TPTP Problem File: ITP036^1.p
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%------------------------------------------------------------------------------
% File : ITP036^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_1096__7244526_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_1096__7244526_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 468 ( 195 unt; 113 typ; 0 def)
% Number of atoms : 826 ( 271 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2724 ( 36 ~; 7 |; 43 &;2295 @)
% ( 0 <=>; 343 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 214 ( 214 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 92 usr; 12 con; 0-3 aty)
% Number of variables : 997 ( 101 ^; 881 !; 15 ?; 997 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:47:10.370
%------------------------------------------------------------------------------
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thf(fact_0_hpsafe__Choice_OIH_I2_J,axiom,
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denota1997846518gree_c @ ( produc394644079real_c @ mu3 @ mu2 ) @ mu @ ( sup_su1349021703um_c_c @ ( static_MBV_a_b_c @ b ) @ v ) ).
% agree
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060mu_062_H_H_O_A_092_060lbrakk_062_I_I_092_060nu_0622_M_A_092_060nu_0622_H_J_M_A_092_060mu_062_H_H_J_A_092_060in_062_Aprog__sem_AJ_Ab_059_AVagree_A_I_092_060mu_062_M_A_092_060mu_062_H_J_A_092_060mu_062_H_H_A_IMBV_Ab_A_092_060union_062_AV_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Mu: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_22 @ nu_2 ) @ Mu ) @ ( denota904684656_a_b_c @ j @ b ) )
=> ~ ( denota1997846518gree_c @ ( produc394644079real_c @ mu3 @ mu2 ) @ Mu @ ( sup_su1349021703um_c_c @ ( static_MBV_a_b_c @ b ) @ v ) ) ) ).
% \<open>\<And>thesis. (\<And>\<mu>''. \<lbrakk>((\<nu>2, \<nu>2'), \<mu>'') \<in> prog_sem J b; Vagree (\<mu>, \<mu>') \<mu>'' (MBV b \<union> V)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_VA,axiom,
denota1997846518gree_c @ ( produc394644079real_c @ nu_12 @ nu_1 ) @ ( produc394644079real_c @ nu_22 @ nu_2 ) @ v ).
% VA
thf(fact_10_sup__Pair__Pair,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,C: finite1398487019real_c,D: finite1398487019real_c] :
( ( sup_su1639086475real_c @ ( produc394644079real_c @ A @ B ) @ ( produc394644079real_c @ C @ D ) )
= ( produc394644079real_c @ ( sup_su2141622551real_c @ A @ C ) @ ( sup_su2141622551real_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_11_sup__Pair__Pair,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
( ( sup_su1672617323um_c_c @ ( produc1439061071um_c_c @ A @ B ) @ ( produc1439061071um_c_c @ C @ D ) )
= ( produc1439061071um_c_c @ ( sup_su1349021703um_c_c @ A @ C ) @ ( sup_su1349021703um_c_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_12_sup__Pair__Pair,axiom,
! [A: produc190496183real_c,B: produc190496183real_c,C: produc190496183real_c,D: produc190496183real_c] :
( ( sup_su1399591867real_c @ ( produc1687028567real_c @ A @ B ) @ ( produc1687028567real_c @ C @ D ) )
= ( produc1687028567real_c @ ( sup_su1639086475real_c @ A @ C ) @ ( sup_su1639086475real_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_13_sup__Pair__Pair,axiom,
! [A: set_Sum_sum_c_c,B: set_Su1783761653um_b_c,C: set_Sum_sum_c_c,D: set_Su1783761653um_b_c] :
( ( sup_su421398139um_b_c @ ( produc919784577um_b_c @ A @ B ) @ ( produc919784577um_b_c @ C @ D ) )
= ( produc919784577um_b_c @ ( sup_su1349021703um_c_c @ A @ C ) @ ( sup_su1968964297um_b_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_14_sup__Pair__Pair,axiom,
! [A: set_Su1783761653um_b_c,B: set_Sum_sum_c_c,C: set_Su1783761653um_b_c,D: set_Sum_sum_c_c] :
( ( sup_su1731217579um_c_c @ ( produc616453753um_c_c @ A @ B ) @ ( produc616453753um_c_c @ C @ D ) )
= ( produc616453753um_c_c @ ( sup_su1968964297um_b_c @ A @ C ) @ ( sup_su1349021703um_c_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_15_sup__Pair__Pair,axiom,
! [A: set_Sum_sum_c_c,B: produc190496183real_c,C: set_Sum_sum_c_c,D: produc190496183real_c] :
( ( sup_su2023888701real_c @ ( produc439851843real_c @ A @ B ) @ ( produc439851843real_c @ C @ D ) )
= ( produc439851843real_c @ ( sup_su1349021703um_c_c @ A @ C ) @ ( sup_su1639086475real_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_16_sup__Pair__Pair,axiom,
! [A: produc190496183real_c,B: set_Sum_sum_c_c,C: produc190496183real_c,D: set_Sum_sum_c_c] :
( ( sup_su1347303021um_c_c @ ( produc639456571um_c_c @ A @ B ) @ ( produc639456571um_c_c @ C @ D ) )
= ( produc639456571um_c_c @ ( sup_su1639086475real_c @ A @ C ) @ ( sup_su1349021703um_c_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_17_sup__Pair__Pair,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c,D: set_Su1783761653um_b_c] :
( ( sup_su1580927803um_b_c @ ( produc754249687um_b_c @ A @ B ) @ ( produc754249687um_b_c @ C @ D ) )
= ( produc754249687um_b_c @ ( sup_su1968964297um_b_c @ A @ C ) @ ( sup_su1968964297um_b_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_18_sup__Pair__Pair,axiom,
! [A: set_Sum_sum_c_c,B: produc214722967um_c_c,C: set_Sum_sum_c_c,D: produc214722967um_c_c] :
( ( sup_su505296157um_c_c @ ( produc1333497635um_c_c @ A @ B ) @ ( produc1333497635um_c_c @ C @ D ) )
= ( produc1333497635um_c_c @ ( sup_su1349021703um_c_c @ A @ C ) @ ( sup_su1672617323um_c_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_19_sup__Pair__Pair,axiom,
! [A: produc190496183real_c,B: set_Su1783761653um_b_c,C: produc190496183real_c,D: set_Su1783761653um_b_c] :
( ( sup_su1009753337um_b_c @ ( produc2010919061um_b_c @ A @ B ) @ ( produc2010919061um_b_c @ C @ D ) )
= ( produc2010919061um_b_c @ ( sup_su1639086475real_c @ A @ C ) @ ( sup_su1968964297um_b_c @ B @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_20_sem,axiom,
member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_12 @ nu_1 ) @ ( produc394644079real_c @ mu3 @ mu2 ) ) @ ( denota904684656_a_b_c @ i @ ( choice_a_b_c @ a @ b ) ) ).
% sem
thf(fact_21_sub,axiom,
ord_le1772180283um_c_c @ ( static_FVP_a_b_c @ ( choice_a_b_c @ a @ b ) ) @ v ).
% sub
thf(fact_22_False,axiom,
~ ( member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_12 @ nu_1 ) @ ( produc394644079real_c @ mu3 @ mu2 ) ) @ ( denota904684656_a_b_c @ i @ a ) ) ).
% False
thf(fact_23_hp_Oinject_I6_J,axiom,
! [X61: hp_a_b_c,X62: hp_a_b_c,Y61: hp_a_b_c,Y62: hp_a_b_c] :
( ( ( choice_a_b_c @ X61 @ X62 )
= ( choice_a_b_c @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% hp.inject(6)
thf(fact_24_UnCI,axiom,
! [C: produc190496183real_c,B2: set_Pr1389752855real_c,A2: set_Pr1389752855real_c] :
( ( ~ ( member1895684704real_c @ C @ B2 )
=> ( member1895684704real_c @ C @ A2 ) )
=> ( member1895684704real_c @ C @ ( sup_su1857314283real_c @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_25_UnCI,axiom,
! [C: produc1037099239real_c,B2: set_Pr242060487real_c,A2: set_Pr242060487real_c] :
( ( ~ ( member616151824real_c @ C @ B2 )
=> ( member616151824real_c @ C @ A2 ) )
=> ( member616151824real_c @ C @ ( sup_su880588443real_c @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_26_UnCI,axiom,
! [C: sum_su737395349um_b_c,B2: set_Su1783761653um_b_c,A2: set_Su1783761653um_b_c] :
( ( ~ ( member1893232190um_b_c @ C @ B2 )
=> ( member1893232190um_b_c @ C @ A2 ) )
=> ( member1893232190um_b_c @ C @ ( sup_su1968964297um_b_c @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_27_UnCI,axiom,
! [C: produc640052711real_c,B2: set_Pr754230471real_c,A2: set_Pr754230471real_c] :
( ( ~ ( member346808080real_c @ C @ B2 )
=> ( member346808080real_c @ C @ A2 ) )
=> ( member346808080real_c @ C @ ( sup_su588839579real_c @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_28_UnCI,axiom,
! [C: sum_sum_c_c,B2: set_Sum_sum_c_c,A2: set_Sum_sum_c_c] :
( ( ~ ( member_Sum_sum_c_c @ C @ B2 )
=> ( member_Sum_sum_c_c @ C @ A2 ) )
=> ( member_Sum_sum_c_c @ C @ ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_29_Un__iff,axiom,
! [C: produc190496183real_c,A2: set_Pr1389752855real_c,B2: set_Pr1389752855real_c] :
( ( member1895684704real_c @ C @ ( sup_su1857314283real_c @ A2 @ B2 ) )
= ( ( member1895684704real_c @ C @ A2 )
| ( member1895684704real_c @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_30_Un__iff,axiom,
! [C: produc1037099239real_c,A2: set_Pr242060487real_c,B2: set_Pr242060487real_c] :
( ( member616151824real_c @ C @ ( sup_su880588443real_c @ A2 @ B2 ) )
= ( ( member616151824real_c @ C @ A2 )
| ( member616151824real_c @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_31_Un__iff,axiom,
! [C: sum_su737395349um_b_c,A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( member1893232190um_b_c @ C @ ( sup_su1968964297um_b_c @ A2 @ B2 ) )
= ( ( member1893232190um_b_c @ C @ A2 )
| ( member1893232190um_b_c @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_32_Un__iff,axiom,
! [C: produc640052711real_c,A2: set_Pr754230471real_c,B2: set_Pr754230471real_c] :
( ( member346808080real_c @ C @ ( sup_su588839579real_c @ A2 @ B2 ) )
= ( ( member346808080real_c @ C @ A2 )
| ( member346808080real_c @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_33_Un__iff,axiom,
! [C: sum_sum_c_c,A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ ( sup_su1349021703um_c_c @ A2 @ B2 ) )
= ( ( member_Sum_sum_c_c @ C @ A2 )
| ( member_Sum_sum_c_c @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_34_sup_Oidem,axiom,
! [A: produc190496183real_c] :
( ( sup_su1639086475real_c @ A @ A )
= A ) ).
