TPTP Problem File: SLH0727^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Separation_Logic_Unbounded/0003_FixedPoint/prob_00786_023490__6899064_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1613 ( 580 unt; 331 typ; 0 def)
% Number of atoms : 4032 (1282 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13063 ( 290 ~; 38 |; 356 &;10888 @)
% ( 0 <=>;1491 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 47 ( 46 usr)
% Number of type conns : 2992 (2992 >; 0 *; 0 +; 0 <<)
% Number of symbols : 288 ( 285 usr; 25 con; 0-7 aty)
% Number of variables : 4651 ( 544 ^;3956 !; 151 ?;4651 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:36.663
%------------------------------------------------------------------------------
% Could-be-implicit typings (46)
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% Explicit typings (285)
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member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
member5498148017924304208tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
member1180172933830803072_a_c_d: produc5278197477302038359_a_c_d > set_Pr1275464188344874039_a_c_d > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060Delta_062____,type,
delta: ( c > d ) > set_a ).
thf(sy_v_a____,type,
a2: a ).
thf(sy_v_b____,type,
b2: a ).
thf(sy_v_mult,type,
mult: b > a > a ).
thf(sy_v_one,type,
one: b ).
thf(sy_v_plus,type,
plus: a > a > option_a ).
thf(sy_v_s____,type,
s: c > d ).
thf(sy_v_sadd,type,
sadd: b > b > b ).
thf(sy_v_sinv,type,
sinv: b > b ).
thf(sy_v_smult,type,
smult: b > b > b ).
thf(sy_v_valid,type,
valid: a > $o ).
% Relevant facts (1276)
thf(fact_0_commutative,axiom,
! [A: a,B: a] :
( ( plus @ A @ B )
= ( plus @ B @ A ) ) ).
% commutative
thf(fact_1_can__divide,axiom,
! [P: b,A: a,B: a] :
( ( ( mult @ P @ A )
= ( mult @ P @ B ) )
=> ( A = B ) ) ).
% can_divide
thf(fact_2_sadd__comm,axiom,
! [P: b,Q: b] :
( ( sadd @ P @ Q )
= ( sadd @ Q @ P ) ) ).
% sadd_comm
thf(fact_3_asso1,axiom,
! [A: a,B: a,Ab: a,C: a,Bc: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ( ( plus @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( plus @ Ab @ C )
= ( plus @ A @ Bc ) ) ) ).
% asso1
thf(fact_4_move__sum,axiom,
! [A: a,A1: a,A2: a,B: a,B1: a,B2: a,X: a,X1: a,X2: a] :
( ( ( some_a @ A )
= ( plus @ A1 @ A2 ) )
=> ( ( ( some_a @ B )
= ( plus @ B1 @ B2 ) )
=> ( ( ( some_a @ X )
= ( plus @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( plus @ A1 @ B1 ) )
=> ( ( ( some_a @ X2 )
= ( plus @ A2 @ B2 ) )
=> ( ( some_a @ X )
= ( plus @ X1 @ X2 ) ) ) ) ) ) ) ).
% move_sum
thf(fact_5_one__neutral,axiom,
! [A: a] :
( ( mult @ one @ A )
= A ) ).
% one_neutral
thf(fact_6_plus__mult,axiom,
! [A: a,B: a,C: a,P: b] :
( ( ( some_a @ A )
= ( plus @ B @ C ) )
=> ( ( some_a @ ( mult @ P @ A ) )
= ( plus @ ( mult @ P @ B ) @ ( mult @ P @ C ) ) ) ) ).
% plus_mult
thf(fact_7__092_060open_062a_A_092_060in_062_A_092_060Delta_062_As_A_092_060and_062_Ab_A_092_060in_062_A_092_060Delta_062_As_092_060close_062,axiom,
( ( member_a @ a2 @ ( delta @ s ) )
& ( member_a @ b2 @ ( delta @ s ) ) ) ).
% \<open>a \<in> \<Delta> s \<and> b \<in> \<Delta> s\<close>
thf(fact_8_distrib__mult,axiom,
! [P: b,Q: b,X: a] :
( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X ) )
= ( plus @ ( mult @ P @ X ) @ ( mult @ Q @ X ) ) ) ).
% distrib_mult
thf(fact_9_unique__inv,axiom,
! [A: a,P: b,B: a] :
( ( A
= ( mult @ P @ B ) )
= ( B
= ( mult @ ( sinv @ P ) @ A ) ) ) ).
% unique_inv
thf(fact_10__092_060open_062sem__combinable_A_092_060Delta_062_092_060close_062,axiom,
sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ delta ).
% \<open>sem_combinable \<Delta>\<close>
thf(fact_11_sem__combinableE,axiom,
! [Delta: ( c > d ) > set_a,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ Delta )
=> ( ( member_a @ A @ ( Delta @ S ) )
=> ( ( member_a @ B @ ( Delta @ S ) )
=> ( ( ( some_a @ X )
= ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
=> ( ( ( sadd @ P @ Q )
= one )
=> ( member_a @ X @ ( Delta @ S ) ) ) ) ) ) ) ).
% sem_combinableE
thf(fact_12_sem__combinableI,axiom,
! [Delta: ( c > d ) > set_a] :
( ! [S2: c > d,P2: b,Q2: b,A3: a,B3: a,X3: a] :
( ( ( ( sadd @ P2 @ Q2 )
= one )
& ( member_a @ A3 @ ( Delta @ S2 ) )
& ( member_a @ B3 @ ( Delta @ S2 ) )
& ( ( some_a @ X3 )
= ( plus @ ( mult @ P2 @ A3 ) @ ( mult @ Q2 @ B3 ) ) ) )
=> ( member_a @ X3 @ ( Delta @ S2 ) ) )
=> ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ Delta ) ) ).
% sem_combinableI
thf(fact_13_sem__combinable__def,axiom,
! [Delta: ( c > d ) > set_a] :
( ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ Delta )
= ( ! [S3: c > d,P3: b,Q3: b,A4: a,B4: a,X4: a] :
( ( ( ( sadd @ P3 @ Q3 )
= one )
& ( member_a @ A4 @ ( Delta @ S3 ) )
& ( member_a @ B4 @ ( Delta @ S3 ) )
& ( ( some_a @ X4 )
= ( plus @ ( mult @ P3 @ A4 ) @ ( mult @ Q3 @ B4 ) ) ) )
=> ( member_a @ X4 @ ( Delta @ S3 ) ) ) ) ) ).
% sem_combinable_def
thf(fact_14_compatible__iff,axiom,
! [A: a,B: a,P: b] :
( ( pre_compatible_a @ plus @ A @ B )
= ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% compatible_iff
thf(fact_15_compatible__imp,axiom,
! [A: a,B: a,P: b] :
( ( pre_compatible_a @ plus @ A @ B )
=> ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% compatible_imp
thf(fact_16_compatible__multiples,axiom,
! [P: b,A: a,Q: b,B: a] :
( ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) )
=> ( pre_compatible_a @ plus @ A @ B ) ) ).
% compatible_multiples
thf(fact_17_larger__same,axiom,
! [A: a,B: a,P: b] :
( ( pre_larger_a @ plus @ A @ B )
= ( pre_larger_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% larger_same
thf(fact_18_asso2,axiom,
! [A: a,B: a,Ab: a,C: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ plus @ B @ C ) )
=> ~ ( pre_compatible_a @ plus @ Ab @ C ) ) ).
