produc394644079real_c @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_106_Pair__mono,axiom,
! [X: real,X5: real,Y: real,Y6: real] :
( ( ord_less_eq_real @ X @ X5 )
=> ( ( ord_less_eq_real @ Y @ Y6 )
=> ( ord_le1342644953l_real @ ( produc705216881l_real @ X @ Y ) @ ( produc705216881l_real @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_107_fst__mono,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( ord_le850691415real_c @ X @ Y )
=> ( ord_le775706699real_c @ ( produc2010422875real_c @ X ) @ ( produc2010422875real_c @ Y ) ) ) ).
% fst_mono
thf(fact_108_snd__mono,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( ord_le850691415real_c @ X @ Y )
=> ( ord_le775706699real_c @ ( produc314122909real_c @ X ) @ ( produc314122909real_c @ Y ) ) ) ).
% snd_mono
thf(fact_109_less__eq__prod__def,axiom,
( ord_le850691415real_c
= ( ^ [X3: produc190496183real_c,Y3: produc190496183real_c] :
( ( ord_le775706699real_c @ ( produc2010422875real_c @ X3 ) @ ( produc2010422875real_c @ Y3 ) )
& ( ord_le775706699real_c @ ( produc314122909real_c @ X3 ) @ ( produc314122909real_c @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_110_less__eq__prod__def,axiom,
( ord_le1342644953l_real
= ( ^ [X3: produc957004601l_real,Y3: produc957004601l_real] :
( ( ord_less_eq_real @ ( produc1422991709l_real @ X3 ) @ ( produc1422991709l_real @ Y3 ) )
& ( ord_less_eq_real @ ( produc1407670175l_real @ X3 ) @ ( produc1407670175l_real @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_111_ODE__vars__lr,axiom,
! [X: c,I: denota231621370t_unit,ODE: oDE_a_c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ X ) @ ( denota1293057250_a_b_c @ I @ ODE ) )
= ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ X ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% ODE_vars_lr
thf(fact_112__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Aab_A_061_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
( ( osafe_a_c @ oDEa )
=> ( ( fsafe_a_b_c @ phi )
=> ( ( ode = oDEa )
=> ( ( p = phi )
=> ( ( ord_less_eq_real @ zero_zero_real @ t )
=> ( ( ( produc394644079real_c @ aa @ ba )
= ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ t ) ) )
=> ( ( initia1631504802real_c @ sol
@ ^ [X6: real] : ( denota1275485728_a_b_c @ i @ oDEa )
@ ( set_or656347191t_real @ zero_zero_real @ t )
@ ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ X3 ) @ ( denota968303861_a_b_c @ i @ phi ) ) ) )
=> ( ( denota256060419gree_c @ ( sol @ zero_zero_real ) @ ab
@ ( collect_c
@ ^ [Uu: c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_BVO_a_c @ oDEa ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( static_FVO_a_c @ oDEa ) ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_FVF_a_b_c @ phi ) ) ) ) )
=> ( ab
= ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> ab = (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i)\<close>
thf(fact_113__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_A_061_Amk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_At_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
( ( osafe_a_c @ oDEa )
=> ( ( fsafe_a_b_c @ phi )
=> ( ( ode = oDEa )
=> ( ( p = phi )
=> ( ( ord_less_eq_real @ zero_zero_real @ t )
=> ( ( ( produc394644079real_c @ aa @ ba )
= ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ t ) ) )
=> ( ( initia1631504802real_c @ sol
@ ^ [X6: real] : ( denota1275485728_a_b_c @ i @ oDEa )
@ ( set_or656347191t_real @ zero_zero_real @ t )
@ ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ X3 ) @ ( denota968303861_a_b_c @ i @ phi ) ) ) )
=> ( ( denota256060419gree_c @ ( sol @ zero_zero_real ) @ ab
@ ( collect_c
@ ^ [Uu: c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_BVO_a_c @ oDEa ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( static_FVO_a_c @ oDEa ) ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_FVF_a_b_c @ phi ) ) ) ) )
=> ( ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ t ) )
= ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ t ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> mk_v I ODE (ab, bb) (sol t) = mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol t $ i else ab $ i)\<close>
thf(fact_114__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A0_A_092_060le_062_At_092_060close_062,axiom,
( ( osafe_a_c @ oDEa )
=> ( ( fsafe_a_b_c @ phi )
=> ( ( ode = oDEa )
=> ( ( p = phi )
=> ( ( ord_less_eq_real @ zero_zero_real @ t )
=> ( ( ( produc394644079real_c @ aa @ ba )
= ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ t ) ) )
=> ( ( initia1631504802real_c @ sol
@ ^ [X6: real] : ( denota1275485728_a_b_c @ i @ oDEa )
@ ( set_or656347191t_real @ zero_zero_real @ t )
@ ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ X3 ) @ ( denota968303861_a_b_c @ i @ phi ) ) ) )
=> ( ( denota256060419gree_c @ ( sol @ zero_zero_real ) @ ab
@ ( collect_c
@ ^ [Uu: c] :
( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_BVO_a_c @ oDEa ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( static_FVO_a_c @ oDEa ) ) )
| ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_FVF_a_b_c @ phi ) ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ t ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> 0 \<le> t\<close>
thf(fact_115_mkV,axiom,
( member1321351885real_c @ sol
@ ( pi_rea1111636068real_c @ ( set_or656347191t_real @ zero_zero_real @ t )
@ ^ [Uu: real] :
( collec230941376real_c
@ ^ [X3: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ X3 ) @ ( denota968303861_a_b_c @ i @ phi ) ) ) ) ) ).
% mkV
thf(fact_116_compl__le__compl__iff,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X ) @ ( uminus1381786404um_c_c @ Y ) )
= ( ord_le1772180283um_c_c @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_117_iff__sem,axiom,
! [Nu3: produc190496183real_c,I: denota231621370t_unit,A4: formula_a_b_c,B4: formula_a_b_c] :
( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ ( equiv_a_b_c @ A4 @ B4 ) ) )
= ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ A4 ) )
= ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I @ B4 ) ) ) ) ).
% iff_sem
thf(fact_118_double__compl,axiom,
! [X: set_Sum_sum_c_c] :
( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ X ) )
= X ) ).
% double_compl
thf(fact_119_compl__eq__compl__iff,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ( uminus1381786404um_c_c @ X )
= ( uminus1381786404um_c_c @ Y ) )
= ( X = Y ) ) ).
% compl_eq_compl_iff
thf(fact_120_VSagree__sub,axiom,
! [A4: set_c,B4: set_c,Nu3: finite1398487019real_c,Omega: finite1398487019real_c] :
( ( ord_less_eq_set_c @ A4 @ B4 )
=> ( ( denota256060419gree_c @ Nu3 @ Omega @ B4 )
=> ( denota256060419gree_c @ Nu3 @ Omega @ A4 ) ) ) ).
% VSagree_sub
thf(fact_121_VSagree__supset,axiom,
! [B4: set_c,A4: set_c,Nu3: finite1398487019real_c,Nu4: finite1398487019real_c] :
( ( ord_less_eq_set_c @ B4 @ A4 )
=> ( ( denota256060419gree_c @ Nu3 @ Nu4 @ A4 )
=> ( denota256060419gree_c @ Nu3 @ Nu4 @ B4 ) ) ) ).
% VSagree_supset
thf(fact_122_VSagree__refl,axiom,
! [Nu3: finite1398487019real_c,A4: set_c] : ( denota256060419gree_c @ Nu3 @ Nu3 @ A4 ) ).
% VSagree_refl
thf(fact_123_VSagree__def,axiom,
( denota256060419gree_c
= ( ^ [Nu: finite1398487019real_c,Nu2: finite1398487019real_c,V: set_c] :
! [X3: c] :
( ( member_c @ X3 @ V )
=> ( ( finite772340578real_c @ Nu @ X3 )
= ( finite772340578real_c @ Nu2 @ X3 ) ) ) ) ) ).
% VSagree_def
thf(fact_124_compl__mono,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y ) @ ( uminus1381786404um_c_c @ X ) ) ) ).
% compl_mono
thf(fact_125_compl__le__swap1,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y @ ( uminus1381786404um_c_c @ X ) )
=> ( ord_le1772180283um_c_c @ X @ ( uminus1381786404um_c_c @ Y ) ) ) ).