% sup.idem
thf(fact_35_sup_Oidem,axiom,
! [A: produc640052711real_c] :
( ( sup_su1399591867real_c @ A @ A )
= A ) ).
% sup.idem
thf(fact_36_sup_Oidem,axiom,
! [A: produc214722967um_c_c] :
( ( sup_su1672617323um_c_c @ A @ A )
= A ) ).
% sup.idem
thf(fact_37_sup_Oidem,axiom,
! [A: set_Pr754230471real_c] :
( ( sup_su588839579real_c @ A @ A )
= A ) ).
% sup.idem
thf(fact_38_sup_Oidem,axiom,
! [A: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ A @ A )
= A ) ).
% sup.idem
thf(fact_39_sup_Oidem,axiom,
! [A: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A @ A )
= A ) ).
% sup.idem
thf(fact_40_sup__idem,axiom,
! [X: produc190496183real_c] :
( ( sup_su1639086475real_c @ X @ X )
= X ) ).
% sup_idem
thf(fact_41_sup__idem,axiom,
! [X: produc640052711real_c] :
( ( sup_su1399591867real_c @ X @ X )
= X ) ).
% sup_idem
thf(fact_42_sup__idem,axiom,
! [X: produc214722967um_c_c] :
( ( sup_su1672617323um_c_c @ X @ X )
= X ) ).
% sup_idem
thf(fact_43_sup__idem,axiom,
! [X: set_Pr754230471real_c] :
( ( sup_su588839579real_c @ X @ X )
= X ) ).
% sup_idem
thf(fact_44_sup__idem,axiom,
! [X: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ X @ X )
= X ) ).
% sup_idem
thf(fact_45_sup__idem,axiom,
! [X: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ X @ X )
= X ) ).
% sup_idem
thf(fact_46_sup_Oleft__idem,axiom,
! [A: produc190496183real_c,B: produc190496183real_c] :
( ( sup_su1639086475real_c @ A @ ( sup_su1639086475real_c @ A @ B ) )
= ( sup_su1639086475real_c @ A @ B ) ) ).
% sup.left_idem
thf(fact_47_sup_Oleft__idem,axiom,
! [A: produc640052711real_c,B: produc640052711real_c] :
( ( sup_su1399591867real_c @ A @ ( sup_su1399591867real_c @ A @ B ) )
= ( sup_su1399591867real_c @ A @ B ) ) ).
% sup.left_idem
thf(fact_48_sup_Oleft__idem,axiom,
! [A: produc214722967um_c_c,B: produc214722967um_c_c] :
( ( sup_su1672617323um_c_c @ A @ ( sup_su1672617323um_c_c @ A @ B ) )
= ( sup_su1672617323um_c_c @ A @ B ) ) ).
% sup.left_idem
thf(fact_49_sup_Oleft__idem,axiom,
! [A: set_Pr754230471real_c,B: set_Pr754230471real_c] :
( ( sup_su588839579real_c @ A @ ( sup_su588839579real_c @ A @ B ) )
= ( sup_su588839579real_c @ A @ B ) ) ).
% sup.left_idem
thf(fact_50_sup_Oleft__idem,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ A @ ( sup_su1968964297um_b_c @ A @ B ) )
= ( sup_su1968964297um_b_c @ A @ B ) ) ).
% sup.left_idem
thf(fact_51_sup_Oleft__idem,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A @ ( sup_su1349021703um_c_c @ A @ B ) )
= ( sup_su1349021703um_c_c @ A @ B ) ) ).
% sup.left_idem
thf(fact_52_subsetI,axiom,
! [A2: set_Pr1389752855real_c,B2: set_Pr1389752855real_c] :
( ! [X2: produc190496183real_c] :
( ( member1895684704real_c @ X2 @ A2 )
=> ( member1895684704real_c @ X2 @ B2 ) )
=> ( ord_le977353143real_c @ A2 @ B2 ) ) ).
% subsetI
thf(fact_53_subsetI,axiom,
! [A2: set_Pr242060487real_c,B2: set_Pr242060487real_c] :
( ! [X2: produc1037099239real_c] :
( ( member616151824real_c @ X2 @ A2 )
=> ( member616151824real_c @ X2 @ B2 ) )
=> ( ord_le100418663real_c @ A2 @ B2 ) ) ).
% subsetI
thf(fact_54_subsetI,axiom,
! [A2: set_Pr754230471real_c,B2: set_Pr754230471real_c] :
( ! [X2: produc640052711real_c] :
( ( member346808080real_c @ X2 @ A2 )
=> ( member346808080real_c @ X2 @ B2 ) )
=> ( ord_le1988344935real_c @ A2 @ B2 ) ) ).
% subsetI
thf(fact_55_subsetI,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ! [X2: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X2 @ A2 )
=> ( member_Sum_sum_c_c @ X2 @ B2 ) )
=> ( ord_le1772180283um_c_c @ A2 @ B2 ) ) ).
% subsetI
thf(fact_56_subsetI,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ! [X2: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X2 @ A2 )
=> ( member1893232190um_b_c @ X2 @ B2 ) )
=> ( ord_le929608341um_b_c @ A2 @ B2 ) ) ).
% subsetI
thf(fact_57_subset__antisym,axiom,
! [A2: set_Pr754230471real_c,B2: set_Pr754230471real_c] :
( ( ord_le1988344935real_c @ A2 @ B2 )
=> ( ( ord_le1988344935real_c @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_58_subset__antisym,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ( ( ord_le1772180283um_c_c @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_59_subset__antisym,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ( ( ord_le929608341um_b_c @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_60_sup_Oright__idem,axiom,
! [A: produc190496183real_c,B: produc190496183real_c] :
( ( sup_su1639086475real_c @ ( sup_su1639086475real_c @ A @ B ) @ B )
= ( sup_su1639086475real_c @ A @ B ) ) ).
% sup.right_idem
thf(fact_61_sup_Oright__idem,axiom,
! [A: produc640052711real_c,B: produc640052711real_c] :
( ( sup_su1399591867real_c @ ( sup_su1399591867real_c @ A @ B ) @ B )
= ( sup_su1399591867real_c @ A @ B ) ) ).
% sup.right_idem
thf(fact_62_sup_Oright__idem,axiom,
! [A: produc214722967um_c_c,B: produc214722967um_c_c] :
( ( sup_su1672617323um_c_c @ ( sup_su1672617323um_c_c @ A @ B ) @ B )
= ( sup_su1672617323um_c_c @ A @ B ) ) ).
% sup.right_idem
thf(fact_63_sup_Oright__idem,axiom,
! [A: set_Pr754230471real_c,B: set_Pr754230471real_c] :
( ( sup_su588839579real_c @ ( sup_su588839579real_c @ A @ B ) @ B )
= ( sup_su588839579real_c @ A @ B ) ) ).
% sup.right_idem
thf(fact_64_sup_Oright__idem,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ ( sup_su1968964297um_b_c @ A @ B ) @ B )
= ( sup_su1968964297um_b_c @ A @ B ) ) ).
% sup.right_idem
thf(fact_65_sup_Oright__idem,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ ( sup_su1349021703um_c_c @ A @ B ) @ B )
= ( sup_su1349021703um_c_c @ A @ B ) ) ).
% sup.right_idem
thf(fact_66_sup__left__idem,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( sup_su1639086475real_c @ X @ ( sup_su1639086475real_c @ X @ Y ) )
= ( sup_su1639086475real_c @ X @ Y ) ) ).
% sup_left_idem
thf(fact_67_sup__left__idem,axiom,
! [X: produc640052711real_c,Y: produc640052711real_c] :
( ( sup_su1399591867real_c @ X @ ( sup_su1399591867real_c @ X @ Y ) )
= ( sup_su1399591867real_c @ X @ Y ) ) ).
% sup_left_idem
thf(fact_68_sup__left__idem,axiom,
! [X: produc214722967um_c_c,Y: produc214722967um_c_c] :
( ( sup_su1672617323um_c_c @ X @ ( sup_su1672617323um_c_c @ X @ Y ) )
= ( sup_su1672617323um_c_c @ X @ Y ) ) ).
% sup_left_idem
thf(fact_69_sup__left__idem,axiom,
! [X: set_Pr754230471real_c,Y: set_Pr754230471real_c] :
( ( sup_su588839579real_c @ X @ ( sup_su588839579real_c @ X @ Y ) )
= ( sup_su588839579real_c @ X @ Y ) ) ).
% sup_left_idem
thf(fact_70_sup__left__idem,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ X @ ( sup_su1968964297um_b_c @ X @ Y ) )
= ( sup_su1968964297um_b_c @ X @ Y ) ) ).
% sup_left_idem
thf(fact_71_sup__left__idem,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ X @ ( sup_su1349021703um_c_c @ X @ Y ) )
= ( sup_su1349021703um_c_c @ X @ Y ) ) ).
% sup_left_idem
thf(fact_72_sub2,axiom,
ord_le1772180283um_c_c @ ( static_FVP_a_b_c @ b ) @ v ).
% sub2
thf(fact_73_sub1,axiom,
ord_le1772180283um_c_c @ ( static_FVP_a_b_c @ a ) @ v ).
% sub1
thf(fact_74_Pair__le,axiom,
! [A: produc190496183real_c,B: produc190496183real_c,C: produc190496183real_c,D: produc190496183real_c] :
( ( ord_le652477831real_c @ ( produc1687028567real_c @ A @ B ) @ ( produc1687028567real_c @ C @ D ) )
= ( ( ord_le850691415real_c @ A @ C )
& ( ord_le850691415real_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_75_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_76_Pair__le,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
( ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ A @ B ) @ ( produc1439061071um_c_c @ C @ D ) )
= ( ( ord_le1772180283um_c_c @ A @ C )
& ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_77_Pair__le,axiom,
! [A: set_Sum_sum_c_c,B: set_Su1783761653um_b_c,C: set_Sum_sum_c_c,D: set_Su1783761653um_b_c] :
( ( ord_le542695343um_b_c @ ( produc919784577um_b_c @ A @ B ) @ ( produc919784577um_b_c @ C @ D ) )
= ( ( ord_le1772180283um_c_c @ A @ C )
& ( ord_le929608341um_b_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_78_Pair__le,axiom,
! [A: set_Su1783761653um_b_c,B: set_Sum_sum_c_c,C: set_Su1783761653um_b_c,D: set_Sum_sum_c_c] :
( ( ord_le1852514783um_c_c @ ( produc616453753um_c_c @ A @ B ) @ ( produc616453753um_c_c @ C @ D ) )
= ( ( ord_le929608341um_b_c @ A @ C )
& ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_79_Pair__le,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c,D: set_Su1783761653um_b_c] :
( ( ord_le907560199um_b_c @ ( produc754249687um_b_c @ A @ B ) @ ( produc754249687um_b_c @ C @ D ) )
= ( ( ord_le929608341um_b_c @ A @ C )
& ( ord_le929608341um_b_c @ B @ D ) ) ) ).