% asso2
thf(fact_19_asso3,axiom,
! [A: a,B: a,C: a,Bc: a] :
( ~ ( pre_compatible_a @ plus @ A @ B )
=> ( ( ( plus @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ plus @ A @ Bc ) ) ) ).
% asso3
thf(fact_20_larger__def,axiom,
! [A: a,B: a] :
( ( pre_larger_a @ plus @ A @ B )
= ( ? [C2: a] :
( ( some_a @ A )
= ( plus @ B @ C2 ) ) ) ) ).
% larger_def
thf(fact_21_larger__first__sum,axiom,
! [Y: a,A: a,B: a,X: a] :
( ( ( some_a @ Y )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ X @ Y )
=> ? [A5: a] :
( ( ( some_a @ X )
= ( plus @ A5 @ B ) )
& ( pre_larger_a @ plus @ A5 @ A ) ) ) ) ).
% larger_first_sum
thf(fact_22_sum__both__larger,axiom,
! [X5: a,A6: a,B5: a,X: a,A: a,B: a] :
( ( ( some_a @ X5 )
= ( plus @ A6 @ B5 ) )
=> ( ( ( some_a @ X )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ A6 @ A )
=> ( ( pre_larger_a @ plus @ B5 @ B )
=> ( pre_larger_a @ plus @ X5 @ X ) ) ) ) ) ).
% sum_both_larger
thf(fact_23_larger__implies__compatible,axiom,
! [X: a,Y: a] :
( ( pre_larger_a @ plus @ X @ Y )
=> ( pre_compatible_a @ plus @ X @ Y ) ) ).
% larger_implies_compatible
thf(fact_24_compatible__smaller,axiom,
! [A: a,B: a,X: a] :
( ( pre_larger_a @ plus @ A @ B )
=> ( ( pre_compatible_a @ plus @ X @ A )
=> ( pre_compatible_a @ plus @ X @ B ) ) ) ).
% compatible_smaller
thf(fact_25_logic_Osem__combinable_Ocong,axiom,
sem_co7516848414490435095_b_c_d = sem_co7516848414490435095_b_c_d ).
% logic.sem_combinable.cong
thf(fact_26_sinv__inverse,axiom,
! [P: b] :
( ( smult @ P @ ( sinv @ P ) )
= one ) ).
% sinv_inverse
thf(fact_27_option_Oinject,axiom,
! [X2: a,Y2: a] :
( ( ( some_a @ X2 )
= ( some_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_28_subset__antisym,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ A7 )
=> ( A7 = B6 ) ) ) ).
% subset_antisym
thf(fact_29_subset__antisym,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ( ord_le746702958409616551od_a_a @ B6 @ A7 )
=> ( A7 = B6 ) ) ) ).
% subset_antisym
thf(fact_30_subset__antisym,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ A7 )
=> ( A7 = B6 ) ) ) ).
% subset_antisym
thf(fact_31_subsetI,axiom,
! [A7: set_option_a,B6: set_option_a] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ A7 )
=> ( member_option_a @ X3 @ B6 ) )
=> ( ord_le1955136853071979460tion_a @ A7 @ B6 ) ) ).
% subsetI
thf(fact_32_subsetI,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
=> ( member_set_a @ X3 @ B6 ) )
=> ( ord_le3724670747650509150_set_a @ A7 @ B6 ) ) ).
% subsetI
thf(fact_33_subsetI,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A7 )
=> ( member1426531477525435216od_a_a @ X3 @ B6 ) )
=> ( ord_le746702958409616551od_a_a @ A7 @ B6 ) ) ).
% subsetI
thf(fact_34_subsetI,axiom,
! [A7: set_a,B6: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ X3 @ B6 ) )
=> ( ord_less_eq_set_a @ A7 @ B6 ) ) ).
% subsetI
thf(fact_35_dual__order_Orefl,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ A ) ).
% dual_order.refl
thf(fact_36_dual__order_Orefl,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_37_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_38_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_39_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_40_order__refl,axiom,
! [X: a > $o] : ( ord_less_eq_a_o @ X @ X ) ).
% order_refl
thf(fact_41_order__refl,axiom,
! [X: $o > set_a] : ( ord_less_eq_o_set_a @ X @ X ) ).
% order_refl
thf(fact_42_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_43_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_44_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_45_compatible__def,axiom,
! [A: a,B: a] :
( ( pre_compatible_a @ plus @ A @ B )
= ( ( plus @ A @ B )
!= none_a ) ) ).
% compatible_def
thf(fact_46_pre__logic_Olarger__def,axiom,
( pre_larger_a
= ( ^ [Plus: a > a > option_a,A4: a,B4: a] :
? [C2: a] :
( ( some_a @ A4 )
= ( Plus @ B4 @ C2 ) ) ) ) ).
% pre_logic.larger_def
thf(fact_47_can__factorize,axiom,
! [Q: b,P: b] :
? [R: b] :
( Q
= ( smult @ R @ P ) ) ).
% can_factorize
thf(fact_48_smult__asso,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ ( smult @ P @ Q ) @ R2 )
= ( smult @ P @ ( smult @ Q @ R2 ) ) ) ).
% smult_asso
thf(fact_49_smult__comm,axiom,
! [P: b,Q: b] :
( ( smult @ P @ Q )
= ( smult @ Q @ P ) ) ).
% smult_comm
thf(fact_50_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P4: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: set_a,P4: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: option_a,P4: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: a,P4: a > $o] :
( ( member_a @ A @ ( collect_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A7: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A7: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A7: set_option_a] :
( ( collect_option_a
@ ^ [X4: option_a] : ( member_option_a @ X4 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A7: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_58_Collect__cong,axiom,
! [P4: set_a > $o,Q4: set_a > $o] :
( ! [X3: set_a] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collect_set_a @ P4 )
= ( collect_set_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_59_Collect__cong,axiom,
! [P4: option_a > $o,Q4: option_a > $o] :
( ! [X3: option_a] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collect_option_a @ P4 )
= ( collect_option_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_60_Collect__cong,axiom,
! [P4: a > $o,Q4: a > $o] :
( ! [X3: a] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collect_a @ P4 )
= ( collect_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_61_double__mult,axiom,
! [P: b,Q: b,A: a] :
( ( mult @ P @ ( mult @ Q @ A ) )
= ( mult @ ( smult @ P @ Q ) @ A ) ) ).
% double_mult
thf(fact_62_smult__distrib,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ P @ ( sadd @ Q @ R2 ) )
= ( sadd @ ( smult @ P @ Q ) @ ( smult @ P @ R2 ) ) ) ).
% smult_distrib
thf(fact_63_sone__neutral,axiom,
! [P: b] :
( ( smult @ one @ P )
= P ) ).
% sone_neutral
thf(fact_64_not__None__eq,axiom,
! [X: option_a] :
( ( X != none_a )
= ( ? [Y3: a] :
( X
= ( some_a @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_65_not__Some__eq,axiom,
! [X: option_a] :
( ( ! [Y3: a] :
( X
!= ( some_a @ Y3 ) ) )
= ( X = none_a ) ) ).
% not_Some_eq
thf(fact_66_option_Odistinct_I1_J,axiom,
! [X2: a] :
( none_a
!= ( some_a @ X2 ) ) ).
% option.distinct(1)
thf(fact_67_option_OdiscI,axiom,
! [Option: option_a,X2: a] :
( ( Option
= ( some_a @ X2 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_68_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X22: a] :
( Y
!= ( some_a @ X22 ) ) ) ).