% compl_le_swap1
thf(fact_126_compl__le__swap2,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y ) @ X )
=> ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_127_image__uminus__atLeastAtMost,axiom,
! [X: real,Y: real] :
( ( image_real_real @ uminus_uminus_real @ ( set_or656347191t_real @ X @ Y ) )
= ( set_or656347191t_real @ ( uminus_uminus_real @ Y ) @ ( uminus_uminus_real @ X ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_128_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_129_Compl__subset__Compl__iff,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ A4 ) @ ( uminus1381786404um_c_c @ B4 ) )
= ( ord_le1772180283um_c_c @ B4 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_130_Compl__anti__mono,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A4 @ B4 )
=> ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ B4 ) @ ( uminus1381786404um_c_c @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_131_image__eqI,axiom,
! [B: c,F: c > c,X: c,A4: set_c] :
( ( B
= ( F @ X ) )
=> ( ( member_c @ X @ A4 )
=> ( member_c @ B @ ( image_c_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_132_image__eqI,axiom,
! [B: c,F: sum_sum_c_c > c,X: sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( B
= ( F @ X ) )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_c @ B @ ( image_Sum_sum_c_c_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_133_image__eqI,axiom,
! [B: sum_sum_c_c,F: c > sum_sum_c_c,X: c,A4: set_c] :
( ( B
= ( F @ X ) )
=> ( ( member_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ B @ ( image_c_Sum_sum_c_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_134_image__eqI,axiom,
! [B: sum_sum_c_d,F: c > sum_sum_c_d,X: c,A4: set_c] :
( ( B
= ( F @ X ) )
=> ( ( member_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ B @ ( image_c_Sum_sum_c_d @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_135_image__eqI,axiom,
! [B: c,F: sum_sum_c_d > c,X: sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ( B
= ( F @ X ) )
=> ( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_c @ B @ ( image_Sum_sum_c_d_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_136_image__eqI,axiom,
! [B: sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,X: sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( B
= ( F @ X ) )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ B @ ( image_666880337um_c_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_137_image__eqI,axiom,
! [B: sum_sum_c_d,F: sum_sum_c_c > sum_sum_c_d,X: sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( B
= ( F @ X ) )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ B @ ( image_675141842um_c_d @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_138_image__eqI,axiom,
! [B: real > finite1398487019real_c,F: c > real > finite1398487019real_c,X: c,A4: set_c] :
( ( B
= ( F @ X ) )
=> ( ( member_c @ X @ A4 )
=> ( member1321351885real_c @ B @ ( image_1226855921real_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_139_image__eqI,axiom,
! [B: sum_sum_c_c,F: sum_sum_c_d > sum_sum_c_c,X: sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ( B
= ( F @ X ) )
=> ( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_Sum_sum_c_c @ B @ ( image_1558941394um_c_c @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_140_image__eqI,axiom,
! [B: sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,X: sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ( B
= ( F @ X ) )
=> ( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_Sum_sum_c_d @ B @ ( image_1567202899um_c_d @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_141_subsetI,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ X4 @ B4 ) )
=> ( ord_le1772180283um_c_c @ A4 @ B4 ) ) ).
% subsetI
thf(fact_142_subsetI,axiom,
! [A4: set_c,B4: set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_c @ X4 @ B4 ) )
=> ( ord_less_eq_set_c @ A4 @ B4 ) ) ).
% subsetI
thf(fact_143_subsetI,axiom,
! [A4: set_Sum_sum_c_d,B4: set_Sum_sum_c_d] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ X4 @ B4 ) )
=> ( ord_le764260924um_c_d @ A4 @ B4 ) ) ).
% subsetI
thf(fact_144_subsetI,axiom,
! [A4: set_Pr1389752855real_c,B4: set_Pr1389752855real_c] :
( ! [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A4 )
=> ( member1895684704real_c @ X4 @ B4 ) )
=> ( ord_le977353143real_c @ A4 @ B4 ) ) ).
% subsetI
thf(fact_145_subsetI,axiom,
! [A4: set_re2066790124real_c,B4: set_re2066790124real_c] :
( ! [X4: real > finite1398487019real_c] :
( ( member1321351885real_c @ X4 @ A4 )
=> ( member1321351885real_c @ X4 @ B4 ) )
=> ( ord_le35002188real_c @ A4 @ B4 ) ) ).
% subsetI
thf(fact_146_Compl__eq__Compl__iff,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ( ( uminus1381786404um_c_c @ A4 )
= ( uminus1381786404um_c_c @ B4 ) )
= ( A4 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_147_Compl__iff,axiom,
! [C: c,A4: set_c] :
( ( member_c @ C @ ( uminus_uminus_set_c @ A4 ) )
= ( ~ ( member_c @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_148_Compl__iff,axiom,
! [C: sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A4 ) )
= ( ~ ( member_Sum_sum_c_d @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_149_Compl__iff,axiom,
! [C: produc190496183real_c,A4: set_Pr1389752855real_c] :
( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A4 ) )
= ( ~ ( member1895684704real_c @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_150_Compl__iff,axiom,
! [C: real > finite1398487019real_c,A4: set_re2066790124real_c] :
( ( member1321351885real_c @ C @ ( uminus946614837real_c @ A4 ) )
= ( ~ ( member1321351885real_c @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_151_Compl__iff,axiom,
! [C: sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A4 ) )
= ( ~ ( member_Sum_sum_c_c @ C @ A4 ) ) ) ).
% Compl_iff
thf(fact_152_ComplI,axiom,
! [C: c,A4: set_c] :
( ~ ( member_c @ C @ A4 )
=> ( member_c @ C @ ( uminus_uminus_set_c @ A4 ) ) ) ).
% ComplI
thf(fact_153_ComplI,axiom,
! [C: sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ~ ( member_Sum_sum_c_d @ C @ A4 )
=> ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A4 ) ) ) ).
% ComplI
thf(fact_154_ComplI,axiom,
! [C: produc190496183real_c,A4: set_Pr1389752855real_c] :
( ~ ( member1895684704real_c @ C @ A4 )
=> ( member1895684704real_c @ C @ ( uminus232257166real_c @ A4 ) ) ) ).
% ComplI
thf(fact_155_ComplI,axiom,
! [C: real > finite1398487019real_c,A4: set_re2066790124real_c] :
( ~ ( member1321351885real_c @ C @ A4 )
=> ( member1321351885real_c @ C @ ( uminus946614837real_c @ A4 ) ) ) ).
% ComplI
thf(fact_156_ComplI,axiom,
! [C: sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ~ ( member_Sum_sum_c_c @ C @ A4 )
=> ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A4 ) ) ) ).
% ComplI
thf(fact_157_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_158_atLeastAtMost__iff,axiom,
! [I3: produc190496183real_c,L: produc190496183real_c,U: produc190496183real_c] :
( ( member1895684704real_c @ I3 @ ( set_or385782508real_c @ L @ U ) )
= ( ( ord_le850691415real_c @ L @ I3 )
& ( ord_le850691415real_c @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_159_atLeastAtMost__iff,axiom,
! [I3: real > finite1398487019real_c,L: real > finite1398487019real_c,U: real > finite1398487019real_c] :
( ( member1321351885real_c @ I3 @ ( set_or1308937537real_c @ L @ U ) )
= ( ( ord_le584953750real_c @ L @ I3 )
& ( ord_le584953750real_c @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_160_atLeastAtMost__iff,axiom,
! [I3: real,L: real,U: real] :
( ( member_real @ I3 @ ( set_or656347191t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I3 )
& ( ord_less_eq_real @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_161_imageI,axiom,
! [X: c,A4: set_c,F: c > c] :
( ( member_c @ X @ A4 )
=> ( member_c @ ( F @ X ) @ ( image_c_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_162_imageI,axiom,
! [X: sum_sum_c_c,A4: set_Sum_sum_c_c,F: sum_sum_c_c > c] :
( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_c @ ( F @ X ) @ ( image_Sum_sum_c_c_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_163_imageI,axiom,
! [X: c,A4: set_c,F: c > sum_sum_c_c] :
( ( member_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ ( image_c_Sum_sum_c_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_164_imageI,axiom,
! [X: c,A4: set_c,F: c > sum_sum_c_d] :
( ( member_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ ( image_c_Sum_sum_c_d @ F @ A4 ) ) ) ).
% imageI
thf(fact_165_imageI,axiom,
! [X: sum_sum_c_d,A4: set_Sum_sum_c_d,F: sum_sum_c_d > c] :
( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_c @ ( F @ X ) @ ( image_Sum_sum_c_d_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_166_imageI,axiom,
! [X: sum_sum_c_c,A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ ( image_666880337um_c_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_167_imageI,axiom,
! [X: sum_sum_c_c,A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d] :
( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ ( image_675141842um_c_d @ F @ A4 ) ) ) ).
% imageI
thf(fact_168_imageI,axiom,
! [X: c,A4: set_c,F: c > real > finite1398487019real_c] :
( ( member_c @ X @ A4 )
=> ( member1321351885real_c @ ( F @ X ) @ ( image_1226855921real_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_169_imageI,axiom,
! [X: sum_sum_c_d,A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c] :
( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ ( image_1558941394um_c_c @ F @ A4 ) ) ) ).
% imageI
thf(fact_170_imageI,axiom,
! [X: sum_sum_c_d,A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ ( image_1567202899um_c_d @ F @ A4 ) ) ) ).