% Pair_le
thf(fact_80_sup_Obounded__iff,axiom,
! [B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ B @ C ) @ A )
= ( ( ord_le1772180283um_c_c @ B @ A )
& ( ord_le1772180283um_c_c @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_81_sup_Obounded__iff,axiom,
! [B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ B @ C ) @ A )
= ( ( ord_le929608341um_b_c @ B @ A )
& ( ord_le929608341um_b_c @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_82_le__sup__iff,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ X @ Y ) @ Z )
= ( ( ord_le1772180283um_c_c @ X @ Z )
& ( ord_le1772180283um_c_c @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_83_le__sup__iff,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c,Z: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ X @ Y ) @ Z )
= ( ( ord_le929608341um_b_c @ X @ Z )
& ( ord_le929608341um_b_c @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_84_Un__subset__iff,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ A2 @ B2 ) @ C2 )
= ( ( ord_le1772180283um_c_c @ A2 @ C2 )
& ( ord_le1772180283um_c_c @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_85_Un__subset__iff,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c,C2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ A2 @ B2 ) @ C2 )
= ( ( ord_le929608341um_b_c @ A2 @ C2 )
& ( ord_le929608341um_b_c @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_86_hpsafe__Choice__simps,axiom,
! [A: hp_a_b_c,B: hp_a_b_c] :
( ( hpsafe_a_b_c @ ( choice_a_b_c @ A @ B ) )
= ( ( hpsafe_a_b_c @ A )
& ( hpsafe_a_b_c @ B ) ) ) ).
% hpsafe_Choice_simps
thf(fact_87_sem2,axiom,
member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_12 @ nu_1 ) @ ( produc394644079real_c @ mu3 @ mu2 ) ) @ ( denota904684656_a_b_c @ i @ b ) ).
% sem2
thf(fact_88_eitherSem,axiom,
( ( member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_12 @ nu_1 ) @ ( produc394644079real_c @ mu3 @ mu2 ) ) @ ( denota904684656_a_b_c @ i @ a ) )
| ( member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_12 @ nu_1 ) @ ( produc394644079real_c @ mu3 @ mu2 ) ) @ ( denota904684656_a_b_c @ i @ b ) ) ) ).
% eitherSem
thf(fact_89_in__mono,axiom,
! [A2: set_Pr754230471real_c,B2: set_Pr754230471real_c,X: produc640052711real_c] :
( ( ord_le1988344935real_c @ A2 @ B2 )
=> ( ( member346808080real_c @ X @ A2 )
=> ( member346808080real_c @ X @ B2 ) ) ) ).
% in_mono
thf(fact_90_in__mono,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,X: sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ( ( member_Sum_sum_c_c @ X @ A2 )
=> ( member_Sum_sum_c_c @ X @ B2 ) ) ) ).
% in_mono
thf(fact_91_in__mono,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c,X: sum_su737395349um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ( ( member1893232190um_b_c @ X @ A2 )
=> ( member1893232190um_b_c @ X @ B2 ) ) ) ).
% in_mono
thf(fact_92_subsetD,axiom,
! [A2: set_Pr754230471real_c,B2: set_Pr754230471real_c,C: produc640052711real_c] :
( ( ord_le1988344935real_c @ A2 @ B2 )
=> ( ( member346808080real_c @ C @ A2 )
=> ( member346808080real_c @ C @ B2 ) ) ) ).
% subsetD
thf(fact_93_subsetD,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,C: sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ( ( member_Sum_sum_c_c @ C @ A2 )
=> ( member_Sum_sum_c_c @ C @ B2 ) ) ) ).
% subsetD
thf(fact_94_subsetD,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c,C: sum_su737395349um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ( ( member1893232190um_b_c @ C @ A2 )
=> ( member1893232190um_b_c @ C @ B2 ) ) ) ).
% subsetD
thf(fact_95_equalityE,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( A2 = B2 )
=> ~ ( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ~ ( ord_le1772180283um_c_c @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_96_equalityE,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( A2 = B2 )
=> ~ ( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ~ ( ord_le929608341um_b_c @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_97_subset__eq,axiom,
( ord_le1988344935real_c
= ( ^ [A3: set_Pr754230471real_c,B3: set_Pr754230471real_c] :
! [X3: produc640052711real_c] :
( ( member346808080real_c @ X3 @ A3 )
=> ( member346808080real_c @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_98_subset__eq,axiom,
( ord_le1772180283um_c_c
= ( ^ [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
! [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A3 )
=> ( member_Sum_sum_c_c @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_99_subset__eq,axiom,
( ord_le929608341um_b_c
= ( ^ [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c] :
! [X3: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ X3 @ A3 )
=> ( member1893232190um_b_c @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_100_equalityD1,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( A2 = B2 )
=> ( ord_le1772180283um_c_c @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_101_equalityD1,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( A2 = B2 )
=> ( ord_le929608341um_b_c @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_102_equalityD2,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( A2 = B2 )
=> ( ord_le1772180283um_c_c @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_103_equalityD2,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( A2 = B2 )
=> ( ord_le929608341um_b_c @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_104_subset__iff,axiom,
( ord_le1988344935real_c
= ( ^ [A3: set_Pr754230471real_c,B3: set_Pr754230471real_c] :
! [T: produc640052711real_c] :
( ( member346808080real_c @ T @ A3 )
=> ( member346808080real_c @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_105_subset__iff,axiom,
( ord_le1772180283um_c_c
= ( ^ [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
! [T: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ T @ A3 )
=> ( member_Sum_sum_c_c @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_106_subset__iff,axiom,
( ord_le929608341um_b_c
= ( ^ [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c] :
! [T: sum_su737395349um_b_c] :
( ( member1893232190um_b_c @ T @ A3 )
=> ( member1893232190um_b_c @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_107_subset__refl,axiom,
! [A2: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A2 @ A2 ) ).
% subset_refl
thf(fact_108_subset__refl,axiom,
! [A2: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ A2 @ A2 ) ).
% subset_refl
thf(fact_109_Collect__mono,axiom,
! [P: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
( ! [X2: sum_sum_c_c] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P ) @ ( collect_Sum_sum_c_c @ Q ) ) ) ).
% Collect_mono
thf(fact_110_Collect__mono,axiom,
! [P: sum_su737395349um_b_c > $o,Q: sum_su737395349um_b_c > $o] :
( ! [X2: sum_su737395349um_b_c] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le929608341um_b_c @ ( collec2086029184um_b_c @ P ) @ ( collec2086029184um_b_c @ Q ) ) ) ).
% Collect_mono
thf(fact_111_subset__trans,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ( ( ord_le1772180283um_c_c @ B2 @ C2 )
=> ( ord_le1772180283um_c_c @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_112_subset__trans,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c,C2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ( ( ord_le929608341um_b_c @ B2 @ C2 )
=> ( ord_le929608341um_b_c @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_113_set__eq__subset,axiom,
( ( ^ [Y2: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] : Y2 = Z2 )
= ( ^ [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A3 @ B3 )
& ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_114_set__eq__subset,axiom,
( ( ^ [Y2: set_Su1783761653um_b_c,Z2: set_Su1783761653um_b_c] : Y2 = Z2 )
= ( ^ [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A3 @ B3 )
& ( ord_le929608341um_b_c @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_115_mem__Collect__eq,axiom,
! [A: produc640052711real_c,P: produc640052711real_c > $o] :
( ( member346808080real_c @ A @ ( collec1126017874real_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A2: set_Pr754230471real_c] :
( ( collec1126017874real_c
@ ^ [X3: produc640052711real_c] : ( member346808080real_c @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_117_Collect__mono__iff,axiom,
! [P: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
( ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P ) @ ( collect_Sum_sum_c_c @ Q ) )
= ( ! [X3: sum_sum_c_c] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_118_Collect__mono__iff,axiom,
! [P: sum_su737395349um_b_c > $o,Q: sum_su737395349um_b_c > $o] :
( ( ord_le929608341um_b_c @ ( collec2086029184um_b_c @ P ) @ ( collec2086029184um_b_c @ Q ) )
= ( ! [X3: sum_su737395349um_b_c] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_119_Pair__mono,axiom,
! [X: produc190496183real_c,X4: produc190496183real_c,Y: produc190496183real_c,Y3: produc190496183real_c] :
( ( ord_le850691415real_c @ X @ X4 )
=> ( ( ord_le850691415real_c @ Y @ Y3 )
=> ( ord_le652477831real_c @ ( produc1687028567real_c @ X @ Y ) @ ( produc1687028567real_c @ X4 @ Y3 ) ) ) ) ).
% Pair_mono
thf(fact_120_Pair__mono,axiom,
! [X: finite1398487019real_c,X4: finite1398487019real_c,Y: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( ord_le775706699real_c @ X @ X4 )
=> ( ( ord_le775706699real_c @ Y @ Y3 )
=> ( ord_le850691415real_c @ ( produc394644079real_c @ X @ Y ) @ ( produc394644079real_c @ X4 @ Y3 ) ) ) ) ).
% Pair_mono
thf(fact_121_Pair__mono,axiom,
! [X: set_Sum_sum_c_c,X4: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ X4 )
=> ( ( ord_le1772180283um_c_c @ Y @ Y3 )
=> ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ X @ Y ) @ ( produc1439061071um_c_c @ X4 @ Y3 ) ) ) ) ).
% Pair_mono
thf(fact_122_Pair__mono,axiom,
! [X: set_Sum_sum_c_c,X4: set_Sum_sum_c_c,Y: set_Su1783761653um_b_c,Y3: set_Su1783761653um_b_c] :
( ( ord_le1772180283um_c_c @ X @ X4 )
=> ( ( ord_le929608341um_b_c @ Y @ Y3 )
=> ( ord_le542695343um_b_c @ ( produc919784577um_b_c @ X @ Y ) @ ( produc919784577um_b_c @ X4 @ Y3 ) ) ) ) ).
% Pair_mono
thf(fact_123_Pair__mono,axiom,
! [X: set_Su1783761653um_b_c,X4: set_Su1783761653um_b_c,Y: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
( ( ord_le929608341um_b_c @ X @ X4 )
=> ( ( ord_le1772180283um_c_c @ Y @ Y3 )
=> ( ord_le1852514783um_c_c @ ( produc616453753um_c_c @ X @ Y ) @ ( produc616453753um_c_c @ X4 @ Y3 ) ) ) ) ).