% option.exhaust
thf(fact_69_split__option__ex,axiom,
( ( ^ [P5: option_a > $o] :
? [X6: option_a] : ( P5 @ X6 ) )
= ( ^ [P6: option_a > $o] :
( ( P6 @ none_a )
| ? [X4: a] : ( P6 @ ( some_a @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_70_split__option__all,axiom,
( ( ^ [P5: option_a > $o] :
! [X6: option_a] : ( P5 @ X6 ) )
= ( ^ [P6: option_a > $o] :
( ( P6 @ none_a )
& ! [X4: a] : ( P6 @ ( some_a @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_71_combine__options__cases,axiom,
! [X: option_a,P4: option_a > option_a > $o,Y: option_a] :
( ( ( X = none_a )
=> ( P4 @ X @ Y ) )
=> ( ( ( Y = none_a )
=> ( P4 @ X @ Y ) )
=> ( ! [A3: a,B3: a] :
( ( X
= ( some_a @ A3 ) )
=> ( ( Y
= ( some_a @ B3 ) )
=> ( P4 @ X @ Y ) ) )
=> ( P4 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_72_pre__logic_Ocompatible__def,axiom,
( pre_compatible_a
= ( ^ [Plus: a > a > option_a,A4: a,B4: a] :
( ( Plus @ A4 @ B4 )
!= none_a ) ) ) ).
% pre_logic.compatible_def
thf(fact_73_less__eq__set__def,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
( ord_le8889544289526388249on_a_o
@ ^ [X4: option_a] : ( member_option_a @ X4 @ A8 )
@ ^ [X4: option_a] : ( member_option_a @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_74_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A8 )
@ ^ [X4: set_a] : ( member_set_a @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_75_less__eq__set__def,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ord_le1591150415168442102_a_a_o
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A8 )
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_76_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A8 )
@ ^ [X4: a] : ( member_a @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_77_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a > $o,Z: a > $o] : ( Y4 = Z ) )
= ( ^ [X4: a > $o,Y3: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y3 )
& ( ord_less_eq_a_o @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_78_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > set_a,Z: $o > set_a] : ( Y4 = Z ) )
= ( ^ [X4: $o > set_a,Y3: $o > set_a] :
( ( ord_less_eq_o_set_a @ X4 @ Y3 )
& ( ord_less_eq_o_set_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_79_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [X4: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_80_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y3 )
& ( ord_le746702958409616551od_a_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_81_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_82_ord__eq__le__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( A = B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_83_ord__eq__le__trans,axiom,
! [A: $o > set_a,B: $o > set_a,C: $o > set_a] :
( ( A = B )
=> ( ( ord_less_eq_o_set_a @ B @ C )
=> ( ord_less_eq_o_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_84_ord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_85_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A = B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_86_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_87_ord__le__eq__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_88_ord__le__eq__trans,axiom,
! [A: $o > set_a,B: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_89_ord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_90_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( B = C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_91_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_92_order__antisym,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_93_order__antisym,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_94_order__antisym,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_95_order__antisym,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_96_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_97_order_Otrans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ord_less_eq_a_o @ A @ C ) ) ) ).
% order.trans
thf(fact_98_order_Otrans,axiom,
! [A: $o > set_a,B: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( ord_less_eq_o_set_a @ B @ C )
=> ( ord_less_eq_o_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_99_order_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_100_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% order.trans
thf(fact_101_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_102_order__trans,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ Y @ Z2 )
=> ( ord_less_eq_a_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_103_order__trans,axiom,
! [X: $o > set_a,Y: $o > set_a,Z2: $o > set_a] :
( ( ord_less_eq_o_set_a @ X @ Y )
=> ( ( ord_less_eq_o_set_a @ Y @ Z2 )
=> ( ord_less_eq_o_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_104_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_105_order__trans,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z2 )
=> ( ord_le746702958409616551od_a_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_106_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_107_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a > $o,Z: a > $o] : ( Y4 = Z ) )
= ( ^ [A4: a > $o,B4: a > $o] :
( ( ord_less_eq_a_o @ B4 @ A4 )
& ( ord_less_eq_a_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_108_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > set_a,Z: $o > set_a] : ( Y4 = Z ) )
= ( ^ [A4: $o > set_a,B4: $o > set_a] :
( ( ord_less_eq_o_set_a @ B4 @ A4 )
& ( ord_less_eq_o_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_109_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_110_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B4 @ A4 )
& ( ord_le746702958409616551od_a_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_111_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_112_dual__order_Oantisym,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_113_dual__order_Oantisym,axiom,
! [B: $o > set_a,A: $o > set_a] :
( ( ord_less_eq_o_set_a @ B @ A )
=> ( ( ord_less_eq_o_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_114_dual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_115_dual__order_Oantisym,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_116_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_117_dual__order_Otrans,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
=> ( ( ord_less_eq_a_o @ C @ B )
=> ( ord_less_eq_a_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_118_dual__order_Otrans,axiom,
! [B: $o > set_a,A: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_o_set_a @ B @ A )
=> ( ( ord_less_eq_o_set_a @ C @ B )
=> ( ord_less_eq_o_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_119_dual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_120_dual__order_Otrans,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B )
=> ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_121_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_122_antisym,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_123_antisym,axiom,
! [A: $o > set_a,B: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( ord_less_eq_o_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_124_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_126_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_127_le__funD,axiom,
! [F: a > $o,G: a > $o,X: a] :
( ( ord_less_eq_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_128_le__funD,axiom,
! [F: $o > set_a,G: $o > set_a,X: $o] :
( ( ord_less_eq_o_set_a @ F @ G )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_129_le__funE,axiom,
! [F: a > $o,G: a > $o,X: a] :
( ( ord_less_eq_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_130_le__funE,axiom,
! [F: $o > set_a,G: $o > set_a,X: $o] :
( ( ord_less_eq_o_set_a @ F @ G )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_131_le__funI,axiom,
! [F: a > $o,G: a > $o] :
( ! [X3: a] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_a_o @ F @ G ) ) ).
% le_funI
thf(fact_132_le__funI,axiom,
! [F: $o > set_a,G: $o > set_a] :
( ! [X3: $o] : ( ord_less_eq_set_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_o_set_a @ F @ G ) ) ).