% imageI
thf(fact_171_image__iff,axiom,
! [Z2: sum_sum_c_c,F: c > sum_sum_c_c,A4: set_c] :
( ( member_Sum_sum_c_c @ Z2 @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
= ( ? [X3: c] :
( ( member_c @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_172_image__iff,axiom,
! [Z2: sum_sum_c_d,F: c > sum_sum_c_d,A4: set_c] :
( ( member_Sum_sum_c_d @ Z2 @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
= ( ? [X3: c] :
( ( member_c @ X3 @ A4 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_173_bex__imageD,axiom,
! [F: c > sum_sum_c_c,A4: set_c,P: sum_sum_c_c > $o] :
( ? [X7: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X7 @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: c] :
( ( member_c @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_174_bex__imageD,axiom,
! [F: c > sum_sum_c_d,A4: set_c,P: sum_sum_c_d > $o] :
( ? [X7: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X7 @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
& ( P @ X7 ) )
=> ? [X4: c] :
( ( member_c @ X4 @ A4 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_175_image__cong,axiom,
! [M: set_c,N: set_c,F: c > sum_sum_c_c,G: c > sum_sum_c_c] :
( ( M = N )
=> ( ! [X4: c] :
( ( member_c @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_c_Sum_sum_c_c @ F @ M )
= ( image_c_Sum_sum_c_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_176_image__cong,axiom,
! [M: set_c,N: set_c,F: c > sum_sum_c_d,G: c > sum_sum_c_d] :
( ( M = N )
=> ( ! [X4: c] :
( ( member_c @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_c_Sum_sum_c_d @ F @ M )
= ( image_c_Sum_sum_c_d @ G @ N ) ) ) ) ).
% image_cong
thf(fact_177_ball__imageD,axiom,
! [F: c > sum_sum_c_c,A4: set_c,P: sum_sum_c_c > $o] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: c] :
( ( member_c @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_178_ball__imageD,axiom,
! [F: c > sum_sum_c_d,A4: set_c,P: sum_sum_c_d > $o] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
=> ( P @ X4 ) )
=> ! [X7: c] :
( ( member_c @ X7 @ A4 )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_179_rev__image__eqI,axiom,
! [X: c,A4: set_c,B: c,F: c > c] :
( ( member_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_c @ B @ ( image_c_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_180_rev__image__eqI,axiom,
! [X: sum_sum_c_c,A4: set_Sum_sum_c_c,B: c,F: sum_sum_c_c > c] :
( ( member_Sum_sum_c_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_c @ B @ ( image_Sum_sum_c_c_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_181_rev__image__eqI,axiom,
! [X: c,A4: set_c,B: sum_sum_c_c,F: c > sum_sum_c_c] :
( ( member_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_Sum_sum_c_c @ B @ ( image_c_Sum_sum_c_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_182_rev__image__eqI,axiom,
! [X: c,A4: set_c,B: sum_sum_c_d,F: c > sum_sum_c_d] :
( ( member_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_Sum_sum_c_d @ B @ ( image_c_Sum_sum_c_d @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_183_rev__image__eqI,axiom,
! [X: sum_sum_c_d,A4: set_Sum_sum_c_d,B: c,F: sum_sum_c_d > c] :
( ( member_Sum_sum_c_d @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_c @ B @ ( image_Sum_sum_c_d_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_184_rev__image__eqI,axiom,
! [X: sum_sum_c_c,A4: set_Sum_sum_c_c,B: sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_Sum_sum_c_c @ B @ ( image_666880337um_c_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_185_rev__image__eqI,axiom,
! [X: sum_sum_c_c,A4: set_Sum_sum_c_c,B: sum_sum_c_d,F: sum_sum_c_c > sum_sum_c_d] :
( ( member_Sum_sum_c_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_Sum_sum_c_d @ B @ ( image_675141842um_c_d @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_186_rev__image__eqI,axiom,
! [X: c,A4: set_c,B: real > finite1398487019real_c,F: c > real > finite1398487019real_c] :
( ( member_c @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member1321351885real_c @ B @ ( image_1226855921real_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_187_rev__image__eqI,axiom,
! [X: sum_sum_c_d,A4: set_Sum_sum_c_d,B: sum_sum_c_c,F: sum_sum_c_d > sum_sum_c_c] :
( ( member_Sum_sum_c_d @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_Sum_sum_c_c @ B @ ( image_1558941394um_c_c @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_188_rev__image__eqI,axiom,
! [X: sum_sum_c_d,A4: set_Sum_sum_c_d,B: sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X @ A4 )
=> ( ( B
= ( F @ X ) )
=> ( member_Sum_sum_c_d @ B @ ( image_1567202899um_c_d @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_189_Collect__mono__iff,axiom,
! [P: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ( ord_le1327118209real_c @ ( collec230941376real_c @ P ) @ ( collec230941376real_c @ Q ) )
= ( ! [X3: finite1398487019real_c] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_190_Collect__mono__iff,axiom,
! [P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) )
= ( ! [X3: c] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_191_Collect__mono,axiom,
! [P: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
( ! [X4: finite1398487019real_c] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le1327118209real_c @ ( collec230941376real_c @ P ) @ ( collec230941376real_c @ Q ) ) ) ).
% Collect_mono
thf(fact_192_Collect__mono,axiom,
! [P: c > $o,Q: c > $o] :
( ! [X4: c] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) ) ) ).
% Collect_mono
thf(fact_193_subset__iff,axiom,
( ord_le1772180283um_c_c
= ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
! [T2: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ T2 @ A5 )
=> ( member_Sum_sum_c_c @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_194_subset__iff,axiom,
( ord_less_eq_set_c
= ( ^ [A5: set_c,B5: set_c] :
! [T2: c] :
( ( member_c @ T2 @ A5 )
=> ( member_c @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_195_subset__iff,axiom,
( ord_le764260924um_c_d
= ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
! [T2: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ T2 @ A5 )
=> ( member_Sum_sum_c_d @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_196_subset__iff,axiom,
( ord_le977353143real_c
= ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
! [T2: produc190496183real_c] :
( ( member1895684704real_c @ T2 @ A5 )
=> ( member1895684704real_c @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_197_subset__iff,axiom,
( ord_le35002188real_c
= ( ^ [A5: set_re2066790124real_c,B5: set_re2066790124real_c] :
! [T2: real > finite1398487019real_c] :
( ( member1321351885real_c @ T2 @ A5 )
=> ( member1321351885real_c @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_198_subset__eq,axiom,
( ord_le1772180283um_c_c
= ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
! [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A5 )
=> ( member_Sum_sum_c_c @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_199_subset__eq,axiom,
( ord_less_eq_set_c
= ( ^ [A5: set_c,B5: set_c] :
! [X3: c] :
( ( member_c @ X3 @ A5 )
=> ( member_c @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_200_subset__eq,axiom,
( ord_le764260924um_c_d
= ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
! [X3: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ A5 )
=> ( member_Sum_sum_c_d @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_201_subset__eq,axiom,
( ord_le977353143real_c
= ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
! [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ A5 )
=> ( member1895684704real_c @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_202_subset__eq,axiom,
( ord_le35002188real_c
= ( ^ [A5: set_re2066790124real_c,B5: set_re2066790124real_c] :
! [X3: real > finite1398487019real_c] :
( ( member1321351885real_c @ X3 @ A5 )
=> ( member1321351885real_c @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_203_subsetD,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c,C: sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A4 @ B4 )
=> ( ( member_Sum_sum_c_c @ C @ A4 )
=> ( member_Sum_sum_c_c @ C @ B4 ) ) ) ).
% subsetD
thf(fact_204_subsetD,axiom,
! [A4: set_c,B4: set_c,C: c] :
( ( ord_less_eq_set_c @ A4 @ B4 )
=> ( ( member_c @ C @ A4 )
=> ( member_c @ C @ B4 ) ) ) ).
% subsetD
thf(fact_205_subsetD,axiom,
! [A4: set_Sum_sum_c_d,B4: set_Sum_sum_c_d,C: sum_sum_c_d] :
( ( ord_le764260924um_c_d @ A4 @ B4 )
=> ( ( member_Sum_sum_c_d @ C @ A4 )
=> ( member_Sum_sum_c_d @ C @ B4 ) ) ) ).
% subsetD
thf(fact_206_subsetD,axiom,
! [A4: set_Pr1389752855real_c,B4: set_Pr1389752855real_c,C: produc190496183real_c] :
( ( ord_le977353143real_c @ A4 @ B4 )
=> ( ( member1895684704real_c @ C @ A4 )
=> ( member1895684704real_c @ C @ B4 ) ) ) ).
% subsetD
thf(fact_207_subsetD,axiom,
! [A4: set_re2066790124real_c,B4: set_re2066790124real_c,C: real > finite1398487019real_c] :
( ( ord_le35002188real_c @ A4 @ B4 )
=> ( ( member1321351885real_c @ C @ A4 )
=> ( member1321351885real_c @ C @ B4 ) ) ) ).
% subsetD
thf(fact_208_in__mono,axiom,
! [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c,X: sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A4 @ B4 )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ X @ B4 ) ) ) ).
% in_mono
thf(fact_209_in__mono,axiom,
! [A4: set_c,B4: set_c,X: c] :
( ( ord_less_eq_set_c @ A4 @ B4 )
=> ( ( member_c @ X @ A4 )
=> ( member_c @ X @ B4 ) ) ) ).