% Pair_mono
thf(fact_124_Pair__mono,axiom,
! [X: set_Su1783761653um_b_c,X4: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c,Y3: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X @ X4 )
=> ( ( ord_le929608341um_b_c @ Y @ Y3 )
=> ( ord_le907560199um_b_c @ ( produc754249687um_b_c @ X @ Y ) @ ( produc754249687um_b_c @ X4 @ Y3 ) ) ) ) ).
% Pair_mono
thf(fact_125_sup_OcoboundedI2,axiom,
! [C: set_Sum_sum_c_c,B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ C @ B )
=> ( ord_le1772180283um_c_c @ C @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_126_sup_OcoboundedI2,axiom,
! [C: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ C @ B )
=> ( ord_le929608341um_b_c @ C @ ( sup_su1968964297um_b_c @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_127_sup_OcoboundedI1,axiom,
! [C: set_Sum_sum_c_c,A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ C @ A )
=> ( ord_le1772180283um_c_c @ C @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_128_sup_OcoboundedI1,axiom,
! [C: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ C @ A )
=> ( ord_le929608341um_b_c @ C @ ( sup_su1968964297um_b_c @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_129_sup_Oabsorb__iff2,axiom,
( ord_le1772180283um_c_c
= ( ^ [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_130_sup_Oabsorb__iff2,axiom,
( ord_le929608341um_b_c
= ( ^ [A4: set_Su1783761653um_b_c,B4: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_131_sup_Oabsorb__iff1,axiom,
( ord_le1772180283um_c_c
= ( ^ [B4: set_Sum_sum_c_c,A4: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_132_sup_Oabsorb__iff1,axiom,
( ord_le929608341um_b_c
= ( ^ [B4: set_Su1783761653um_b_c,A4: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_133_sup_Ocobounded2,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ B @ ( sup_su1349021703um_c_c @ A @ B ) ) ).
% sup.cobounded2
thf(fact_134_sup_Ocobounded2,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ B @ ( sup_su1968964297um_b_c @ A @ B ) ) ).
% sup.cobounded2
thf(fact_135_sup_Ocobounded1,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A @ ( sup_su1349021703um_c_c @ A @ B ) ) ).
% sup.cobounded1
thf(fact_136_sup_Ocobounded1,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ A @ ( sup_su1968964297um_b_c @ A @ B ) ) ).
% sup.cobounded1
thf(fact_137_sup_Oorder__iff,axiom,
( ord_le1772180283um_c_c
= ( ^ [B4: set_Sum_sum_c_c,A4: set_Sum_sum_c_c] :
( A4
= ( sup_su1349021703um_c_c @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_138_sup_Oorder__iff,axiom,
( ord_le929608341um_b_c
= ( ^ [B4: set_Su1783761653um_b_c,A4: set_Su1783761653um_b_c] :
( A4
= ( sup_su1968964297um_b_c @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_139_sup_OboundedI,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( ord_le1772180283um_c_c @ C @ A )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_140_sup_OboundedI,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( ( ord_le929608341um_b_c @ C @ A )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_141_sup_OboundedE,axiom,
! [B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ B @ C ) @ A )
=> ~ ( ( ord_le1772180283um_c_c @ B @ A )
=> ~ ( ord_le1772180283um_c_c @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_142_sup_OboundedE,axiom,
! [B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ B @ C ) @ A )
=> ~ ( ( ord_le929608341um_b_c @ B @ A )
=> ~ ( ord_le929608341um_b_c @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_143_sup__absorb2,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ( sup_su1349021703um_c_c @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_144_sup__absorb2,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X @ Y )
=> ( ( sup_su1968964297um_b_c @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_145_sup__absorb1,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y @ X )
=> ( ( sup_su1349021703um_c_c @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_146_sup__absorb1,axiom,
! [Y: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ Y @ X )
=> ( ( sup_su1968964297um_b_c @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_147_sup_Oabsorb2,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( sup_su1349021703um_c_c @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_148_sup_Oabsorb2,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( sup_su1968964297um_b_c @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_149_sup_Oabsorb1,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( sup_su1349021703um_c_c @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_150_sup_Oabsorb1,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( ( sup_su1968964297um_b_c @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_151_sup__unique,axiom,
! [F: set_Sum_sum_c_c > set_Sum_sum_c_c > set_Sum_sum_c_c,X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c,Z3: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y4 @ X2 )
=> ( ( ord_le1772180283um_c_c @ Z3 @ X2 )
=> ( ord_le1772180283um_c_c @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su1349021703um_c_c @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_152_sup__unique,axiom,
! [F: set_Su1783761653um_b_c > set_Su1783761653um_b_c > set_Su1783761653um_b_c,X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] :
( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c,Z3: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ Y4 @ X2 )
=> ( ( ord_le929608341um_b_c @ Z3 @ X2 )
=> ( ord_le929608341um_b_c @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_su1968964297um_b_c @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_153_sup_OorderI,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( A
= ( sup_su1349021703um_c_c @ A @ B ) )
=> ( ord_le1772180283um_c_c @ B @ A ) ) ).
% sup.orderI
thf(fact_154_sup_OorderI,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( A
= ( sup_su1968964297um_b_c @ A @ B ) )
=> ( ord_le929608341um_b_c @ B @ A ) ) ).
% sup.orderI
thf(fact_155_sup_OorderE,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( A
= ( sup_su1349021703um_c_c @ A @ B ) ) ) ).
% sup.orderE
thf(fact_156_sup_OorderE,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( A
= ( sup_su1968964297um_b_c @ A @ B ) ) ) ).
% sup.orderE
thf(fact_157_le__iff__sup,axiom,
( ord_le1772180283um_c_c
= ( ^ [X3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_158_le__iff__sup,axiom,
( ord_le929608341um_b_c
= ( ^ [X3: set_Su1783761653um_b_c,Y5: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_159_sup__least,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y @ X )
=> ( ( ord_le1772180283um_c_c @ Z @ X )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_160_sup__least,axiom,
! [Y: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c,Z: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ Y @ X )
=> ( ( ord_le929608341um_b_c @ Z @ X )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_161_sup__mono,axiom,
! [A: set_Sum_sum_c_c,C: set_Sum_sum_c_c,B: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ C )
=> ( ( ord_le1772180283um_c_c @ B @ D )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ A @ B ) @ ( sup_su1349021703um_c_c @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_162_sup__mono,axiom,
! [A: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,D: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ C )
=> ( ( ord_le929608341um_b_c @ B @ D )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ A @ B ) @ ( sup_su1968964297um_b_c @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_163_sup_Omono,axiom,
! [C: set_Sum_sum_c_c,A: set_Sum_sum_c_c,D: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ C @ A )
=> ( ( ord_le1772180283um_c_c @ D @ B )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ C @ D ) @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_164_sup_Omono,axiom,
! [C: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,D: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ C @ A )
=> ( ( ord_le929608341um_b_c @ D @ B )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ C @ D ) @ ( sup_su1968964297um_b_c @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_165_le__supI2,axiom,
! [X: set_Sum_sum_c_c,B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ B )
=> ( ord_le1772180283um_c_c @ X @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ).
% le_supI2
thf(fact_166_le__supI2,axiom,
! [X: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X @ B )
=> ( ord_le929608341um_b_c @ X @ ( sup_su1968964297um_b_c @ A @ B ) ) ) ).
% le_supI2
thf(fact_167_le__supI1,axiom,
! [X: set_Sum_sum_c_c,A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ A )
=> ( ord_le1772180283um_c_c @ X @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ).
% le_supI1
thf(fact_168_le__supI1,axiom,
! [X: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X @ A )
=> ( ord_le929608341um_b_c @ X @ ( sup_su1968964297um_b_c @ A @ B ) ) ) ).
% le_supI1
thf(fact_169_sup__ge2,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ Y @ ( sup_su1349021703um_c_c @ X @ Y ) ) ).
% sup_ge2
thf(fact_170_sup__ge2,axiom,
! [Y: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ Y @ ( sup_su1968964297um_b_c @ X @ Y ) ) ).
% sup_ge2
thf(fact_171_sup__ge1,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X @ ( sup_su1349021703um_c_c @ X @ Y ) ) ).
% sup_ge1
thf(fact_172_sup__ge1,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ X @ ( sup_su1968964297um_b_c @ X @ Y ) ) ).
% sup_ge1
thf(fact_173_le__supI,axiom,
! [A: set_Sum_sum_c_c,X: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ X )
=> ( ( ord_le1772180283um_c_c @ B @ X )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_174_le__supI,axiom,
! [A: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ X )
=> ( ( ord_le929608341um_b_c @ B @ X )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_175_le__supE,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,X: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ A @ B ) @ X )
=> ~ ( ( ord_le1772180283um_c_c @ A @ X )
=> ~ ( ord_le1772180283um_c_c @ B @ X ) ) ) ).
% le_supE
thf(fact_176_le__supE,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ A @ B ) @ X )
=> ~ ( ( ord_le929608341um_b_c @ A @ X )
=> ~ ( ord_le929608341um_b_c @ B @ X ) ) ) ).