% le_funI
thf(fact_133_le__fun__def,axiom,
( ord_less_eq_a_o
= ( ^ [F2: a > $o,G2: a > $o] :
! [X4: a] : ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_134_le__fun__def,axiom,
( ord_less_eq_o_set_a
= ( ^ [F2: $o > set_a,G2: $o > set_a] :
! [X4: $o] : ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_135_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a > $o,Z: a > $o] : ( Y4 = Z ) )
= ( ^ [A4: a > $o,B4: a > $o] :
( ( ord_less_eq_a_o @ A4 @ B4 )
& ( ord_less_eq_a_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_136_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > set_a,Z: $o > set_a] : ( Y4 = Z ) )
= ( ^ [A4: $o > set_a,B4: $o > set_a] :
( ( ord_less_eq_o_set_a @ A4 @ B4 )
& ( ord_less_eq_o_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_137_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_138_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B4 )
& ( ord_le746702958409616551od_a_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_139_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_140_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: set_a,F: ( a > $o ) > set_a,B: a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: a > $o,F: set_a > a > $o,B: set_a,C: set_a] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: set_a,F: ( $o > set_a ) > set_a,B: $o > set_a,C: $o > set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_set_a @ B @ C )
=> ( ! [X3: $o > set_a,Y5: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: set_a,F: set_Product_prod_a_a > set_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ! [X3: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o,B: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: a > $o,F: set_set_a > a > $o,B: set_set_a,C: set_set_a] :
( ( ord_less_eq_a_o @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A: $o > set_a,F: set_a > $o > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_o_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_o_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_a_o @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > $o > set_a,C: $o > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_o_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_o_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_less_eq_a_o @ ( F @ B ) @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_set_a,C: set_set_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst2,axiom,
! [A: $o > set_a,B: $o > set_a,F: ( $o > set_a ) > set_a,C: set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: $o > set_a,Y5: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_160_order__eq__refl,axiom,
! [X: a > $o,Y: a > $o] :
( ( X = Y )
=> ( ord_less_eq_a_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_161_order__eq__refl,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( X = Y )
=> ( ord_less_eq_o_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_162_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_163_order__eq__refl,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X = Y )
=> ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_164_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_165_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_166_ord__eq__le__subst,axiom,
! [A: a > $o,F: set_a > a > $o,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_167_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_168_ord__eq__le__subst,axiom,
! [A: set_a,F: ( a > $o ) > set_a,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_169_ord__eq__le__subst,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_170_ord__eq__le__subst,axiom,
! [A: $o > set_a,F: set_a > $o > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_o_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_171_ord__eq__le__subst,axiom,
! [A: set_Product_prod_a_a,F: set_a > set_Product_prod_a_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_172_ord__eq__le__subst,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_173_ord__eq__le__subst,axiom,
! [A: set_set_a,F: ( a > $o ) > set_set_a,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a_o @ B @ C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_174_ord__eq__le__subst,axiom,
! [A: set_a,F: ( $o > set_a ) > set_a,B: $o > set_a,C: $o > set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_set_a @ B @ C )
=> ( ! [X3: $o > set_a,Y5: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_175_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_176_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_177_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_178_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_179_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_180_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > $o > set_a,C: $o > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_o_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_o_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_181_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_182_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_less_eq_a_o @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_183_ord__le__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_set_a,C: set_set_a] :
( ( ord_less_eq_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: a > $o,Y5: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_184_ord__le__eq__subst,axiom,
! [A: $o > set_a,B: $o > set_a,F: ( $o > set_a ) > set_a,C: set_a] :
( ( ord_less_eq_o_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: $o > set_a,Y5: $o > set_a] :
( ( ord_less_eq_o_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_185_order__antisym__conv,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ Y @ X )
=> ( ( ord_less_eq_a_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_186_order__antisym__conv,axiom,
! [Y: $o > set_a,X: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y @ X )
=> ( ( ord_less_eq_o_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_187_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_188_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( ord_le746702958409616551od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_189_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_190_in__mono,axiom,
! [A7: set_option_a,B6: set_option_a,X: option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ B6 )
=> ( ( member_option_a @ X @ A7 )
=> ( member_option_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_191_in__mono,axiom,
! [A7: set_set_a,B6: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ( member_set_a @ X @ A7 )
=> ( member_set_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_192_in__mono,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ( member1426531477525435216od_a_a @ X @ A7 )
=> ( member1426531477525435216od_a_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_193_in__mono,axiom,
! [A7: set_a,B6: set_a,X: a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( member_a @ X @ A7 )
=> ( member_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_194_subsetD,axiom,
! [A7: set_option_a,B6: set_option_a,C: option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ B6 )
=> ( ( member_option_a @ C @ A7 )
=> ( member_option_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_195_subsetD,axiom,
! [A7: set_set_a,B6: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ( member_set_a @ C @ A7 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_196_subsetD,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ( member1426531477525435216od_a_a @ C @ A7 )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_197_subsetD,axiom,
! [A7: set_a,B6: set_a,C: a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( member_a @ C @ A7 )
=> ( member_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_198_equalityE,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( A7 = B6 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ~ ( ord_le3724670747650509150_set_a @ B6 @ A7 ) ) ) ).
% equalityE
thf(fact_199_equalityE,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( A7 = B6 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ~ ( ord_le746702958409616551od_a_a @ B6 @ A7 ) ) ) ).
% equalityE
thf(fact_200_equalityE,axiom,
! [A7: set_a,B6: set_a] :
( ( A7 = B6 )
=> ~ ( ( ord_less_eq_set_a @ A7 @ B6 )
=> ~ ( ord_less_eq_set_a @ B6 @ A7 ) ) ) ).
% equalityE
thf(fact_201_subset__eq,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
! [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
=> ( member_option_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_202_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A8 )
=> ( member_set_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_203_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A8 )
=> ( member1426531477525435216od_a_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_204_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A8 )
=> ( member_a @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_205_equalityD1,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( A7 = B6 )
=> ( ord_le3724670747650509150_set_a @ A7 @ B6 ) ) ).
% equalityD1
thf(fact_206_equalityD1,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( A7 = B6 )
=> ( ord_le746702958409616551od_a_a @ A7 @ B6 ) ) ).
% equalityD1
thf(fact_207_equalityD1,axiom,
! [A7: set_a,B6: set_a] :
( ( A7 = B6 )
=> ( ord_less_eq_set_a @ A7 @ B6 ) ) ).
% equalityD1
thf(fact_208_equalityD2,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( A7 = B6 )
=> ( ord_le3724670747650509150_set_a @ B6 @ A7 ) ) ).
% equalityD2
thf(fact_209_equalityD2,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( A7 = B6 )
=> ( ord_le746702958409616551od_a_a @ B6 @ A7 ) ) ).
% equalityD2
thf(fact_210_equalityD2,axiom,
! [A7: set_a,B6: set_a] :
( ( A7 = B6 )
=> ( ord_less_eq_set_a @ B6 @ A7 ) ) ).
% equalityD2
thf(fact_211_subset__iff,axiom,
( ord_le1955136853071979460tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
! [T: option_a] :
( ( member_option_a @ T @ A8 )
=> ( member_option_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_212_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A8 )
=> ( member_set_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_213_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A8 )
=> ( member1426531477525435216od_a_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_214_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
! [T: a] :
( ( member_a @ T @ A8 )
=> ( member_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_215_subset__refl,axiom,
! [A7: set_set_a] : ( ord_le3724670747650509150_set_a @ A7 @ A7 ) ).
% subset_refl
thf(fact_216_subset__refl,axiom,
! [A7: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A7 @ A7 ) ).
% subset_refl
thf(fact_217_subset__refl,axiom,
! [A7: set_a] : ( ord_less_eq_set_a @ A7 @ A7 ) ).
% subset_refl
thf(fact_218_Collect__mono,axiom,
! [P4: option_a > $o,Q4: option_a > $o] :
( ! [X3: option_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le1955136853071979460tion_a @ ( collect_option_a @ P4 ) @ ( collect_option_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_219_Collect__mono,axiom,
! [P4: set_a > $o,Q4: set_a > $o] :
( ! [X3: set_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P4 ) @ ( collect_set_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_220_Collect__mono,axiom,
! [P4: product_prod_a_a > $o,Q4: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P4 ) @ ( collec3336397797384452498od_a_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_221_Collect__mono,axiom,
! [P4: a > $o,Q4: a > $o] :
( ! [X3: a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_222_subset__trans,axiom,
! [A7: set_set_a,B6: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C3 )
=> ( ord_le3724670747650509150_set_a @ A7 @ C3 ) ) ) ).
% subset_trans
thf(fact_223_subset__trans,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a,C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ( ord_le746702958409616551od_a_a @ B6 @ C3 )
=> ( ord_le746702958409616551od_a_a @ A7 @ C3 ) ) ) ).