% in_mono
thf(fact_210_in__mono,axiom,
! [A4: set_Sum_sum_c_d,B4: set_Sum_sum_c_d,X: sum_sum_c_d] :
( ( ord_le764260924um_c_d @ A4 @ B4 )
=> ( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_Sum_sum_c_d @ X @ B4 ) ) ) ).
% in_mono
thf(fact_211_in__mono,axiom,
! [A4: set_Pr1389752855real_c,B4: set_Pr1389752855real_c,X: produc190496183real_c] :
( ( ord_le977353143real_c @ A4 @ B4 )
=> ( ( member1895684704real_c @ X @ A4 )
=> ( member1895684704real_c @ X @ B4 ) ) ) ).
% in_mono
thf(fact_212_in__mono,axiom,
! [A4: set_re2066790124real_c,B4: set_re2066790124real_c,X: real > finite1398487019real_c] :
( ( ord_le35002188real_c @ A4 @ B4 )
=> ( ( member1321351885real_c @ X @ A4 )
=> ( member1321351885real_c @ X @ B4 ) ) ) ).
% in_mono
thf(fact_213_double__complement,axiom,
! [A4: set_Sum_sum_c_c] :
( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_214_ComplD,axiom,
! [C: c,A4: set_c] :
( ( member_c @ C @ ( uminus_uminus_set_c @ A4 ) )
=> ~ ( member_c @ C @ A4 ) ) ).
% ComplD
thf(fact_215_ComplD,axiom,
! [C: sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A4 ) )
=> ~ ( member_Sum_sum_c_d @ C @ A4 ) ) ).
% ComplD
thf(fact_216_ComplD,axiom,
! [C: produc190496183real_c,A4: set_Pr1389752855real_c] :
( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A4 ) )
=> ~ ( member1895684704real_c @ C @ A4 ) ) ).
% ComplD
thf(fact_217_ComplD,axiom,
! [C: real > finite1398487019real_c,A4: set_re2066790124real_c] :
( ( member1321351885real_c @ C @ ( uminus946614837real_c @ A4 ) )
=> ~ ( member1321351885real_c @ C @ A4 ) ) ).
% ComplD
thf(fact_218_ComplD,axiom,
! [C: sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A4 ) )
=> ~ ( member_Sum_sum_c_c @ C @ A4 ) ) ).
% ComplD
thf(fact_219_imageE,axiom,
! [B: c,F: c > c,A4: set_c] :
( ( member_c @ B @ ( image_c_c @ F @ A4 ) )
=> ~ ! [X4: c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_220_imageE,axiom,
! [B: sum_sum_c_c,F: c > sum_sum_c_c,A4: set_c] :
( ( member_Sum_sum_c_c @ B @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
=> ~ ! [X4: c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_221_imageE,axiom,
! [B: c,F: sum_sum_c_c > c,A4: set_Sum_sum_c_c] :
( ( member_c @ B @ ( image_Sum_sum_c_c_c @ F @ A4 ) )
=> ~ ! [X4: sum_sum_c_c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_c_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_222_imageE,axiom,
! [B: c,F: sum_sum_c_d > c,A4: set_Sum_sum_c_d] :
( ( member_c @ B @ ( image_Sum_sum_c_d_c @ F @ A4 ) )
=> ~ ! [X4: sum_sum_c_d] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_c_d @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_223_imageE,axiom,
! [B: sum_sum_c_d,F: c > sum_sum_c_d,A4: set_c] :
( ( member_Sum_sum_c_d @ B @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
=> ~ ! [X4: c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_224_imageE,axiom,
! [B: sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ B @ ( image_666880337um_c_c @ F @ A4 ) )
=> ~ ! [X4: sum_sum_c_c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_c_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_225_imageE,axiom,
! [B: sum_sum_c_c,F: sum_sum_c_d > sum_sum_c_c,A4: set_Sum_sum_c_d] :
( ( member_Sum_sum_c_c @ B @ ( image_1558941394um_c_c @ F @ A4 ) )
=> ~ ! [X4: sum_sum_c_d] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_c_d @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_226_imageE,axiom,
! [B: c,F: ( real > finite1398487019real_c ) > c,A4: set_re2066790124real_c] :
( ( member_c @ B @ ( image_266490115al_c_c @ F @ A4 ) )
=> ~ ! [X4: real > finite1398487019real_c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member1321351885real_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_227_imageE,axiom,
! [B: sum_sum_c_d,F: sum_sum_c_c > sum_sum_c_d,A4: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_d @ B @ ( image_675141842um_c_d @ F @ A4 ) )
=> ~ ! [X4: sum_sum_c_c] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_c_c @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_228_imageE,axiom,
! [B: sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,A4: set_Sum_sum_c_d] :
( ( member_Sum_sum_c_d @ B @ ( image_1567202899um_c_d @ F @ A4 ) )
=> ~ ! [X4: sum_sum_c_d] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_Sum_sum_c_d @ X4 @ A4 ) ) ) ).
% imageE
thf(fact_229_image__image,axiom,
! [F: sum_sum_c_c > sum_sum_c_c,G: c > sum_sum_c_c,A4: set_c] :
( ( image_666880337um_c_c @ F @ ( image_c_Sum_sum_c_c @ G @ A4 ) )
= ( image_c_Sum_sum_c_c
@ ^ [X3: c] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_230_image__image,axiom,
! [F: sum_sum_c_c > sum_sum_c_d,G: c > sum_sum_c_c,A4: set_c] :
( ( image_675141842um_c_d @ F @ ( image_c_Sum_sum_c_c @ G @ A4 ) )
= ( image_c_Sum_sum_c_d
@ ^ [X3: c] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_231_image__image,axiom,
! [F: sum_sum_c_d > sum_sum_c_c,G: c > sum_sum_c_d,A4: set_c] :
( ( image_1558941394um_c_c @ F @ ( image_c_Sum_sum_c_d @ G @ A4 ) )
= ( image_c_Sum_sum_c_c
@ ^ [X3: c] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_232_image__image,axiom,
! [F: sum_sum_c_d > sum_sum_c_d,G: c > sum_sum_c_d,A4: set_c] :
( ( image_1567202899um_c_d @ F @ ( image_c_Sum_sum_c_d @ G @ A4 ) )
= ( image_c_Sum_sum_c_d
@ ^ [X3: c] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_233_image__image,axiom,
! [F: c > sum_sum_c_c,G: c > c,A4: set_c] :
( ( image_c_Sum_sum_c_c @ F @ ( image_c_c @ G @ A4 ) )
= ( image_c_Sum_sum_c_c
@ ^ [X3: c] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_234_image__image,axiom,
! [F: c > sum_sum_c_d,G: c > c,A4: set_c] :
( ( image_c_Sum_sum_c_d @ F @ ( image_c_c @ G @ A4 ) )
= ( image_c_Sum_sum_c_d
@ ^ [X3: c] : ( F @ ( G @ X3 ) )
@ A4 ) ) ).