% le_supE
thf(fact_177_inf__sup__ord_I3_J,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X @ ( sup_su1349021703um_c_c @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_178_inf__sup__ord_I3_J,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ X @ ( sup_su1968964297um_b_c @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_179_inf__sup__ord_I4_J,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ Y @ ( sup_su1349021703um_c_c @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_180_inf__sup__ord_I4_J,axiom,
! [Y: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ Y @ ( sup_su1968964297um_b_c @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_181_subset__Un__eq,axiom,
( ord_le1772180283um_c_c
= ( ^ [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_182_subset__Un__eq,axiom,
( ord_le929608341um_b_c
= ( ^ [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c] :
( ( sup_su1968964297um_b_c @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_183_subset__UnE,axiom,
! [C2: set_Sum_sum_c_c,A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ C2 @ ( sup_su1349021703um_c_c @ A2 @ B2 ) )
=> ~ ! [A5: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A5 @ A2 )
=> ! [B5: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B5 @ B2 )
=> ( C2
!= ( sup_su1349021703um_c_c @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_184_subset__UnE,axiom,
! [C2: set_Su1783761653um_b_c,A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ C2 @ ( sup_su1968964297um_b_c @ A2 @ B2 ) )
=> ~ ! [A5: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A5 @ A2 )
=> ! [B5: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B5 @ B2 )
=> ( C2
!= ( sup_su1968964297um_b_c @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_185_Un__absorb2,axiom,
! [B2: set_Sum_sum_c_c,A2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B2 @ A2 )
=> ( ( sup_su1349021703um_c_c @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_186_Un__absorb2,axiom,
! [B2: set_Su1783761653um_b_c,A2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B2 @ A2 )
=> ( ( sup_su1968964297um_b_c @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_187_Un__absorb1,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ( ( sup_su1349021703um_c_c @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_188_Un__absorb1,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ( ( sup_su1968964297um_b_c @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_189_Un__upper2,axiom,
! [B2: set_Sum_sum_c_c,A2: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ B2 @ ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_190_Un__upper2,axiom,
! [B2: set_Su1783761653um_b_c,A2: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ B2 @ ( sup_su1968964297um_b_c @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_191_Un__upper1,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A2 @ ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_192_Un__upper1,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ A2 @ ( sup_su1968964297um_b_c @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_193_Un__least,axiom,
! [A2: set_Sum_sum_c_c,C2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ C2 )
=> ( ( ord_le1772180283um_c_c @ B2 @ C2 )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_194_Un__least,axiom,
! [A2: set_Su1783761653um_b_c,C2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ C2 )
=> ( ( ord_le929608341um_b_c @ B2 @ C2 )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_195_Un__mono,axiom,
! [A2: set_Sum_sum_c_c,C2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,D2: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A2 @ C2 )
=> ( ( ord_le1772180283um_c_c @ B2 @ D2 )
=> ( ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ A2 @ B2 ) @ ( sup_su1349021703um_c_c @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_196_Un__mono,axiom,
! [A2: set_Su1783761653um_b_c,C2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c,D2: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A2 @ C2 )
=> ( ( ord_le929608341um_b_c @ B2 @ D2 )
=> ( ord_le929608341um_b_c @ ( sup_su1968964297um_b_c @ A2 @ B2 ) @ ( sup_su1968964297um_b_c @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_197_hpsafe__fsafe_Ohpsafe__Choice,axiom,
! [A: hp_a_b_c,B: hp_a_b_c] :
( ( hpsafe_a_b_c @ A )
=> ( ( hpsafe_a_b_c @ B )
=> ( hpsafe_a_b_c @ ( choice_a_b_c @ A @ B ) ) ) ) ).
% hpsafe_fsafe.hpsafe_Choice
thf(fact_198_sup__left__commute,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ X @ ( sup_su1349021703um_c_c @ Y @ Z ) )
= ( sup_su1349021703um_c_c @ Y @ ( sup_su1349021703um_c_c @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_199_sup_Oleft__commute,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ B @ ( sup_su1349021703um_c_c @ A @ C ) )
= ( sup_su1349021703um_c_c @ A @ ( sup_su1349021703um_c_c @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_200_sup__commute,axiom,
( sup_su1349021703um_c_c
= ( ^ [X3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] : ( sup_su1349021703um_c_c @ Y5 @ X3 ) ) ) ).
% sup_commute
thf(fact_201_sup_Ocommute,axiom,
( sup_su1349021703um_c_c
= ( ^ [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] : ( sup_su1349021703um_c_c @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_202_sup__assoc,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ ( sup_su1349021703um_c_c @ X @ Y ) @ Z )
= ( sup_su1349021703um_c_c @ X @ ( sup_su1349021703um_c_c @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_203_sup_Oassoc,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ ( sup_su1349021703um_c_c @ A @ B ) @ C )
= ( sup_su1349021703um_c_c @ A @ ( sup_su1349021703um_c_c @ B @ C ) ) ) ).
% sup.assoc
thf(fact_204_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Sum_sum_c_c,K: set_Sum_sum_c_c,B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( B2
= ( sup_su1349021703um_c_c @ K @ B ) )
=> ( ( sup_su1349021703um_c_c @ A @ B2 )
= ( sup_su1349021703um_c_c @ K @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_205_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Sum_sum_c_c,K: set_Sum_sum_c_c,A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( A2
= ( sup_su1349021703um_c_c @ K @ A ) )
=> ( ( sup_su1349021703um_c_c @ A2 @ B )
= ( sup_su1349021703um_c_c @ K @ ( sup_su1349021703um_c_c @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_206_inf__sup__aci_I5_J,axiom,
( sup_su1349021703um_c_c
= ( ^ [X3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] : ( sup_su1349021703um_c_c @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_207_inf__sup__aci_I6_J,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ ( sup_su1349021703um_c_c @ X @ Y ) @ Z )
= ( sup_su1349021703um_c_c @ X @ ( sup_su1349021703um_c_c @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_208_inf__sup__aci_I7_J,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ X @ ( sup_su1349021703um_c_c @ Y @ Z ) )
= ( sup_su1349021703um_c_c @ Y @ ( sup_su1349021703um_c_c @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_209_inf__sup__aci_I8_J,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ X @ ( sup_su1349021703um_c_c @ X @ Y ) )
= ( sup_su1349021703um_c_c @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_210_Un__left__commute,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A2 @ ( sup_su1349021703um_c_c @ B2 @ C2 ) )
= ( sup_su1349021703um_c_c @ B2 @ ( sup_su1349021703um_c_c @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_211_Un__left__absorb,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A2 @ ( sup_su1349021703um_c_c @ A2 @ B2 ) )
= ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_212_Un__commute,axiom,
( sup_su1349021703um_c_c
= ( ^ [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] : ( sup_su1349021703um_c_c @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_213_Un__absorb,axiom,
! [A2: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_214_Un__assoc,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
( ( sup_su1349021703um_c_c @ ( sup_su1349021703um_c_c @ A2 @ B2 ) @ C2 )
= ( sup_su1349021703um_c_c @ A2 @ ( sup_su1349021703um_c_c @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_215_ball__Un,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,P: sum_sum_c_c > $o] :
( ( ! [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ ( sup_su1349021703um_c_c @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_216_bex__Un,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,P: sum_sum_c_c > $o] :
( ( ? [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ ( sup_su1349021703um_c_c @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: sum_sum_c_c] :
( ( member_Sum_sum_c_c @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_217_UnI2,axiom,
! [C: produc640052711real_c,B2: set_Pr754230471real_c,A2: set_Pr754230471real_c] :
( ( member346808080real_c @ C @ B2 )
=> ( member346808080real_c @ C @ ( sup_su588839579real_c @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_218_UnI2,axiom,
! [C: sum_sum_c_c,B2: set_Sum_sum_c_c,A2: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ B2 )
=> ( member_Sum_sum_c_c @ C @ ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_219_UnI1,axiom,
! [C: produc640052711real_c,A2: set_Pr754230471real_c,B2: set_Pr754230471real_c] :
( ( member346808080real_c @ C @ A2 )
=> ( member346808080real_c @ C @ ( sup_su588839579real_c @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_220_UnI1,axiom,
! [C: sum_sum_c_c,A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ A2 )
=> ( member_Sum_sum_c_c @ C @ ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_221_UnE,axiom,
! [C: produc640052711real_c,A2: set_Pr754230471real_c,B2: set_Pr754230471real_c] :
( ( member346808080real_c @ C @ ( sup_su588839579real_c @ A2 @ B2 ) )
=> ( ~ ( member346808080real_c @ C @ A2 )
=> ( member346808080real_c @ C @ B2 ) ) ) ).
% UnE
thf(fact_222_UnE,axiom,
! [C: sum_sum_c_c,A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( member_Sum_sum_c_c @ C @ ( sup_su1349021703um_c_c @ A2 @ B2 ) )
=> ( ~ ( member_Sum_sum_c_c @ C @ A2 )
=> ( member_Sum_sum_c_c @ C @ B2 ) ) ) ).
% UnE
thf(fact_223_IA,axiom,
denota69106024_a_b_c @ i @ j @ ( static_SIGP_a_b_c @ ( choice_a_b_c @ a @ b ) ) ).
% IA
thf(fact_224_IH1,axiom,
! [I: denota231621370t_unit,J: denota231621370t_unit] :
( ! [Nu: produc190496183real_c,Nu2: produc190496183real_c,Mu2: produc190496183real_c,V: set_Sum_sum_c_c] :
( ( denota69106024_a_b_c @ I @ J @ ( static_SIGP_a_b_c @ a ) )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ V )
=> ( ( ord_le1772180283um_c_c @ ( static_FVP_a_b_c @ a ) @ V )
=> ( ( member346808080real_c @ ( produc1687028567real_c @ Nu @ Mu2 ) @ ( denota904684656_a_b_c @ I @ a ) )
=> ? [Mu3: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ Nu2 @ Mu3 ) @ ( denota904684656_a_b_c @ J @ a ) )
& ( denota1997846518gree_c @ Mu2 @ Mu3 @ ( sup_su1349021703um_c_c @ ( static_MBV_a_b_c @ a ) @ V ) ) ) ) ) ) )
& ( coinci772602909_a_b_c @ a @ I ) ) ).
% IH1
thf(fact_225_IH2,axiom,
! [I: denota231621370t_unit,J: denota231621370t_unit] :
( ! [Nu: produc190496183real_c,Nu2: produc190496183real_c,Mu2: produc190496183real_c,V: set_Sum_sum_c_c] :
( ( denota69106024_a_b_c @ I @ J @ ( static_SIGP_a_b_c @ b ) )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ V )
=> ( ( ord_le1772180283um_c_c @ ( static_FVP_a_b_c @ b ) @ V )
=> ( ( member346808080real_c @ ( produc1687028567real_c @ Nu @ Mu2 ) @ ( denota904684656_a_b_c @ I @ b ) )
=> ? [Mu3: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ Nu2 @ Mu3 ) @ ( denota904684656_a_b_c @ J @ b ) )
& ( denota1997846518gree_c @ Mu2 @ Mu3 @ ( sup_su1349021703um_c_c @ ( static_MBV_a_b_c @ b ) @ V ) ) ) ) ) ) )
& ( coinci772602909_a_b_c @ b @ I ) ) ).
% IH2
thf(fact_226_coincide__hp_H__def,axiom,
( coinci1852866155_a_b_c
= ( ^ [Alpha: hp_a_b_c] :
! [I2: denota231621370t_unit,J2: denota231621370t_unit] :
( ( coinci354700372_a_b_c @ Alpha @ I2 @ J2 )
& ( coinci772602909_a_b_c @ Alpha @ I2 ) ) ) ) ).
% coincide_hp'_def
thf(fact_227_FVP_Osimps_I6_J,axiom,
! [Alpha2: hp_a_b_c,Beta: hp_a_b_c] :
( ( static_FVP_a_b_c @ ( choice_a_b_c @ Alpha2 @ Beta ) )
= ( sup_su1349021703um_c_c @ ( static_FVP_a_b_c @ Alpha2 ) @ ( static_FVP_a_b_c @ Beta ) ) ) ).