% subset_trans
thf(fact_224_subset__trans,axiom,
! [A7: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ord_less_eq_set_a @ A7 @ C3 ) ) ) ).
% subset_trans
thf(fact_225_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A8: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A8 @ B7 )
& ( ord_le3724670747650509150_set_a @ B7 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_226_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y4 = Z ) )
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A8 @ B7 )
& ( ord_le746702958409616551od_a_a @ B7 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_227_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A8: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A8 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_228_Collect__mono__iff,axiom,
! [P4: option_a > $o,Q4: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ ( collect_option_a @ P4 ) @ ( collect_option_a @ Q4 ) )
= ( ! [X4: option_a] :
( ( P4 @ X4 )
=> ( Q4 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_229_Collect__mono__iff,axiom,
! [P4: set_a > $o,Q4: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P4 ) @ ( collect_set_a @ Q4 ) )
= ( ! [X4: set_a] :
( ( P4 @ X4 )
=> ( Q4 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_230_Collect__mono__iff,axiom,
! [P4: product_prod_a_a > $o,Q4: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P4 ) @ ( collec3336397797384452498od_a_a @ Q4 ) )
= ( ! [X4: product_prod_a_a] :
( ( P4 @ X4 )
=> ( Q4 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_231_Collect__mono__iff,axiom,
! [P4: a > $o,Q4: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) )
= ( ! [X4: a] :
( ( P4 @ X4 )
=> ( Q4 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_232_pre__logic_Olarger_Ocong,axiom,
pre_larger_a = pre_larger_a ).
% pre_logic.larger.cong
thf(fact_233_pre__logic_Ocompatible_Ocong,axiom,
pre_compatible_a = pre_compatible_a ).
% pre_logic.compatible.cong
thf(fact_234_Collect__subset,axiom,
! [A7: set_option_a,P4: option_a > $o] :
( ord_le1955136853071979460tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
& ( P4 @ X4 ) ) )
@ A7 ) ).
% Collect_subset
thf(fact_235_Collect__subset,axiom,
! [A7: set_set_a,P4: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ A7 )
& ( P4 @ X4 ) ) )
@ A7 ) ).
% Collect_subset
thf(fact_236_Collect__subset,axiom,
! [A7: set_Product_prod_a_a,P4: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A7 )
& ( P4 @ X4 ) ) )
@ A7 ) ).
% Collect_subset
thf(fact_237_Collect__subset,axiom,
! [A7: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A7 )
& ( P4 @ X4 ) ) )
@ A7 ) ).
% Collect_subset
thf(fact_238_logic__axioms,axiom,
logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).
% logic_axioms
thf(fact_239_pred__subset__eq,axiom,
! [R3: set_option_a,S4: set_option_a] :
( ( ord_le8889544289526388249on_a_o
@ ^ [X4: option_a] : ( member_option_a @ X4 @ R3 )
@ ^ [X4: option_a] : ( member_option_a @ X4 @ S4 ) )
= ( ord_le1955136853071979460tion_a @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_240_pred__subset__eq,axiom,
! [R3: set_set_a,S4: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ R3 )
@ ^ [X4: set_a] : ( member_set_a @ X4 @ S4 ) )
= ( ord_le3724670747650509150_set_a @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_241_pred__subset__eq,axiom,
! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
( ( ord_le1591150415168442102_a_a_o
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ R3 )
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ S4 ) )
= ( ord_le746702958409616551od_a_a @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_242_pred__subset__eq,axiom,
! [R3: set_a,S4: set_a] :
( ( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R3 )
@ ^ [X4: a] : ( member_a @ X4 @ S4 ) )
= ( ord_less_eq_set_a @ R3 @ S4 ) ) ).
% pred_subset_eq
thf(fact_243_conj__subset__def,axiom,
! [A7: set_option_a,P4: option_a > $o,Q4: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ A7
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) )
= ( ( ord_le1955136853071979460tion_a @ A7 @ ( collect_option_a @ P4 ) )
& ( ord_le1955136853071979460tion_a @ A7 @ ( collect_option_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_244_conj__subset__def,axiom,
! [A7: set_set_a,P4: set_a > $o,Q4: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A7
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) )
= ( ( ord_le3724670747650509150_set_a @ A7 @ ( collect_set_a @ P4 ) )
& ( ord_le3724670747650509150_set_a @ A7 @ ( collect_set_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_245_conj__subset__def,axiom,
! [A7: set_Product_prod_a_a,P4: product_prod_a_a > $o,Q4: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A7
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) )
= ( ( ord_le746702958409616551od_a_a @ A7 @ ( collec3336397797384452498od_a_a @ P4 ) )
& ( ord_le746702958409616551od_a_a @ A7 @ ( collec3336397797384452498od_a_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_246_conj__subset__def,axiom,
! [A7: set_a,P4: a > $o,Q4: a > $o] :
( ( ord_less_eq_set_a @ A7
@ ( collect_a
@ ^ [X4: a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) )
= ( ( ord_less_eq_set_a @ A7 @ ( collect_a @ P4 ) )
& ( ord_less_eq_set_a @ A7 @ ( collect_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_247_prop__restrict,axiom,
! [X: option_a,Z3: set_option_a,X7: set_option_a,P4: option_a > $o] :
( ( member_option_a @ X @ Z3 )
=> ( ( ord_le1955136853071979460tion_a @ Z3
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ X7 )
& ( P4 @ X4 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_248_prop__restrict,axiom,
! [X: set_a,Z3: set_set_a,X7: set_set_a,P4: set_a > $o] :
( ( member_set_a @ X @ Z3 )
=> ( ( ord_le3724670747650509150_set_a @ Z3
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ X7 )
& ( P4 @ X4 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_249_prop__restrict,axiom,
! [X: product_prod_a_a,Z3: set_Product_prod_a_a,X7: set_Product_prod_a_a,P4: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X @ Z3 )
=> ( ( ord_le746702958409616551od_a_a @ Z3
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ X7 )
& ( P4 @ X4 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_250_prop__restrict,axiom,
! [X: a,Z3: set_a,X7: set_a,P4: a > $o] :
( ( member_a @ X @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ X7 )
& ( P4 @ X4 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_251_Collect__restrict,axiom,
! [X7: set_option_a,P4: option_a > $o] :
( ord_le1955136853071979460tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ X7 )
& ( P4 @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_252_Collect__restrict,axiom,
! [X7: set_set_a,P4: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ X7 )
& ( P4 @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_253_Collect__restrict,axiom,
! [X7: set_Product_prod_a_a,P4: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ X7 )
& ( P4 @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_254_Collect__restrict,axiom,
! [X7: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ X7 )