% image_image
thf(fact_235_Compr__image__eq,axiom,
! [F: c > c,A4: set_c,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_c_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_c_c @ F
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_236_Compr__image__eq,axiom,
! [F: c > sum_sum_c_c,A4: set_c,P: sum_sum_c_c > $o] :
( ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_c_Sum_sum_c_c @ F
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_237_Compr__image__eq,axiom,
! [F: c > sum_sum_c_d,A4: set_c,P: sum_sum_c_d > $o] :
( ( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_c_Sum_sum_c_d @ F
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_238_Compr__image__eq,axiom,
! [F: c > finite1398487019real_c,A4: set_c,P: finite1398487019real_c > $o] :
( ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] :
( ( member1261661570real_c @ X3 @ ( image_687300774real_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_687300774real_c @ F
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_239_Compr__image__eq,axiom,
! [F: sum_sum_c_c > c,A4: set_Sum_sum_c_c,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_Sum_sum_c_c_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_Sum_sum_c_c_c @ F
@ ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_240_Compr__image__eq,axiom,
! [F: sum_sum_c_d > c,A4: set_Sum_sum_c_d,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_Sum_sum_c_d_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_Sum_sum_c_d_c @ F
@ ( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_241_Compr__image__eq,axiom,
! [F: finite1398487019real_c > c,A4: set_Fi1407883041real_c,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_1260608398al_c_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_1260608398al_c_c @ F
@ ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] :
( ( member1261661570real_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_242_Compr__image__eq,axiom,
! [F: sum_sum_c_c > sum_sum_c_c,A4: set_Sum_sum_c_c,P: sum_sum_c_c > $o] :
( ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ ( image_666880337um_c_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_666880337um_c_c @ F
@ ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_243_Compr__image__eq,axiom,
! [F: sum_sum_c_d > sum_sum_c_c,A4: set_Sum_sum_c_d,P: sum_sum_c_c > $o] :
( ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ ( image_1558941394um_c_c @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_1558941394um_c_c @ F
@ ( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_244_Compr__image__eq,axiom,
! [F: sum_sum_c_c > sum_sum_c_d,A4: set_Sum_sum_c_c,P: sum_sum_c_d > $o] :
( ( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ ( image_675141842um_c_d @ F @ A4 ) )
& ( P @ X3 ) ) )
= ( image_675141842um_c_d @ F
@ ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A4 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_245_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
@ ^ [X3: sum_sum_c_c] : ( member_Sum_sum_c_c @ X3 @ A5 )
@ ^ [X3: sum_sum_c_c] : ( member_Sum_sum_c_c @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_246_less__eq__set__def,axiom,
( ord_less_eq_set_c
= ( ^ [A5: set_c,B5: set_c] :
( ord_less_eq_c_o
@ ^ [X3: c] : ( member_c @ X3 @ A5 )
@ ^ [X3: c] : ( member_c @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_247_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
@ ^ [X3: sum_sum_c_d] : ( member_Sum_sum_c_d @ X3 @ A5 )
@ ^ [X3: sum_sum_c_d] : ( member_Sum_sum_c_d @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_248_less__eq__set__def,axiom,
( ord_le977353143real_c
= ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
( ord_le1132241894al_c_o
@ ^ [X3: produc190496183real_c] : ( member1895684704real_c @ X3 @ A5 )
@ ^ [X3: produc190496183real_c] : ( member1895684704real_c @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_249_less__eq__set__def,axiom,
( ord_le35002188real_c
= ( ^ [A5: set_re2066790124real_c,B5: set_re2066790124real_c] :
( ord_le818951225al_c_o
@ ^ [X3: real > finite1398487019real_c] : ( member1321351885real_c @ X3 @ A5 )
@ ^ [X3: real > finite1398487019real_c] : ( member1321351885real_c @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_250_Collect__subset,axiom,
! [A4: set_Sum_sum_c_c,P: sum_sum_c_c > $o] :
( ord_le1772180283um_c_c
@ ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_251_Collect__subset,axiom,
! [A4: set_Sum_sum_c_d,P: sum_sum_c_d > $o] :
( ord_le764260924um_c_d
@ ( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_252_Collect__subset,axiom,
! [A4: set_Pr1389752855real_c,P: produc190496183real_c > $o] :
( ord_le977353143real_c
@ ( collec1643251106real_c
@ ^ [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_253_Collect__subset,axiom,
! [A4: set_re2066790124real_c,P: ( real > finite1398487019real_c ) > $o] :
( ord_le35002188real_c
@ ( collec913357835real_c
@ ^ [X3: real > finite1398487019real_c] :
( ( member1321351885real_c @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_254_Collect__subset,axiom,
! [A4: set_Fi1407883041real_c,P: finite1398487019real_c > $o] :
( ord_le1327118209real_c
@ ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] :
( ( member1261661570real_c @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_255_Collect__subset,axiom,
! [A4: set_c,P: c > $o] :
( ord_less_eq_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A4 )
& ( P @ X3 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_256_Collect__neg__eq,axiom,
! [P: finite1398487019real_c > $o] :
( ( collec230941376real_c
@ ^ [X3: finite1398487019real_c] :
~ ( P @ X3 ) )
= ( uminus102402154real_c @ ( collec230941376real_c @ P ) ) ) ).
% Collect_neg_eq
thf(fact_257_Collect__neg__eq,axiom,
! [P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
~ ( P @ X3 ) )
= ( uminus_uminus_set_c @ ( collect_c @ P ) ) ) ).
% Collect_neg_eq
thf(fact_258_Collect__neg__eq,axiom,
! [P: sum_sum_c_c > $o] :
( ( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
~ ( P @ X3 ) )
= ( uminus1381786404um_c_c @ ( collect_Sum_sum_c_c @ P ) ) ) ).
% Collect_neg_eq
thf(fact_259_Compl__eq,axiom,
( uminus373867045um_c_d
= ( ^ [A5: set_Sum_sum_c_d] :
( collect_Sum_sum_c_d
@ ^ [X3: sum_sum_c_d] :
~ ( member_Sum_sum_c_d @ X3 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_260_Compl__eq,axiom,
( uminus232257166real_c
= ( ^ [A5: set_Pr1389752855real_c] :
( collec1643251106real_c
@ ^ [X3: produc190496183real_c] :
~ ( member1895684704real_c @ X3 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_261_Compl__eq,axiom,
( uminus946614837real_c
= ( ^ [A5: set_re2066790124real_c] :
( collec913357835real_c
@ ^ [X3: real > finite1398487019real_c] :
~ ( member1321351885real_c @ X3 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_262_Compl__eq,axiom,
( uminus102402154real_c
= ( ^ [A5: set_Fi1407883041real_c] :
( collec230941376real_c
@ ^ [X3: finite1398487019real_c] :
~ ( member1261661570real_c @ X3 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_263_Compl__eq,axiom,
( uminus_uminus_set_c
= ( ^ [A5: set_c] :
( collect_c
@ ^ [X3: c] :
~ ( member_c @ X3 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_264_Compl__eq,axiom,
( uminus1381786404um_c_c
= ( ^ [A5: set_Sum_sum_c_c] :
( collect_Sum_sum_c_c
@ ^ [X3: sum_sum_c_c] :
~ ( member_Sum_sum_c_c @ X3 @ A5 ) ) ) ) ).
% Compl_eq
thf(fact_265_uminus__set__def,axiom,
( uminus373867045um_c_d
= ( ^ [A5: set_Sum_sum_c_d] :
( collect_Sum_sum_c_d
@ ( uminus916499104_c_d_o
@ ^ [X3: sum_sum_c_d] : ( member_Sum_sum_c_d @ X3 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_266_uminus__set__def,axiom,
( uminus232257166real_c
= ( ^ [A5: set_Pr1389752855real_c] :
( collec1643251106real_c
@ ( uminus32155087al_c_o
@ ^ [X3: produc190496183real_c] : ( member1895684704real_c @ X3 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_267_uminus__set__def,axiom,
( uminus946614837real_c
= ( ^ [A5: set_re2066790124real_c] :
( collec913357835real_c
@ ( uminus1756072592al_c_o
@ ^ [X3: real > finite1398487019real_c] : ( member1321351885real_c @ X3 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_268_uminus__set__def,axiom,
( uminus102402154real_c
= ( ^ [A5: set_Fi1407883041real_c] :
( collec230941376real_c
@ ( uminus23580315al_c_o
@ ^ [X3: finite1398487019real_c] : ( member1261661570real_c @ X3 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_269_uminus__set__def,axiom,
( uminus_uminus_set_c
= ( ^ [A5: set_c] :
( collect_c
@ ( uminus_uminus_c_o
@ ^ [X3: c] : ( member_c @ X3 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_270_uminus__set__def,axiom,
( uminus1381786404um_c_c
= ( ^ [A5: set_Sum_sum_c_c] :
( collect_Sum_sum_c_c
@ ( uminus88729697_c_c_o
@ ^ [X3: sum_sum_c_c] : ( member_Sum_sum_c_c @ X3 @ A5 ) ) ) ) ) ).
% uminus_set_def
thf(fact_271_image__mono,axiom,
! [A4: set_c,B4: set_c,F: c > sum_sum_c_c] :
( ( ord_less_eq_set_c @ A4 @ B4 )
=> ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A4 ) @ ( image_c_Sum_sum_c_c @ F @ B4 ) ) ) ).
% image_mono
thf(fact_272_image__mono,axiom,
! [A4: set_c,B4: set_c,F: c > sum_sum_c_d] :
( ( ord_less_eq_set_c @ A4 @ B4 )
=> ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A4 ) @ ( image_c_Sum_sum_c_d @ F @ B4 ) ) ) ).