% FVP.simps(6)
thf(fact_228_agree__supset,axiom,
! [B2: set_Sum_sum_c_c,A2: set_Sum_sum_c_c,Nu3: produc190496183real_c,Nu4: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ B2 @ A2 )
=> ( ( denota1997846518gree_c @ Nu3 @ Nu4 @ A2 )
=> ( denota1997846518gree_c @ Nu3 @ Nu4 @ B2 ) ) ) ).
% agree_supset
thf(fact_229_agree__sub,axiom,
! [A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c,Nu3: produc190496183real_c,Omega: produc190496183real_c] :
( ( ord_le1772180283um_c_c @ A2 @ B2 )
=> ( ( denota1997846518gree_c @ Nu3 @ Omega @ B2 )
=> ( denota1997846518gree_c @ Nu3 @ Omega @ A2 ) ) ) ).
% agree_sub
thf(fact_230_old_Oprod_Oinject,axiom,
! [A: produc190496183real_c,B: produc190496183real_c,A6: produc190496183real_c,B6: produc190496183real_c] :
( ( ( produc1687028567real_c @ A @ B )
= ( produc1687028567real_c @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_231_old_Oprod_Oinject,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,A6: finite1398487019real_c,B6: finite1398487019real_c] :
( ( ( produc394644079real_c @ A @ B )
= ( produc394644079real_c @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_232_prod_Oinject,axiom,
! [X1: produc190496183real_c,X22: produc190496183real_c,Y1: produc190496183real_c,Y22: produc190496183real_c] :
( ( ( produc1687028567real_c @ X1 @ X22 )
= ( produc1687028567real_c @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_233_prod_Oinject,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c,Y1: finite1398487019real_c,Y22: finite1398487019real_c] :
( ( ( produc394644079real_c @ X1 @ X22 )
= ( produc394644079real_c @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_234_order__refl,axiom,
! [X: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X @ X ) ).
% order_refl
thf(fact_235_order__refl,axiom,
! [X: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ X @ X ) ).
% order_refl
thf(fact_236_Ssub_I2_J,axiom,
ord_le929608341um_b_c @ ( static_SIGP_a_b_c @ b ) @ ( static_SIGP_a_b_c @ ( choice_a_b_c @ a @ b ) ) ).
% Ssub(2)
thf(fact_237_Ssub_I1_J,axiom,
ord_le929608341um_b_c @ ( static_SIGP_a_b_c @ a ) @ ( static_SIGP_a_b_c @ ( choice_a_b_c @ a @ b ) ) ).
% Ssub(1)
thf(fact_238_IA2,axiom,
denota69106024_a_b_c @ i @ j @ ( static_SIGP_a_b_c @ b ) ).
% IA2
thf(fact_239_IA1,axiom,
denota69106024_a_b_c @ i @ j @ ( static_SIGP_a_b_c @ a ) ).
% IA1
thf(fact_240_coincide__hp__def,axiom,
( coinci354700372_a_b_c
= ( ^ [Alpha: hp_a_b_c,I2: denota231621370t_unit,J2: denota231621370t_unit] :
! [Nu5: produc190496183real_c,Nu6: produc190496183real_c,Mu4: produc190496183real_c,V2: set_Sum_sum_c_c] :
( ( denota69106024_a_b_c @ I2 @ J2 @ ( static_SIGP_a_b_c @ Alpha ) )
=> ( ( denota1997846518gree_c @ Nu5 @ Nu6 @ V2 )
=> ( ( ord_le1772180283um_c_c @ ( static_FVP_a_b_c @ Alpha ) @ V2 )
=> ( ( member346808080real_c @ ( produc1687028567real_c @ Nu5 @ Mu4 ) @ ( denota904684656_a_b_c @ I2 @ Alpha ) )
=> ? [Mu5: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ Nu6 @ Mu5 ) @ ( denota904684656_a_b_c @ J2 @ Alpha ) )
& ( denota1997846518gree_c @ Mu4 @ Mu5 @ ( sup_su1349021703um_c_c @ ( static_MBV_a_b_c @ Alpha ) @ V2 ) ) ) ) ) ) ) ) ) ).
% coincide_hp_def
thf(fact_241_Iagree__sub,axiom,
! [A2: set_Su1783761653um_b_c,B2: set_Su1783761653um_b_c,I3: denota231621370t_unit,J3: denota231621370t_unit] :
( ( ord_le929608341um_b_c @ A2 @ B2 )
=> ( ( denota69106024_a_b_c @ I3 @ J3 @ B2 )
=> ( denota69106024_a_b_c @ I3 @ J3 @ A2 ) ) ) ).
% Iagree_sub
thf(fact_242_Iagree__refl,axiom,
! [I3: denota231621370t_unit,A2: set_Su1783761653um_b_c] : ( denota69106024_a_b_c @ I3 @ I3 @ A2 ) ).
% Iagree_refl
thf(fact_243_Iagree__comm,axiom,
! [A2: denota231621370t_unit,B2: denota231621370t_unit,V3: set_Su1783761653um_b_c] :
( ( denota69106024_a_b_c @ A2 @ B2 @ V3 )
=> ( denota69106024_a_b_c @ B2 @ A2 @ V3 ) ) ).
% Iagree_comm
thf(fact_244_SIGP_Osimps_I6_J,axiom,
! [A: hp_a_b_c,B: hp_a_b_c] :
( ( static_SIGP_a_b_c @ ( choice_a_b_c @ A @ B ) )
= ( sup_su1968964297um_b_c @ ( static_SIGP_a_b_c @ A ) @ ( static_SIGP_a_b_c @ B ) ) ) ).
% SIGP.simps(6)
thf(fact_245_dual__order_Oantisym,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( ord_le1772180283um_c_c @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_246_dual__order_Oantisym,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( ( ord_le929608341um_b_c @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_247_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] : Y2 = Z2 )
= ( ^ [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B4 @ A4 )
& ( ord_le1772180283um_c_c @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_248_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Su1783761653um_b_c,Z2: set_Su1783761653um_b_c] : Y2 = Z2 )
= ( ^ [A4: set_Su1783761653um_b_c,B4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B4 @ A4 )
& ( ord_le929608341um_b_c @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_249_dual__order_Otrans,axiom,
! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ B @ A )
=> ( ( ord_le1772180283um_c_c @ C @ B )
=> ( ord_le1772180283um_c_c @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_250_dual__order_Otrans,axiom,
! [B: set_Su1783761653um_b_c,A: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ B @ A )
=> ( ( ord_le929608341um_b_c @ C @ B )
=> ( ord_le929608341um_b_c @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_251_dual__order_Orefl,axiom,
! [A: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A @ A ) ).
% dual_order.refl
thf(fact_252_dual__order_Orefl,axiom,
! [A: set_Su1783761653um_b_c] : ( ord_le929608341um_b_c @ A @ A ) ).
% dual_order.refl
thf(fact_253_order__trans,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ( ord_le1772180283um_c_c @ Y @ Z )
=> ( ord_le1772180283um_c_c @ X @ Z ) ) ) ).
% order_trans
thf(fact_254_order__trans,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c,Z: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X @ Y )
=> ( ( ord_le929608341um_b_c @ Y @ Z )
=> ( ord_le929608341um_b_c @ X @ Z ) ) ) ).
% order_trans
thf(fact_255_order__class_Oorder_Oantisym,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_256_order__class_Oorder_Oantisym,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( ord_le929608341um_b_c @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_257_ord__le__eq__trans,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( B = C )
=> ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_258_ord__le__eq__trans,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( B = C )
=> ( ord_le929608341um_b_c @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_259_ord__eq__le__trans,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( A = B )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_260_ord__eq__le__trans,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( A = B )
=> ( ( ord_le929608341um_b_c @ B @ C )
=> ( ord_le929608341um_b_c @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_261_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] : Y2 = Z2 )
= ( ^ [A4: set_Sum_sum_c_c,B4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A4 @ B4 )
& ( ord_le1772180283um_c_c @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_262_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Su1783761653um_b_c,Z2: set_Su1783761653um_b_c] : Y2 = Z2 )
= ( ^ [A4: set_Su1783761653um_b_c,B4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A4 @ B4 )
& ( ord_le929608341um_b_c @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_263_antisym__conv,axiom,
! [Y: set_Sum_sum_c_c,X: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ Y @ X )
=> ( ( ord_le1772180283um_c_c @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_264_antisym__conv,axiom,
! [Y: set_Su1783761653um_b_c,X: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ Y @ X )
=> ( ( ord_le929608341um_b_c @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_265_order_Otrans,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% order.trans
thf(fact_266_order_Otrans,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( ord_le929608341um_b_c @ B @ C )
=> ( ord_le929608341um_b_c @ A @ C ) ) ) ).
% order.trans
thf(fact_267_eq__refl,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( X = Y )
=> ( ord_le1772180283um_c_c @ X @ Y ) ) ).
% eq_refl
thf(fact_268_eq__refl,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] :
( ( X = Y )
=> ( ord_le929608341um_b_c @ X @ Y ) ) ).