& ( P4 @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_255_subset__CollectI,axiom,
! [B6: set_option_a,A7: set_option_a,Q4: option_a > $o,P4: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ B6 @ A7 )
=> ( ! [X3: option_a] :
( ( member_option_a @ X3 @ B6 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le1955136853071979460tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ B6 )
& ( Q4 @ X4 ) ) )
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_256_subset__CollectI,axiom,
! [B6: set_set_a,A7: set_set_a,Q4: set_a > $o,P4: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B6 @ A7 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ B6 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ B6 )
& ( Q4 @ X4 ) ) )
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_257_subset__CollectI,axiom,
! [B6: set_Product_prod_a_a,A7: set_Product_prod_a_a,Q4: product_prod_a_a > $o,P4: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ B6 @ A7 )
=> ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B6 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B6 )
& ( Q4 @ X4 ) ) )
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_258_subset__CollectI,axiom,
! [B6: set_a,A7: set_a,Q4: a > $o,P4: a > $o] :
( ( ord_less_eq_set_a @ B6 @ A7 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B6 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ B6 )
& ( Q4 @ X4 ) ) )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_259_subset__Collect__iff,axiom,
! [B6: set_option_a,A7: set_option_a,P4: option_a > $o] :
( ( ord_le1955136853071979460tion_a @ B6 @ A7 )
=> ( ( ord_le1955136853071979460tion_a @ B6
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) )
= ( ! [X4: option_a] :
( ( member_option_a @ X4 @ B6 )
=> ( P4 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_260_subset__Collect__iff,axiom,
! [B6: set_set_a,A7: set_set_a,P4: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ B6 @ A7 )
=> ( ( ord_le3724670747650509150_set_a @ B6
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ B6 )
=> ( P4 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_261_subset__Collect__iff,axiom,
! [B6: set_Product_prod_a_a,A7: set_Product_prod_a_a,P4: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ B6 @ A7 )
=> ( ( ord_le746702958409616551od_a_a @ B6
@ ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B6 )
=> ( P4 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_262_subset__Collect__iff,axiom,
! [B6: set_a,A7: set_a,P4: a > $o] :
( ( ord_less_eq_set_a @ B6 @ A7 )
=> ( ( ord_less_eq_set_a @ B6
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A7 )
& ( P4 @ X4 ) ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ B6 )
=> ( P4 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_263_GreatestI2__order,axiom,
! [P4: ( a > $o ) > $o,X: a > $o,Q4: ( a > $o ) > $o] :
( ( P4 @ X )
=> ( ! [Y5: a > $o] :
( ( P4 @ Y5 )
=> ( ord_less_eq_a_o @ Y5 @ X ) )
=> ( ! [X3: a > $o] :
( ( P4 @ X3 )
=> ( ! [Y6: a > $o] :
( ( P4 @ Y6 )
=> ( ord_less_eq_a_o @ Y6 @ X3 ) )
=> ( Q4 @ X3 ) ) )
=> ( Q4 @ ( order_Greatest_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_264_GreatestI2__order,axiom,
! [P4: ( $o > set_a ) > $o,X: $o > set_a,Q4: ( $o > set_a ) > $o] :
( ( P4 @ X )
=> ( ! [Y5: $o > set_a] :
( ( P4 @ Y5 )
=> ( ord_less_eq_o_set_a @ Y5 @ X ) )
=> ( ! [X3: $o > set_a] :
( ( P4 @ X3 )
=> ( ! [Y6: $o > set_a] :
( ( P4 @ Y6 )
=> ( ord_less_eq_o_set_a @ Y6 @ X3 ) )
=> ( Q4 @ X3 ) ) )
=> ( Q4 @ ( order_6114237596796908690_set_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_265_GreatestI2__order,axiom,
! [P4: set_set_a > $o,X: set_set_a,Q4: set_set_a > $o] :
( ( P4 @ X )
=> ( ! [Y5: set_set_a] :
( ( P4 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ Y5 @ X ) )
=> ( ! [X3: set_set_a] :
( ( P4 @ X3 )
=> ( ! [Y6: set_set_a] :
( ( P4 @ Y6 )
=> ( ord_le3724670747650509150_set_a @ Y6 @ X3 ) )
=> ( Q4 @ X3 ) ) )
=> ( Q4 @ ( order_3565860530148683671_set_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_266_GreatestI2__order,axiom,
! [P4: set_Product_prod_a_a > $o,X: set_Product_prod_a_a,Q4: set_Product_prod_a_a > $o] :
( ( P4 @ X )
=> ( ! [Y5: set_Product_prod_a_a] :
( ( P4 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ Y5 @ X ) )
=> ( ! [X3: set_Product_prod_a_a] :
( ( P4 @ X3 )
=> ( ! [Y6: set_Product_prod_a_a] :
( ( P4 @ Y6 )
=> ( ord_le746702958409616551od_a_a @ Y6 @ X3 ) )
=> ( Q4 @ X3 ) ) )
=> ( Q4 @ ( order_6315983676290676192od_a_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_267_GreatestI2__order,axiom,
! [P4: set_a > $o,X: set_a,Q4: set_a > $o] :
( ( P4 @ X )
=> ( ! [Y5: set_a] :
( ( P4 @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( ! [X3: set_a] :
( ( P4 @ X3 )
=> ( ! [Y6: set_a] :
( ( P4 @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X3 ) )
=> ( Q4 @ X3 ) ) )
=> ( Q4 @ ( order_Greatest_set_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_268_Greatest__equality,axiom,
! [P4: ( a > $o ) > $o,X: a > $o] :
( ( P4 @ X )
=> ( ! [Y5: a > $o] :
( ( P4 @ Y5 )
=> ( ord_less_eq_a_o @ Y5 @ X ) )
=> ( ( order_Greatest_a_o @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_269_Greatest__equality,axiom,
! [P4: ( $o > set_a ) > $o,X: $o > set_a] :
( ( P4 @ X )
=> ( ! [Y5: $o > set_a] :
( ( P4 @ Y5 )
=> ( ord_less_eq_o_set_a @ Y5 @ X ) )
=> ( ( order_6114237596796908690_set_a @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_270_Greatest__equality,axiom,
! [P4: set_set_a > $o,X: set_set_a] :
( ( P4 @ X )
=> ( ! [Y5: set_set_a] :
( ( P4 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ Y5 @ X ) )
=> ( ( order_3565860530148683671_set_a @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_271_Greatest__equality,axiom,
! [P4: set_Product_prod_a_a > $o,X: set_Product_prod_a_a] :
( ( P4 @ X )
=> ( ! [Y5: set_Product_prod_a_a] :
( ( P4 @ Y5 )
=> ( ord_le746702958409616551od_a_a @ Y5 @ X ) )
=> ( ( order_6315983676290676192od_a_a @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_272_Greatest__equality,axiom,
! [P4: set_a > $o,X: set_a] :
( ( P4 @ X )
=> ( ! [Y5: set_a] :
( ( P4 @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( ( order_Greatest_set_a @ P4 )
= X ) ) ) ).
% Greatest_equality
thf(fact_273_valid__mono,axiom,
! [A: a,B: a] :
( ( ( valid @ A )
& ( pre_larger_a @ plus @ A @ B ) )
=> ( valid @ B ) ) ).
% valid_mono
thf(fact_274_predicate1I,axiom,
! [P4: a > $o,Q4: a > $o] :
( ! [X3: a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_less_eq_a_o @ P4 @ Q4 ) ) ).
% predicate1I
thf(fact_275_rev__predicate1D,axiom,
! [P4: a > $o,X: a,Q4: a > $o] :
( ( P4 @ X )
=> ( ( ord_less_eq_a_o @ P4 @ Q4 )
=> ( Q4 @ X ) ) ) ).
% rev_predicate1D
thf(fact_276_predicate1D,axiom,
! [P4: a > $o,Q4: a > $o,X: a] :
( ( ord_less_eq_a_o @ P4 @ Q4 )
=> ( ( P4 @ X )
=> ( Q4 @ X ) ) ) ).
% predicate1D
thf(fact_277_logic_Osmult__distrib,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R2: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sadd @ Q @ R2 ) )
= ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R2 ) ) ) ) ).
% logic.smult_distrib
thf(fact_278_logic_Osone__neutral,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ One @ P )
= P ) ) ).