% image_mono
thf(fact_273_image__subsetI,axiom,
! [A4: set_c,F: c > c,B4: set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_less_eq_set_c @ ( image_c_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_274_image__subsetI,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > c,B4: set_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_less_eq_set_c @ ( image_Sum_sum_c_c_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_275_image__subsetI,axiom,
! [A4: set_c,F: c > sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_276_image__subsetI,axiom,
! [A4: set_c,F: c > sum_sum_c_d,B4: set_Sum_sum_c_d] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ B4 ) )
=> ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_277_image__subsetI,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > c,B4: set_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_less_eq_set_c @ ( image_Sum_sum_c_d_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_278_image__subsetI,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_279_image__subsetI,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B4: set_Sum_sum_c_d] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ B4 ) )
=> ( ord_le764260924um_c_d @ ( image_675141842um_c_d @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_280_image__subsetI,axiom,
! [A4: set_c,F: c > real > finite1398487019real_c,B4: set_re2066790124real_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member1321351885real_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_le35002188real_c @ ( image_1226855921real_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_281_image__subsetI,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B4 ) )
=> ( ord_le1772180283um_c_c @ ( image_1558941394um_c_c @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_282_image__subsetI,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,B4: set_Sum_sum_c_d] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ B4 ) )
=> ( ord_le764260924um_c_d @ ( image_1567202899um_c_d @ F @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_283_subset__imageE,axiom,
! [B4: set_Sum_sum_c_c,F: c > sum_sum_c_c,A4: set_c] :
( ( ord_le1772180283um_c_c @ B4 @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
=> ~ ! [C2: set_c] :
( ( ord_less_eq_set_c @ C2 @ A4 )
=> ( B4
!= ( image_c_Sum_sum_c_c @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_284_subset__imageE,axiom,
! [B4: set_Sum_sum_c_d,F: c > sum_sum_c_d,A4: set_c] :
( ( ord_le764260924um_c_d @ B4 @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
=> ~ ! [C2: set_c] :
( ( ord_less_eq_set_c @ C2 @ A4 )
=> ( B4
!= ( image_c_Sum_sum_c_d @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_285_image__subset__iff,axiom,
! [F: c > sum_sum_c_c,A4: set_c,B4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A4 ) @ B4 )
= ( ! [X3: c] :
( ( member_c @ X3 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_286_image__subset__iff,axiom,
! [F: c > sum_sum_c_d,A4: set_c,B4: set_Sum_sum_c_d] :
( ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A4 ) @ B4 )
= ( ! [X3: c] :
( ( member_c @ X3 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_287_subset__image__iff,axiom,
! [B4: set_Sum_sum_c_c,F: c > sum_sum_c_c,A4: set_c] :
( ( ord_le1772180283um_c_c @ B4 @ ( image_c_Sum_sum_c_c @ F @ A4 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A4 )
& ( B4
= ( image_c_Sum_sum_c_c @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_288_subset__image__iff,axiom,
! [B4: set_Sum_sum_c_d,F: c > sum_sum_c_d,A4: set_c] :
( ( ord_le764260924um_c_d @ B4 @ ( image_c_Sum_sum_c_d @ F @ A4 ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A4 )
& ( B4
= ( image_c_Sum_sum_c_d @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_289_repd_Oelims,axiom,
! [X: produc190496183real_c,Xa: c,Xb: real,Y: produc190496183real_c] :
( ( ( denotational_repd_c @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( produc394644079real_c @ ( produc2010422875real_c @ X )
@ ( finite557796307c_real
@ ^ [Y3: c] : ( if_real @ ( Xa = Y3 ) @ Xb @ ( finite772340578real_c @ ( produc314122909real_c @ X ) @ Y3 ) ) ) ) ) ) ).
% repd.elims
thf(fact_290_repd_Osimps,axiom,
( denotational_repd_c
= ( ^ [V3: produc190496183real_c,X3: c,R: real] :
( produc394644079real_c @ ( produc2010422875real_c @ V3 )
@ ( finite557796307c_real
@ ^ [Y3: c] : ( if_real @ ( X3 = Y3 ) @ R @ ( finite772340578real_c @ ( produc314122909real_c @ V3 ) @ Y3 ) ) ) ) ) ) ).
% repd.simps
thf(fact_291_image__subset__iff__funcset,axiom,
! [F2: c > sum_sum_c_c,A4: set_c,B4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F2 @ A4 ) @ B4 )
= ( member_c_Sum_sum_c_c @ F2
@ ( pi_c_Sum_sum_c_c @ A4
@ ^ [Uu: c] : B4 ) ) ) ).
% image_subset_iff_funcset
thf(fact_292_image__subset__iff__funcset,axiom,
! [F2: c > sum_sum_c_d,A4: set_c,B4: set_Sum_sum_c_d] :
( ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F2 @ A4 ) @ B4 )
= ( member_c_Sum_sum_c_d @ F2
@ ( pi_c_Sum_sum_c_d @ A4
@ ^ [Uu: c] : B4 ) ) ) ).
% image_subset_iff_funcset
thf(fact_293_image__subset__iff__funcset,axiom,
! [F2: real > finite1398487019real_c,A4: set_real,B4: set_Fi1407883041real_c] :
( ( ord_le1327118209real_c @ ( image_327677598real_c @ F2 @ A4 ) @ B4 )
= ( member1321351885real_c @ F2
@ ( pi_rea1111636068real_c @ A4
@ ^ [Uu: real] : B4 ) ) ) ).
% image_subset_iff_funcset
thf(fact_294_Pi__I,axiom,
! [A4: set_c,F: c > c,B4: c > set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_c_c @ F @ ( pi_c_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_295_Pi__I,axiom,
! [A4: set_real,F: real > finite1398487019real_c,B4: real > set_Fi1407883041real_c] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( member1261661570real_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member1321351885real_c @ F @ ( pi_rea1111636068real_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_296_Pi__I,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > c,B4: sum_sum_c_c > set_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_Sum_sum_c_c_c @ F @ ( pi_Sum_sum_c_c_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_297_Pi__I,axiom,
! [A4: set_c,F: c > sum_sum_c_c,B4: c > set_Sum_sum_c_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_c_Sum_sum_c_c @ F @ ( pi_c_Sum_sum_c_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_298_Pi__I,axiom,
! [A4: set_c,F: c > sum_sum_c_d,B4: c > set_Sum_sum_c_d] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_c_Sum_sum_c_d @ F @ ( pi_c_Sum_sum_c_d @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_299_Pi__I,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > c,B4: sum_sum_c_d > set_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_Sum_sum_c_d_c @ F @ ( pi_Sum_sum_c_d_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_300_Pi__I,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B4: sum_sum_c_c > set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member1326674048um_c_c @ F @ ( pi_Sum253006871um_c_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_301_Pi__I,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B4: sum_sum_c_c > set_Sum_sum_c_d] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member318754689um_c_d @ F @ ( pi_Sum261268376um_c_d @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_302_Pi__I,axiom,
! [A4: set_c,F: c > real > finite1398487019real_c,B4: c > set_re2066790124real_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member1321351885real_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member469531344real_c @ F @ ( pi_c_r41638571real_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_303_Pi__I,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B4: sum_sum_c_d > set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member71251457um_c_c @ F @ ( pi_Sum1145067928um_c_c @ A4 @ B4 ) ) ) ).
% Pi_I
thf(fact_304_PiE,axiom,
! [F: c > c,A4: set_c,B4: c > set_c,X: c] :
( ( member_c_c @ F @ ( pi_c_c @ A4 @ B4 ) )
=> ( ~ ( member_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_305_PiE,axiom,
! [F: c > sum_sum_c_c,A4: set_c,B4: c > set_Sum_sum_c_c,X: c] :
( ( member_c_Sum_sum_c_c @ F @ ( pi_c_Sum_sum_c_c @ A4 @ B4 ) )
=> ( ~ ( member_Sum_sum_c_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_306_PiE,axiom,
! [F: sum_sum_c_c > c,A4: set_Sum_sum_c_c,B4: sum_sum_c_c > set_c,X: sum_sum_c_c] :
( ( member_Sum_sum_c_c_c @ F @ ( pi_Sum_sum_c_c_c @ A4 @ B4 ) )
=> ( ~ ( member_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_Sum_sum_c_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_307_PiE,axiom,
! [F: sum_sum_c_d > c,A4: set_Sum_sum_c_d,B4: sum_sum_c_d > set_c,X: sum_sum_c_d] :
( ( member_Sum_sum_c_d_c @ F @ ( pi_Sum_sum_c_d_c @ A4 @ B4 ) )
=> ( ~ ( member_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_Sum_sum_c_d @ X @ A4 ) ) ) ).
% PiE
thf(fact_308_PiE,axiom,
! [F: c > sum_sum_c_d,A4: set_c,B4: c > set_Sum_sum_c_d,X: c] :
( ( member_c_Sum_sum_c_d @ F @ ( pi_c_Sum_sum_c_d @ A4 @ B4 ) )
=> ( ~ ( member_Sum_sum_c_d @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_309_PiE,axiom,
! [F: real > finite1398487019real_c,A4: set_real,B4: real > set_Fi1407883041real_c,X: real] :
( ( member1321351885real_c @ F @ ( pi_rea1111636068real_c @ A4 @ B4 ) )
=> ( ~ ( member1261661570real_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_real @ X @ A4 ) ) ) ).
% PiE
thf(fact_310_PiE,axiom,
! [F: sum_sum_c_c > sum_sum_c_c,A4: set_Sum_sum_c_c,B4: sum_sum_c_c > set_Sum_sum_c_c,X: sum_sum_c_c] :
( ( member1326674048um_c_c @ F @ ( pi_Sum253006871um_c_c @ A4 @ B4 ) )
=> ( ~ ( member_Sum_sum_c_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_Sum_sum_c_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_311_PiE,axiom,
! [F: sum_sum_c_d > sum_sum_c_c,A4: set_Sum_sum_c_d,B4: sum_sum_c_d > set_Sum_sum_c_c,X: sum_sum_c_d] :
( ( member71251457um_c_c @ F @ ( pi_Sum1145067928um_c_c @ A4 @ B4 ) )
=> ( ~ ( member_Sum_sum_c_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_Sum_sum_c_d @ X @ A4 ) ) ) ).