% eq_refl
thf(fact_269_antisym,axiom,
! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X @ Y )
=> ( ( ord_le1772180283um_c_c @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_270_antisym,axiom,
! [X: set_Su1783761653um_b_c,Y: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X @ Y )
=> ( ( ord_le929608341um_b_c @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_271_eq__iff,axiom,
( ( ^ [Y2: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] : Y2 = Z2 )
= ( ^ [X3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X3 @ Y5 )
& ( ord_le1772180283um_c_c @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_272_eq__iff,axiom,
( ( ^ [Y2: set_Su1783761653um_b_c,Z2: set_Su1783761653um_b_c] : Y2 = Z2 )
= ( ^ [X3: set_Su1783761653um_b_c,Y5: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X3 @ Y5 )
& ( ord_le929608341um_b_c @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_273_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_274_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_275_ord__le__eq__subst,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,F: set_Su1783761653um_b_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_276_ord__le__eq__subst,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,F: set_Su1783761653um_b_c > set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_277_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_278_ord__eq__le__subst,axiom,
! [A: set_Su1783761653um_b_c,F: set_Sum_sum_c_c > set_Su1783761653um_b_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_279_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_c_c,F: set_Su1783761653um_b_c > set_Sum_sum_c_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( A
= ( F @ B ) )
=> ( ( ord_le929608341um_b_c @ B @ C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_280_ord__eq__le__subst,axiom,
! [A: set_Su1783761653um_b_c,F: set_Su1783761653um_b_c > set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( A
= ( F @ B ) )
=> ( ( ord_le929608341um_b_c @ B @ C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_281_order__subst2,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_282_order__subst2,axiom,
! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le1772180283um_c_c @ A @ B )
=> ( ( ord_le929608341um_b_c @ ( F @ B ) @ C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_283_order__subst2,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,F: set_Su1783761653um_b_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_284_order__subst2,axiom,
! [A: set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,F: set_Su1783761653um_b_c > set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ B )
=> ( ( ord_le929608341um_b_c @ ( F @ B ) @ C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_285_order__subst1,axiom,
! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_286_order__subst1,axiom,
! [A: set_Sum_sum_c_c,F: set_Su1783761653um_b_c > set_Sum_sum_c_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
=> ( ( ord_le929608341um_b_c @ B @ C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le1772180283um_c_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_287_order__subst1,axiom,
! [A: set_Su1783761653um_b_c,F: set_Sum_sum_c_c > set_Su1783761653um_b_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
( ( ord_le929608341um_b_c @ A @ ( F @ B ) )
=> ( ( ord_le1772180283um_c_c @ B @ C )
=> ( ! [X2: set_Sum_sum_c_c,Y4: set_Sum_sum_c_c] :
( ( ord_le1772180283um_c_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_288_order__subst1,axiom,
! [A: set_Su1783761653um_b_c,F: set_Su1783761653um_b_c > set_Su1783761653um_b_c,B: set_Su1783761653um_b_c,C: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ A @ ( F @ B ) )
=> ( ( ord_le929608341um_b_c @ B @ C )
=> ( ! [X2: set_Su1783761653um_b_c,Y4: set_Su1783761653um_b_c] :
( ( ord_le929608341um_b_c @ X2 @ Y4 )
=> ( ord_le929608341um_b_c @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_le929608341um_b_c @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_289_surj__pair,axiom,
! [P2: produc640052711real_c] :
? [X2: produc190496183real_c,Y4: produc190496183real_c] :
( P2
= ( produc1687028567real_c @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_290_surj__pair,axiom,
! [P2: produc190496183real_c] :
? [X2: finite1398487019real_c,Y4: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_291_prod__cases,axiom,
! [P: produc640052711real_c > $o,P2: produc640052711real_c] :
( ! [A7: produc190496183real_c,B7: produc190496183real_c] : ( P @ ( produc1687028567real_c @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_292_prod__cases,axiom,
! [P: produc190496183real_c > $o,P2: produc190496183real_c] :
( ! [A7: finite1398487019real_c,B7: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_293_Pair__inject,axiom,
! [A: produc190496183real_c,B: produc190496183real_c,A6: produc190496183real_c,B6: produc190496183real_c] :
( ( ( produc1687028567real_c @ A @ B )
= ( produc1687028567real_c @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_294_Pair__inject,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,A6: finite1398487019real_c,B6: finite1398487019real_c] :
( ( ( produc394644079real_c @ A @ B )
= ( produc394644079real_c @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_295_old_Oprod_Oexhaust,axiom,
! [Y: produc640052711real_c] :
~ ! [A7: produc190496183real_c,B7: produc190496183real_c] :
( Y
!= ( produc1687028567real_c @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_296_old_Oprod_Oexhaust,axiom,
! [Y: produc190496183real_c] :
~ ! [A7: finite1398487019real_c,B7: finite1398487019real_c] :
( Y
!= ( produc394644079real_c @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_297_old_Oprod_Oinducts,axiom,
! [P: produc640052711real_c > $o,Prod: produc640052711real_c] :
( ! [A7: produc190496183real_c,B7: produc190496183real_c] : ( P @ ( produc1687028567real_c @ A7 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_298_old_Oprod_Oinducts,axiom,
! [P: produc190496183real_c > $o,Prod: produc190496183real_c] :
( ! [A7: finite1398487019real_c,B7: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A7 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_299_agree__comm,axiom,
! [A2: produc190496183real_c,B2: produc190496183real_c,V3: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ A2 @ B2 @ V3 )
=> ( denota1997846518gree_c @ B2 @ A2 @ V3 ) ) ).
% agree_comm
thf(fact_300_agree__refl,axiom,
! [Nu3: produc190496183real_c,A2: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu3 @ Nu3 @ A2 ) ).
% agree_refl
thf(fact_301_prod__cases3,axiom,
! [Y: produc640052711real_c] :
~ ! [A7: produc190496183real_c,B7: finite1398487019real_c,C3: finite1398487019real_c] :
( Y
!= ( produc1687028567real_c @ A7 @ ( produc394644079real_c @ B7 @ C3 ) ) ) ).
% prod_cases3
thf(fact_302_prod__induct3,axiom,
! [P: produc640052711real_c > $o,X: produc640052711real_c] :
( ! [A7: produc190496183real_c,B7: finite1398487019real_c,C3: finite1398487019real_c] : ( P @ ( produc1687028567real_c @ A7 @ ( produc394644079real_c @ B7 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_303_agree__union,axiom,
! [Nu3: produc190496183real_c,Omega: produc190496183real_c,A2: set_Sum_sum_c_c,B2: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ Nu3 @ Omega @ A2 )
=> ( ( denota1997846518gree_c @ Nu3 @ Omega @ B2 )
=> ( denota1997846518gree_c @ Nu3 @ Omega @ ( sup_su1349021703um_c_c @ A2 @ B2 ) ) ) ) ).
% agree_union
thf(fact_304_prog__sem_Osimps_I5_J,axiom,
! [I3: denota231621370t_unit,Alpha2: hp_a_b_c,Beta: hp_a_b_c] :
( ( denota904684656_a_b_c @ I3 @ ( choice_a_b_c @ Alpha2 @ Beta ) )
= ( sup_su588839579real_c @ ( denota904684656_a_b_c @ I3 @ Alpha2 ) @ ( denota904684656_a_b_c @ I3 @ Beta ) ) ) ).
% prog_sem.simps(5)
thf(fact_305_same__fstI,axiom,
! [P: produc190496183real_c > $o,X: produc190496183real_c,Y3: produc190496183real_c,Y: produc190496183real_c,R: produc190496183real_c > set_Pr754230471real_c] :
( ( P @ X )
=> ( ( member346808080real_c @ ( produc1687028567real_c @ Y3 @ Y ) @ ( R @ X ) )
=> ( member616151824real_c @ ( produc544496471real_c @ ( produc1687028567real_c @ X @ Y3 ) @ ( produc1687028567real_c @ X @ Y ) ) @ ( same_f598810924real_c @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_306_same__fstI,axiom,
! [P: finite1398487019real_c > $o,X: finite1398487019real_c,Y3: finite1398487019real_c,Y: finite1398487019real_c,R: finite1398487019real_c > set_Pr1389752855real_c] :
( ( P @ X )
=> ( ( member1895684704real_c @ ( produc394644079real_c @ Y3 @ Y ) @ ( R @ X ) )
=> ( member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ X @ Y3 ) @ ( produc394644079real_c @ X @ Y ) ) @ ( same_f1974156740real_c @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_307_relChain__def,axiom,
( bNF_Ca1667803246um_c_c
= ( ^ [R2: set_Pr754230471real_c,As: produc190496183real_c > set_Sum_sum_c_c] :
! [I4: produc190496183real_c,J4: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ I4 @ J4 ) @ R2 )
=> ( ord_le1772180283um_c_c @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_308_relChain__def,axiom,
( bNF_Ca1693016812um_c_c
= ( ^ [R2: set_Pr1389752855real_c,As: finite1398487019real_c > set_Sum_sum_c_c] :
! [I4: finite1398487019real_c,J4: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ I4 @ J4 ) @ R2 )
=> ( ord_le1772180283um_c_c @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_309_relChain__def,axiom,
( bNF_Ca419199138um_b_c
= ( ^ [R2: set_Pr754230471real_c,As: produc190496183real_c > set_Su1783761653um_b_c] :
! [I4: produc190496183real_c,J4: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ I4 @ J4 ) @ R2 )
=> ( ord_le929608341um_b_c @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_310_relChain__def,axiom,
( bNF_Ca318582436um_b_c
= ( ^ [R2: set_Pr1389752855real_c,As: finite1398487019real_c > set_Su1783761653um_b_c] :
! [I4: finite1398487019real_c,J4: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ I4 @ J4 ) @ R2 )
=> ( ord_le929608341um_b_c @ ( As @ I4 ) @ ( As @ J4 ) ) ) ) ) ).
% relChain_def
thf(fact_311_GreatestI2__order,axiom,
! [P: set_Sum_sum_c_c > $o,X: set_Sum_sum_c_c,Q: set_Sum_sum_c_c > $o] :
( ( P @ X )
=> ( ! [Y4: set_Sum_sum_c_c] :
( ( P @ Y4 )
=> ( ord_le1772180283um_c_c @ Y4 @ X ) )
=> ( ! [X2: set_Sum_sum_c_c] :
( ( P @ X2 )
=> ( ! [Y6: set_Sum_sum_c_c] :
( ( P @ Y6 )
=> ( ord_le1772180283um_c_c @ Y6 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1776530114um_c_c @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_312_GreatestI2__order,axiom,
! [P: set_Su1783761653um_b_c > $o,X: set_Su1783761653um_b_c,Q: set_Su1783761653um_b_c > $o] :
( ( P @ X )
=> ( ! [Y4: set_Su1783761653um_b_c] :
( ( P @ Y4 )
=> ( ord_le929608341um_b_c @ Y4 @ X ) )
=> ( ! [X2: set_Su1783761653um_b_c] :
( ( P @ X2 )
=> ( ! [Y6: set_Su1783761653um_b_c] :
( ( P @ Y6 )
=> ( ord_le929608341um_b_c @ Y6 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_365181262um_b_c @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_313_Greatest__equality,axiom,
! [P: set_Sum_sum_c_c > $o,X: set_Sum_sum_c_c] :
( ( P @ X )
=> ( ! [Y4: set_Sum_sum_c_c] :
( ( P @ Y4 )
=> ( ord_le1772180283um_c_c @ Y4 @ X ) )
=> ( ( order_1776530114um_c_c @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_314_Greatest__equality,axiom,
! [P: set_Su1783761653um_b_c > $o,X: set_Su1783761653um_b_c] :
( ( P @ X )
=> ( ! [Y4: set_Su1783761653um_b_c] :
( ( P @ Y4 )
=> ( ord_le929608341um_b_c @ Y4 @ X ) )
=> ( ( order_365181262um_b_c @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_315_prod_Ocollapse,axiom,
! [Prod: produc640052711real_c] :
( ( produc1687028567real_c @ ( produc876006723real_c @ Prod ) @ ( produc415593093real_c @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_316_prod_Ocollapse,axiom,
! [Prod: produc190496183real_c] :
( ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_317_less__eq__prod__def,axiom,
( ord_le684454199um_c_c
= ( ^ [X3: produc214722967um_c_c,Y5: produc214722967um_c_c] :
( ( ord_le1772180283um_c_c @ ( produc1983230011um_c_c @ X3 ) @ ( produc1983230011um_c_c @ Y5 ) )
& ( ord_le1772180283um_c_c @ ( produc1132019837um_c_c @ X3 ) @ ( produc1132019837um_c_c @ Y5 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_318_less__eq__prod__def,axiom,
( ord_le542695343um_b_c
= ( ^ [X3: produc263671631um_b_c,Y5: produc263671631um_b_c] :
( ( ord_le1772180283um_c_c @ ( produc1758795157um_b_c @ X3 ) @ ( produc1758795157um_b_c @ Y5 ) )
& ( ord_le929608341um_b_c @ ( produc1193764051um_b_c @ X3 ) @ ( produc1193764051um_b_c @ Y5 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_319_less__eq__prod__def,axiom,
( ord_le1852514783um_c_c
= ( ^ [X3: produc1573491071um_c_c,Y5: produc1573491071um_c_c] :
( ( ord_le929608341um_b_c @ ( produc1455464333um_c_c @ X3 ) @ ( produc1455464333um_c_c @ Y5 ) )
& ( ord_le1772180283um_c_c @ ( produc890433227um_c_c @ X3 ) @ ( produc890433227um_c_c @ Y5 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_320_less__eq__prod__def,axiom,
( ord_le907560199um_b_c
= ( ^ [X3: produc1415638375um_b_c,Y5: produc1415638375um_b_c] :
( ( ord_le929608341um_b_c @ ( produc1961869763um_b_c @ X3 ) @ ( produc1961869763um_b_c @ Y5 ) )
& ( ord_le929608341um_b_c @ ( produc932426501um_b_c @ X3 ) @ ( produc932426501um_b_c @ Y5 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_321_sup__prod__def,axiom,
( sup_su1399591867real_c
= ( ^ [X3: produc640052711real_c,Y5: produc640052711real_c] : ( produc1687028567real_c @ ( sup_su1639086475real_c @ ( produc876006723real_c @ X3 ) @ ( produc876006723real_c @ Y5 ) ) @ ( sup_su1639086475real_c @ ( produc415593093real_c @ X3 ) @ ( produc415593093real_c @ Y5 ) ) ) ) ) ).