% logic.sone_neutral
thf(fact_279_logic_Osinv__inverse,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sinv @ P ) )
= One ) ) ).
% logic.sinv_inverse
thf(fact_280_logic_Oone__neutral,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ One @ A )
= A ) ) ).
% logic.one_neutral
thf(fact_281_logic_Odouble__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ P @ ( Mult @ Q @ A ) )
= ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).
% logic.double_mult
thf(fact_282_logic_Ocommutative,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Plus2 @ A @ B )
= ( Plus2 @ B @ A ) ) ) ).
% logic.commutative
thf(fact_283_logic_Ounique__inv,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,P: b,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( A
= ( Mult @ P @ B ) )
= ( B
= ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).
% logic.unique_inv
thf(fact_284_logic_Osmult__comm,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ Q )
= ( Smult @ Q @ P ) ) ) ).
% logic.smult_comm
thf(fact_285_logic_Osmult__asso,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R2: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ ( Smult @ P @ Q ) @ R2 )
= ( Smult @ P @ ( Smult @ Q @ R2 ) ) ) ) ).
% logic.smult_asso
thf(fact_286_logic_Ocan__divide,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Mult @ P @ A )
= ( Mult @ P @ B ) )
=> ( A = B ) ) ) ).
% logic.can_divide
thf(fact_287_logic_Osadd__comm,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Sadd @ P @ Q )
= ( Sadd @ Q @ P ) ) ) ).
% logic.sadd_comm
thf(fact_288_logic_Odistrib__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X ) )
= ( Plus2 @ ( Mult @ P @ X ) @ ( Mult @ Q @ X ) ) ) ) ).
% logic.distrib_mult
thf(fact_289_logic_Oplus__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus2 @ B @ C ) )
=> ( ( some_a @ ( Mult @ P @ A ) )
= ( Plus2 @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).
% logic.plus_mult
thf(fact_290_logic_Omove__sum,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,A1: a,A2: a,B: a,B1: a,B2: a,X: a,X1: a,X2: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus2 @ A1 @ A2 ) )
=> ( ( ( some_a @ B )
= ( Plus2 @ B1 @ B2 ) )
=> ( ( ( some_a @ X )
= ( Plus2 @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( Plus2 @ A1 @ B1 ) )
=> ( ( ( some_a @ X2 )
= ( Plus2 @ A2 @ B2 ) )
=> ( ( some_a @ X )
= ( Plus2 @ X1 @ X2 ) ) ) ) ) ) ) ) ).
% logic.move_sum
thf(fact_291_logic_Oasso1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a,Bc: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus2 @ A @ B )
= ( some_a @ Ab ) )
& ( ( Plus2 @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( Plus2 @ Ab @ C )
= ( Plus2 @ A @ Bc ) ) ) ) ).
% logic.asso1
thf(fact_292_logic_Ovalid__mono,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Valid @ A )
& ( pre_larger_a @ Plus2 @ A @ B ) )
=> ( Valid @ B ) ) ) ).
% logic.valid_mono
thf(fact_293_logic_Olarger__same,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ A @ B )
= ( pre_larger_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.larger_same
thf(fact_294_logic_Ocompatible__iff,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ A @ B )
= ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_iff
thf(fact_295_logic_Ocompatible__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ A @ B )
=> ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_imp
thf(fact_296_logic_Ocompatible__multiples,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,Q: b,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
=> ( pre_compatible_a @ Plus2 @ A @ B ) ) ) ).
% logic.compatible_multiples
thf(fact_297_logic_Osum__both__larger,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X5: a,A6: a,B5: a,X: a,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ X5 )
= ( Plus2 @ A6 @ B5 ) )
=> ( ( ( some_a @ X )
= ( Plus2 @ A @ B ) )
=> ( ( pre_larger_a @ Plus2 @ A6 @ A )
=> ( ( pre_larger_a @ Plus2 @ B5 @ B )
=> ( pre_larger_a @ Plus2 @ X5 @ X ) ) ) ) ) ) ).
% logic.sum_both_larger
thf(fact_298_logic_Olarger__first__sum,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Y: a,A: a,B: a,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ Y )
= ( Plus2 @ A @ B ) )
=> ( ( pre_larger_a @ Plus2 @ X @ Y )
=> ? [A5: a] :
( ( ( some_a @ X )
= ( Plus2 @ A5 @ B ) )
& ( pre_larger_a @ Plus2 @ A5 @ A ) ) ) ) ) ).
% logic.larger_first_sum
thf(fact_299_logic_Oasso2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus2 @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ Plus2 @ B @ C ) )
=> ~ ( pre_compatible_a @ Plus2 @ Ab @ C ) ) ) ).
% logic.asso2
thf(fact_300_logic_Oasso3,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,Bc: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ~ ( pre_compatible_a @ Plus2 @ A @ B )
=> ( ( ( Plus2 @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ Plus2 @ A @ Bc ) ) ) ) ).
% logic.asso3
thf(fact_301_logic_Ocompatible__smaller,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ A @ B )
=> ( ( pre_compatible_a @ Plus2 @ X @ A )
=> ( pre_compatible_a @ Plus2 @ X @ B ) ) ) ) ).
% logic.compatible_smaller
thf(fact_302_logic_Olarger__implies__compatible,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,Y: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ X @ Y )
=> ( pre_compatible_a @ Plus2 @ X @ Y ) ) ) ).
% logic.larger_implies_compatible
thf(fact_303_logic_Osem__combinable__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ Delta )
= ( ! [S3: c > d,P3: b,Q3: b,A4: a,B4: a,X4: a] :
( ( ( ( Sadd @ P3 @ Q3 )
= One )
& ( member_a @ A4 @ ( Delta @ S3 ) )
& ( member_a @ B4 @ ( Delta @ S3 ) )
& ( ( some_a @ X4 )
= ( Plus2 @ ( Mult @ P3 @ A4 ) @ ( Mult @ Q3 @ B4 ) ) ) )
=> ( member_a @ X4 @ ( Delta @ S3 ) ) ) ) ) ) ).
% logic.sem_combinable_def
thf(fact_304_logic_Osem__combinableI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [S2: c > d,P2: b,Q2: b,A3: a,B3: a,X3: a] :
( ( ( ( Sadd @ P2 @ Q2 )
= One )
& ( member_a @ A3 @ ( Delta @ S2 ) )
& ( member_a @ B3 @ ( Delta @ S2 ) )
& ( ( some_a @ X3 )
= ( Plus2 @ ( Mult @ P2 @ A3 ) @ ( Mult @ Q2 @ B3 ) ) ) )
=> ( member_a @ X3 @ ( Delta @ S2 ) ) )
=> ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ Delta ) ) ) ).
% logic.sem_combinableI
thf(fact_305_logic_Osem__combinableE,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ Delta )
=> ( ( member_a @ A @ ( Delta @ S ) )
=> ( ( member_a @ B @ ( Delta @ S ) )
=> ( ( ( some_a @ X )
= ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
=> ( ( ( Sadd @ P @ Q )
= One )
=> ( member_a @ X @ ( Delta @ S ) ) ) ) ) ) ) ) ).