% PiE
thf(fact_312_PiE,axiom,
! [F: ( real > finite1398487019real_c ) > c,A4: set_re2066790124real_c,B4: ( real > finite1398487019real_c ) > set_c,X: real > finite1398487019real_c] :
( ( member527957930al_c_c @ F @ ( pi_rea1228756413al_c_c @ A4 @ B4 ) )
=> ( ~ ( member_c @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member1321351885real_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_313_PiE,axiom,
! [F: sum_sum_c_c > sum_sum_c_d,A4: set_Sum_sum_c_c,B4: sum_sum_c_c > set_Sum_sum_c_d,X: sum_sum_c_c] :
( ( member318754689um_c_d @ F @ ( pi_Sum261268376um_c_d @ A4 @ B4 ) )
=> ( ~ ( member_Sum_sum_c_d @ ( F @ X ) @ ( B4 @ X ) )
=> ~ ( member_Sum_sum_c_c @ X @ A4 ) ) ) ).
% PiE
thf(fact_314_Pi__I_H,axiom,
! [A4: set_c,F: c > c,B4: c > set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_c_c @ F @ ( pi_c_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_315_Pi__I_H,axiom,
! [A4: set_real,F: real > finite1398487019real_c,B4: real > set_Fi1407883041real_c] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( member1261661570real_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member1321351885real_c @ F @ ( pi_rea1111636068real_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_316_Pi__I_H,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > c,B4: sum_sum_c_c > set_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_Sum_sum_c_c_c @ F @ ( pi_Sum_sum_c_c_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_317_Pi__I_H,axiom,
! [A4: set_c,F: c > sum_sum_c_c,B4: c > set_Sum_sum_c_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_c_Sum_sum_c_c @ F @ ( pi_c_Sum_sum_c_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_318_Pi__I_H,axiom,
! [A4: set_c,F: c > sum_sum_c_d,B4: c > set_Sum_sum_c_d] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_c_Sum_sum_c_d @ F @ ( pi_c_Sum_sum_c_d @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_319_Pi__I_H,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > c,B4: sum_sum_c_d > set_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member_Sum_sum_c_d_c @ F @ ( pi_Sum_sum_c_d_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_320_Pi__I_H,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B4: sum_sum_c_c > set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member1326674048um_c_c @ F @ ( pi_Sum253006871um_c_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_321_Pi__I_H,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B4: sum_sum_c_c > set_Sum_sum_c_d] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member318754689um_c_d @ F @ ( pi_Sum261268376um_c_d @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_322_Pi__I_H,axiom,
! [A4: set_c,F: c > real > finite1398487019real_c,B4: c > set_re2066790124real_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member1321351885real_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member469531344real_c @ F @ ( pi_c_r41638571real_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_323_Pi__I_H,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B4: sum_sum_c_d > set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ ( B4 @ X4 ) ) )
=> ( member71251457um_c_c @ F @ ( pi_Sum1145067928um_c_c @ A4 @ B4 ) ) ) ).
% Pi_I'
thf(fact_324_Pi__iff,axiom,
! [F: real > finite1398487019real_c,I: set_real,X8: real > set_Fi1407883041real_c] :
( ( member1321351885real_c @ F @ ( pi_rea1111636068real_c @ I @ X8 ) )
= ( ! [X3: real] :
( ( member_real @ X3 @ I )
=> ( member1261661570real_c @ ( F @ X3 ) @ ( X8 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_325_Pi__mem,axiom,
! [F: c > c,A4: set_c,B4: c > set_c,X: c] :
( ( member_c_c @ F @ ( pi_c_c @ A4 @ B4 ) )
=> ( ( member_c @ X @ A4 )
=> ( member_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_326_Pi__mem,axiom,
! [F: sum_sum_c_c > c,A4: set_Sum_sum_c_c,B4: sum_sum_c_c > set_c,X: sum_sum_c_c] :
( ( member_Sum_sum_c_c_c @ F @ ( pi_Sum_sum_c_c_c @ A4 @ B4 ) )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_327_Pi__mem,axiom,
! [F: c > sum_sum_c_c,A4: set_c,B4: c > set_Sum_sum_c_c,X: c] :
( ( member_c_Sum_sum_c_c @ F @ ( pi_c_Sum_sum_c_c @ A4 @ B4 ) )
=> ( ( member_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_328_Pi__mem,axiom,
! [F: c > sum_sum_c_d,A4: set_c,B4: c > set_Sum_sum_c_d,X: c] :
( ( member_c_Sum_sum_c_d @ F @ ( pi_c_Sum_sum_c_d @ A4 @ B4 ) )
=> ( ( member_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_329_Pi__mem,axiom,
! [F: sum_sum_c_d > c,A4: set_Sum_sum_c_d,B4: sum_sum_c_d > set_c,X: sum_sum_c_d] :
( ( member_Sum_sum_c_d_c @ F @ ( pi_Sum_sum_c_d_c @ A4 @ B4 ) )
=> ( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_330_Pi__mem,axiom,
! [F: real > finite1398487019real_c,A4: set_real,B4: real > set_Fi1407883041real_c,X: real] :
( ( member1321351885real_c @ F @ ( pi_rea1111636068real_c @ A4 @ B4 ) )
=> ( ( member_real @ X @ A4 )
=> ( member1261661570real_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_331_Pi__mem,axiom,
! [F: sum_sum_c_c > sum_sum_c_c,A4: set_Sum_sum_c_c,B4: sum_sum_c_c > set_Sum_sum_c_c,X: sum_sum_c_c] :
( ( member1326674048um_c_c @ F @ ( pi_Sum253006871um_c_c @ A4 @ B4 ) )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_332_Pi__mem,axiom,
! [F: sum_sum_c_c > sum_sum_c_d,A4: set_Sum_sum_c_c,B4: sum_sum_c_c > set_Sum_sum_c_d,X: sum_sum_c_c] :
( ( member318754689um_c_d @ F @ ( pi_Sum261268376um_c_d @ A4 @ B4 ) )
=> ( ( member_Sum_sum_c_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_333_Pi__mem,axiom,
! [F: c > real > finite1398487019real_c,A4: set_c,B4: c > set_re2066790124real_c,X: c] :
( ( member469531344real_c @ F @ ( pi_c_r41638571real_c @ A4 @ B4 ) )
=> ( ( member_c @ X @ A4 )
=> ( member1321351885real_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_334_Pi__mem,axiom,
! [F: sum_sum_c_d > sum_sum_c_c,A4: set_Sum_sum_c_d,B4: sum_sum_c_d > set_Sum_sum_c_c,X: sum_sum_c_d] :
( ( member71251457um_c_c @ F @ ( pi_Sum1145067928um_c_c @ A4 @ B4 ) )
=> ( ( member_Sum_sum_c_d @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ ( B4 @ X ) ) ) ) ).
% Pi_mem
thf(fact_335_Pi__cong,axiom,
! [A4: set_real,F: real > finite1398487019real_c,G: real > finite1398487019real_c,B4: real > set_Fi1407883041real_c] :
( ! [W: real] :
( ( member_real @ W @ A4 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member1321351885real_c @ F @ ( pi_rea1111636068real_c @ A4 @ B4 ) )
= ( member1321351885real_c @ G @ ( pi_rea1111636068real_c @ A4 @ B4 ) ) ) ) ).
% Pi_cong
thf(fact_336_funcsetI,axiom,
! [A4: set_c,F: c > c,B4: set_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ B4 ) )
=> ( member_c_c @ F
@ ( pi_c_c @ A4
@ ^ [Uu: c] : B4 ) ) ) ).
% funcsetI
thf(fact_337_funcsetI,axiom,
! [A4: set_real,F: real > finite1398487019real_c,B4: set_Fi1407883041real_c] :
( ! [X4: real] :
( ( member_real @ X4 @ A4 )
=> ( member1261661570real_c @ ( F @ X4 ) @ B4 ) )
=> ( member1321351885real_c @ F
@ ( pi_rea1111636068real_c @ A4
@ ^ [Uu: real] : B4 ) ) ) ).
% funcsetI
thf(fact_338_funcsetI,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > c,B4: set_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ B4 ) )
=> ( member_Sum_sum_c_c_c @ F
@ ( pi_Sum_sum_c_c_c @ A4
@ ^ [Uu: sum_sum_c_c] : B4 ) ) ) ).
% funcsetI
thf(fact_339_funcsetI,axiom,
! [A4: set_c,F: c > sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B4 ) )
=> ( member_c_Sum_sum_c_c @ F
@ ( pi_c_Sum_sum_c_c @ A4
@ ^ [Uu: c] : B4 ) ) ) ).
% funcsetI
thf(fact_340_funcsetI,axiom,
! [A4: set_c,F: c > sum_sum_c_d,B4: set_Sum_sum_c_d] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ B4 ) )
=> ( member_c_Sum_sum_c_d @ F
@ ( pi_c_Sum_sum_c_d @ A4
@ ^ [Uu: c] : B4 ) ) ) ).