% sup_prod_def
thf(fact_322_sup__prod__def,axiom,
( sup_su1639086475real_c
= ( ^ [X3: produc190496183real_c,Y5: produc190496183real_c] : ( produc394644079real_c @ ( sup_su2141622551real_c @ ( produc2010422875real_c @ X3 ) @ ( produc2010422875real_c @ Y5 ) ) @ ( sup_su2141622551real_c @ ( produc314122909real_c @ X3 ) @ ( produc314122909real_c @ Y5 ) ) ) ) ) ).
% sup_prod_def
thf(fact_323_sup__prod__def,axiom,
( sup_su1672617323um_c_c
= ( ^ [X3: produc214722967um_c_c,Y5: produc214722967um_c_c] : ( produc1439061071um_c_c @ ( sup_su1349021703um_c_c @ ( produc1983230011um_c_c @ X3 ) @ ( produc1983230011um_c_c @ Y5 ) ) @ ( sup_su1349021703um_c_c @ ( produc1132019837um_c_c @ X3 ) @ ( produc1132019837um_c_c @ Y5 ) ) ) ) ) ).
% sup_prod_def
thf(fact_324_fst__conv,axiom,
! [X1: produc190496183real_c,X22: produc190496183real_c] :
( ( produc876006723real_c @ ( produc1687028567real_c @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_325_fst__conv,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c] :
( ( produc2010422875real_c @ ( produc394644079real_c @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_326_fst__eqD,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c,A: produc190496183real_c] :
( ( ( produc876006723real_c @ ( produc1687028567real_c @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_327_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_328_snd__conv,axiom,
! [X1: produc190496183real_c,X22: produc190496183real_c] :
( ( produc415593093real_c @ ( produc1687028567real_c @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_329_snd__conv,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c] :
( ( produc314122909real_c @ ( produc394644079real_c @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_330_snd__eqD,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c,A: produc190496183real_c] :
( ( ( produc415593093real_c @ ( produc1687028567real_c @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_331_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_332_surjective__pairing,axiom,
! [T2: produc640052711real_c] :
( T2
= ( produc1687028567real_c @ ( produc876006723real_c @ T2 ) @ ( produc415593093real_c @ T2 ) ) ) ).
% surjective_pairing
thf(fact_333_surjective__pairing,axiom,
! [T2: produc190496183real_c] :
( T2
= ( produc394644079real_c @ ( produc2010422875real_c @ T2 ) @ ( produc314122909real_c @ T2 ) ) ) ).
% surjective_pairing
thf(fact_334_prod_Oexhaust__sel,axiom,
! [Prod: produc640052711real_c] :
( Prod
= ( produc1687028567real_c @ ( produc876006723real_c @ Prod ) @ ( produc415593093real_c @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_335_prod_Oexhaust__sel,axiom,
! [Prod: produc190496183real_c] :
( Prod
= ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_336_exI__realizer,axiom,
! [P: produc190496183real_c > produc190496183real_c > $o,Y: produc190496183real_c,X: produc190496183real_c] :
( ( P @ Y @ X )
=> ( P @ ( produc415593093real_c @ ( produc1687028567real_c @ X @ Y ) ) @ ( produc876006723real_c @ ( produc1687028567real_c @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_337_exI__realizer,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,Y: finite1398487019real_c,X: finite1398487019real_c] :
( ( P @ Y @ X )
=> ( P @ ( produc314122909real_c @ ( produc394644079real_c @ X @ Y ) ) @ ( produc2010422875real_c @ ( produc394644079real_c @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_338_conjI__realizer,axiom,
! [P: produc190496183real_c > $o,P2: produc190496183real_c,Q: produc190496183real_c > $o,Q2: produc190496183real_c] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc876006723real_c @ ( produc1687028567real_c @ P2 @ Q2 ) ) )
& ( Q @ ( produc415593093real_c @ ( produc1687028567real_c @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_339_conjI__realizer,axiom,
! [P: finite1398487019real_c > $o,P2: finite1398487019real_c,Q: finite1398487019real_c > $o,Q2: finite1398487019real_c] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc2010422875real_c @ ( produc394644079real_c @ P2 @ Q2 ) ) )
& ( Q @ ( produc314122909real_c @ ( produc394644079real_c @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_340_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: produc190496183real_c > produc190496183real_c > $o,X: produc190496183real_c,Y: produc190496183real_c,A: produc640052711real_c] :
( ( P @ X @ Y )
=> ( ( A
= ( produc1687028567real_c @ X @ Y ) )
=> ( P @ ( produc876006723real_c @ A ) @ ( produc415593093real_c @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_341_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,X: finite1398487019real_c,Y: finite1398487019real_c,A: produc190496183real_c] :
( ( P @ X @ Y )
=> ( ( A
= ( produc394644079real_c @ X @ Y ) )
=> ( P @ ( produc2010422875real_c @ A ) @ ( produc314122909real_c @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_342_sndI,axiom,
! [X: produc640052711real_c,Y: produc190496183real_c,Z: produc190496183real_c] :
( ( X
= ( produc1687028567real_c @ Y @ Z ) )
=> ( ( produc415593093real_c @ X )
= Z ) ) ).
% sndI
thf(fact_343_sndI,axiom,
! [X: produc190496183real_c,Y: finite1398487019real_c,Z: finite1398487019real_c] :
( ( X
= ( produc394644079real_c @ Y @ Z ) )
=> ( ( produc314122909real_c @ X )
= Z ) ) ).
% sndI
thf(fact_344_fstI,axiom,
! [X: produc640052711real_c,Y: produc190496183real_c,Z: produc190496183real_c] :
( ( X
= ( produc1687028567real_c @ Y @ Z ) )
=> ( ( produc876006723real_c @ X )
= Y ) ) ).
% fstI
thf(fact_345_fstI,axiom,
! [X: produc190496183real_c,Y: finite1398487019real_c,Z: finite1398487019real_c] :
( ( X
= ( produc394644079real_c @ Y @ Z ) )
=> ( ( produc2010422875real_c @ X )
= Y ) ) ).
% fstI
thf(fact_346_eq__snd__iff,axiom,
! [B: produc190496183real_c,P2: produc640052711real_c] :
( ( B
= ( produc415593093real_c @ P2 ) )
= ( ? [A4: produc190496183real_c] :
( P2
= ( produc1687028567real_c @ A4 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_347_eq__snd__iff,axiom,
! [B: finite1398487019real_c,P2: produc190496183real_c] :
( ( B
= ( produc314122909real_c @ P2 ) )
= ( ? [A4: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ A4 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_348_eq__fst__iff,axiom,
! [A: produc190496183real_c,P2: produc640052711real_c] :
( ( A
= ( produc876006723real_c @ P2 ) )
= ( ? [B4: produc190496183real_c] :
( P2
= ( produc1687028567real_c @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_349_eq__fst__iff,axiom,
! [A: finite1398487019real_c,P2: produc190496183real_c] :
( ( A
= ( produc2010422875real_c @ P2 ) )
= ( ? [B4: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_350_prod_Oswap__def,axiom,
( produc350724343real_c
= ( ^ [P3: produc640052711real_c] : ( produc1687028567real_c @ ( produc415593093real_c @ P3 ) @ ( produc876006723real_c @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_351_prod_Oswap__def,axiom,
( produc704157711real_c
= ( ^ [P3: produc190496183real_c] : ( produc394644079real_c @ ( produc314122909real_c @ P3 ) @ ( produc2010422875real_c @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_352_swap__simp,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( produc350724343real_c @ ( produc1687028567real_c @ X @ Y ) )
= ( produc1687028567real_c @ Y @ X ) ) ).
% swap_simp
thf(fact_353_swap__simp,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( produc704157711real_c @ ( produc394644079real_c @ X @ Y ) )
= ( produc394644079real_c @ Y @ X ) ) ).
% swap_simp
% Conjectures (1)
thf(conj_0,conjecture,
? [Mu6: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ ( produc394644079real_c @ nu_22 @ nu_2 ) @ Mu6 ) @ ( denota904684656_a_b_c @ j @ ( choice_a_b_c @ a @ b ) ) )
& ( denota1997846518gree_c @ ( produc394644079real_c @ mu3 @ mu2 ) @ Mu6 @ ( sup_su1349021703um_c_c @ ( static_MBV_a_b_c @ ( choice_a_b_c @ a @ b ) ) @ v ) ) ) ).
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