% funcsetI
thf(fact_341_funcsetI,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > c,B4: set_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_c @ ( F @ X4 ) @ B4 ) )
=> ( member_Sum_sum_c_d_c @ F
@ ( pi_Sum_sum_c_d_c @ A4
@ ^ [Uu: sum_sum_c_d] : B4 ) ) ) ).
% funcsetI
thf(fact_342_funcsetI,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B4 ) )
=> ( member1326674048um_c_c @ F
@ ( pi_Sum253006871um_c_c @ A4
@ ^ [Uu: sum_sum_c_c] : B4 ) ) ) ).
% funcsetI
thf(fact_343_funcsetI,axiom,
! [A4: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B4: set_Sum_sum_c_d] :
( ! [X4: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X4 @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X4 ) @ B4 ) )
=> ( member318754689um_c_d @ F
@ ( pi_Sum261268376um_c_d @ A4
@ ^ [Uu: sum_sum_c_c] : B4 ) ) ) ).
% funcsetI
thf(fact_344_funcsetI,axiom,
! [A4: set_c,F: c > real > finite1398487019real_c,B4: set_re2066790124real_c] :
( ! [X4: c] :
( ( member_c @ X4 @ A4 )
=> ( member1321351885real_c @ ( F @ X4 ) @ B4 ) )
=> ( member469531344real_c @ F
@ ( pi_c_r41638571real_c @ A4
@ ^ [Uu: c] : B4 ) ) ) ).
% funcsetI
thf(fact_345_funcsetI,axiom,
! [A4: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ! [X4: sum_sum_c_d] :
( ( member_Sum_sum_c_d @ X4 @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X4 ) @ B4 ) )
=> ( member71251457um_c_c @ F
@ ( pi_Sum1145067928um_c_c @ A4
@ ^ [Uu: sum_sum_c_d] : B4 ) ) ) ).
% funcsetI
thf(fact_346_funcset__mem,axiom,
! [F: ( real > finite1398487019real_c ) > sum_sum_c_c,A4: set_re2066790124real_c,B4: set_Sum_sum_c_c,X: real > finite1398487019real_c] :
( ( member354622481um_c_c @ F
@ ( pi_rea1500330408um_c_c @ A4
@ ^ [Uu: real > finite1398487019real_c] : B4 ) )
=> ( ( member1321351885real_c @ X @ A4 )
=> ( member_Sum_sum_c_c @ ( F @ X ) @ B4 ) ) ) ).
% funcset_mem
thf(fact_347_funcset__mem,axiom,
! [F: ( real > finite1398487019real_c ) > c,A4: set_re2066790124real_c,B4: set_c,X: real > finite1398487019real_c] :
( ( member527957930al_c_c @ F
@ ( pi_rea1228756413al_c_c @ A4
@ ^ [Uu: real > finite1398487019real_c] : B4 ) )
=> ( ( member1321351885real_c @ X @ A4 )
=> ( member_c @ ( F @ X ) @ B4 ) ) ) ).
% funcset_mem
thf(fact_348_funcset__mem,axiom,
! [F: ( real > finite1398487019real_c ) > sum_sum_c_d,A4: set_re2066790124real_c,B4: set_Sum_sum_c_d,X: real > finite1398487019real_c] :
( ( member1494186770um_c_d @ F
@ ( pi_rea1508591913um_c_d @ A4
@ ^ [Uu: real > finite1398487019real_c] : B4 ) )
=> ( ( member1321351885real_c @ X @ A4 )
=> ( member_Sum_sum_c_d @ ( F @ X ) @ B4 ) ) ) ).
% funcset_mem
thf(fact_349_funcset__mem,axiom,
! [F: ( real > finite1398487019real_c ) > produc190496183real_c,A4: set_re2066790124real_c,B4: set_Pr1389752855real_c,X: real > finite1398487019real_c] :
( ( member954762593real_c @ F
@ ( pi_rea1379668980real_c @ A4
@ ^ [Uu: real > finite1398487019real_c] : B4 ) )
=> ( ( member1321351885real_c @ X @ A4 )
=> ( member1895684704real_c @ ( F @ X ) @ B4 ) ) ) ).
% funcset_mem
thf(fact_350_funcset__mem,axiom,
! [F: ( real > finite1398487019real_c ) > real > finite1398487019real_c,A4: set_re2066790124real_c,B4: set_re2066790124real_c,X: real > finite1398487019real_c] :
( ( member1851015714real_c @ F
@ ( pi_rea1386151481real_c @ A4
@ ^ [Uu: real > finite1398487019real_c] : B4 ) )
=> ( ( member1321351885real_c @ X @ A4 )
=> ( member1321351885real_c @ ( F @ X ) @ B4 ) ) ) ).
% funcset_mem
thf(fact_351_funcset__mem,axiom,
! [F: real > finite1398487019real_c,A4: set_real,B4: set_Fi1407883041real_c,X: real] :
( ( member1321351885real_c @ F
@ ( pi_rea1111636068real_c @ A4
@ ^ [Uu: real] : B4 ) )
=> ( ( member_real @ X @ A4 )
=> ( member1261661570real_c @ ( F @ X ) @ B4 ) ) ) ).
% funcset_mem
% Helper facts (5)
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_T,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( if_Fin36944805real_c @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_T,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( if_Fin36944805real_c @ $true @ X @ Y )
= X ) ).
% Conjectures (12)
thf(conj_0,hypothesis,
member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( static_FVF_a_b_c @ phi ) ).
thf(conj_1,hypothesis,
member_c @ i2 @ ( denota540094197_a_b_c @ i @ oDEa ) ).
thf(conj_2,hypothesis,
~ ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) ) ).
thf(conj_3,hypothesis,
( ( ~ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
& ~ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) ) )
=> ( ( ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c
@ ( produc2010422875real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ ( sol @ zero_zero_real ) @ i2 ) ) )
& ( ~ ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c
@ ( produc2010422875real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ ab @ i2 ) ) ) ) ) ).
thf(conj_4,hypothesis,
( ( ~ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
& ~ ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) ) )
=> ( ( finite772340578real_c @ ( produc2010422875real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite772340578real_c @ ab @ i2 ) ) ) ).
thf(conj_5,hypothesis,
( ( ~ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
& ~ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) ) )
=> ( ( finite772340578real_c
@ ( produc314122909real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ bb @ i2 ) ) ) ).
thf(conj_6,hypothesis,
( ( ~ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
& ~ ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) ) )
=> ( ( finite772340578real_c @ ( produc314122909real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite772340578real_c @ bb @ i2 ) ) ) ).
thf(conj_7,hypothesis,
( ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c
@ ( produc2010422875real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ ( sol @ x ) @ i2 ) ) )
& ( ~ ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c
@ ( produc2010422875real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ ab @ i2 ) ) ) ) )
& ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c
@ ( produc2010422875real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ ( sol @ x ) @ i2 ) ) )
& ( ~ ( member_c @ i2 @ ( static_FVO_a_c @ oDEa ) )
=> ( ( finite772340578real_c
@ ( produc2010422875real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c @ ab @ i2 ) ) ) ) ) ) ).
thf(conj_8,hypothesis,
( ( ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( finite772340578real_c
@ ( produc314122909real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c
@ ( denota1275485728_a_b_c @ i @ oDEa
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) )
@ i2 ) ) )
& ( ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( finite772340578real_c
@ ( produc314122909real_c
@ ( denota161327353_a_b_c @ i @ oDEa
@ ( produc394644079real_c
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ zero_zero_real ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) )
@ bb )
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) ) )
@ i2 )
= ( finite772340578real_c
@ ( denota1275485728_a_b_c @ i @ oDEa
@ ( finite557796307c_real
@ ^ [I2: c] : ( if_real @ ( member_c @ I2 @ ( static_FVO_a_c @ oDEa ) ) @ ( finite772340578real_c @ ( sol @ x ) @ I2 ) @ ( finite772340578real_c @ ab @ I2 ) ) ) )
@ i2 ) ) ) ) ).
thf(conj_9,hypothesis,
( ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( finite772340578real_c @ ( produc2010422875real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite772340578real_c @ ( sol @ x ) @ i2 ) ) )
& ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( finite772340578real_c @ ( produc2010422875real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite772340578real_c @ ( sol @ x ) @ i2 ) ) ) ) ).
thf(conj_10,hypothesis,
( ( ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( finite772340578real_c @ ( produc314122909real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite772340578real_c @ ( denota1275485728_a_b_c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) )
& ( ( member_Sum_sum_c_c @ ( sum_Inr_c_c @ i2 ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( denota540094197_a_b_c @ i @ oDEa ) ) )
=> ( ( finite772340578real_c @ ( produc314122909real_c @ ( denota161327353_a_b_c @ i @ oDEa @ ( produc394644079real_c @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite772340578real_c @ ( denota1275485728_a_b_c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) ) ) ).
thf(conj_11,conjecture,
( ( finite772340578real_c @ ab @ i2 )
= ( finite772340578real_c @ ( sol @ x ) @ i2 ) ) ).
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