% logic.sem_combinableE
thf(fact_306_logic__def,axiom,
( logic_a_b
= ( ^ [Plus: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
( ! [A4: a,B4: a] :
( ( Plus @ A4 @ B4 )
= ( Plus @ B4 @ A4 ) )
& ! [A4: a,B4: a,Ab2: a,C2: a,Bc2: a] :
( ( ( ( Plus @ A4 @ B4 )
= ( some_a @ Ab2 ) )
& ( ( Plus @ B4 @ C2 )
= ( some_a @ Bc2 ) ) )
=> ( ( Plus @ Ab2 @ C2 )
= ( Plus @ A4 @ Bc2 ) ) )
& ! [A4: a,B4: a,Ab2: a,C2: a] :
( ( ( ( Plus @ A4 @ B4 )
= ( some_a @ Ab2 ) )
& ~ ( pre_compatible_a @ Plus @ B4 @ C2 ) )
=> ~ ( pre_compatible_a @ Plus @ Ab2 @ C2 ) )
& ! [P3: b] :
( ( Smult2 @ P3 @ ( Sinv2 @ P3 ) )
= One2 )
& ! [P3: b] :
( ( Smult2 @ One2 @ P3 )
= P3 )
& ! [P3: b,Q3: b] :
( ( Sadd2 @ P3 @ Q3 )
= ( Sadd2 @ Q3 @ P3 ) )
& ! [P3: b,Q3: b] :
( ( Smult2 @ P3 @ Q3 )
= ( Smult2 @ Q3 @ P3 ) )
& ! [P3: b,Q3: b,R4: b] :
( ( Smult2 @ P3 @ ( Sadd2 @ Q3 @ R4 ) )
= ( Sadd2 @ ( Smult2 @ P3 @ Q3 ) @ ( Smult2 @ P3 @ R4 ) ) )
& ! [P3: b,Q3: b,R4: b] :
( ( Smult2 @ ( Smult2 @ P3 @ Q3 ) @ R4 )
= ( Smult2 @ P3 @ ( Smult2 @ Q3 @ R4 ) ) )
& ! [P3: b,Q3: b,A4: a] :
( ( Mult2 @ P3 @ ( Mult2 @ Q3 @ A4 ) )
= ( Mult2 @ ( Smult2 @ P3 @ Q3 ) @ A4 ) )
& ! [A4: a,B4: a,C2: a,P3: b] :
( ( ( some_a @ A4 )
= ( Plus @ B4 @ C2 ) )
=> ( ( some_a @ ( Mult2 @ P3 @ A4 ) )
= ( Plus @ ( Mult2 @ P3 @ B4 ) @ ( Mult2 @ P3 @ C2 ) ) ) )
& ! [P3: b,Q3: b,X4: a] :
( ( some_a @ ( Mult2 @ ( Sadd2 @ P3 @ Q3 ) @ X4 ) )
= ( Plus @ ( Mult2 @ P3 @ X4 ) @ ( Mult2 @ Q3 @ X4 ) ) )
& ! [A4: a] :
( ( Mult2 @ One2 @ A4 )
= A4 )
& ! [A4: a,B4: a] :
( ( ( Valid2 @ A4 )
& ( pre_larger_a @ Plus @ A4 @ B4 ) )
=> ( Valid2 @ B4 ) ) ) ) ) ).
% logic_def
thf(fact_307_sem__combinable__appliesE,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta ) )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S @ Delta @ A7 )
=> ( ( ( some_a @ X )
= ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
=> ( ( ( sadd @ P @ Q )
= one )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ A7 ) ) ) ) ) ) ).
% sem_combinable_appliesE
thf(fact_308_sem__combinable__equiv,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( sem_co7516848414490435095_b_c_d @ plus @ mult @ sadd @ one @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta ) )
= ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 ) ) ).
% sem_combinable_equiv
thf(fact_309_combinable__instantiate__one,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S @ Delta @ A7 )
=> ( ( ( some_a @ X )
= ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
=> ( ( ( sadd @ P @ Q )
= one )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ A7 ) ) ) ) ) ) ).
% combinable_instantiate_one
thf(fact_310_not__in__fv__def,axiom,
! [A7: assertion_a_b_d_c,S4: set_c] :
( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A7 @ S4 )
= ( ! [Sigma: a,S3: c > d,Delta2: ( c > d ) > set_a,S5: c > d] :
( ( equal_outside_c_d @ S3 @ S5 @ S4 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S3 @ Delta2 @ A7 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S5 @ Delta2 @ A7 ) ) ) ) ) ).
% not_in_fv_def
thf(fact_311_combinable__pure,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A7 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 ) ) ).
% combinable_pure
thf(fact_312_applies__eq__equiv,axiom,
! [X: a,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,S: c > d] :
( ( member_a @ X @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ S ) )
= ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ A7 ) ) ).
% applies_eq_equiv
thf(fact_313_pure__def,axiom,
! [A7: assertion_a_b_d_c] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A7 )
= ( ! [Sigma: a,Sigma2: a,S3: c > d,Delta2: ( c > d ) > set_a,Delta3: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S3 @ Delta2 @ A7 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta3 @ A7 ) ) ) ) ).
% pure_def
thf(fact_314_smaller__interp__applies__cons,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,Delta4: ( c > d ) > set_a,A: a,S: c > d] :
( ( smaller_interp_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta ) @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta4 ) )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S @ Delta4 @ A7 ) ) ) ).
% smaller_interp_applies_cons
thf(fact_315_applies__eq_Oelims,axiom,
! [X: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( ( applies_eq_a_b_d_c @ plus @ mult @ valid @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ plus @ mult @ valid @ Uu @ Xb @ Xa @ X ) ) ) ) ).
% applies_eq.elims
thf(fact_316_applies__eq_Osimps,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,S: c > d] :
( ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ S )
= ( collect_a
@ ^ [Uu: a] :
? [A4: a] :
( ( Uu = A4 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A4 @ S @ Delta @ A7 ) ) ) ) ).
% applies_eq.simps
thf(fact_317_combinableI__old,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ! [A3: a,B3: a,P2: b,Q2: b,X3: a,Sigma3: a,S2: c > d] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A3 @ S2 @ Delta @ A7 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B3 @ S2 @ Delta @ A7 )
& ( ( some_a @ Sigma3 )
= ( plus @ ( mult @ P2 @ A3 ) @ ( mult @ Q2 @ B3 ) ) )
& ( Sigma3
= ( mult @ ( sadd @ P2 @ Q2 ) @ X3 ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X3 @ S2 @ Delta @ A7 ) )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 ) ) ).
% combinableI_old
thf(fact_318_indep__interp__def,axiom,
! [A7: assertion_a_b_d_c] :
( ( indep_interp_a_b_d_c @ plus @ mult @ valid @ A7 )
= ( ! [X4: a,S3: c > d,Delta2: ( c > d ) > set_a,Delta3: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ X4 @ S3 @ Delta2 @ A7 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ X4 @ S3 @ Delta3 @ A7 ) ) ) ) ).
% indep_interp_def
thf(fact_319_logic_Oapplies__eq__equiv,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,S: c > d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( member_a @ X @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ S ) )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ A7 ) ) ) ).
% logic.applies_eq_equiv
thf(fact_320_logic_Oapplies__eq_Ocong,axiom,
applies_eq_a_b_d_c = applies_eq_a_b_d_c ).
% logic.applies_eq.cong
thf(fact_321_logic_Osat_Ocong,axiom,
sat_a_b_c_d = sat_a_b_c_d ).
% logic.sat.cong
thf(fact_322_logic_Opure__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 )
= ( ! [Sigma: a,Sigma2: a,S3: c > d,Delta2: ( c > d ) > set_a,Delta3: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta2 @ A7 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta3 @ A7 ) ) ) ) ) ).
% logic.pure_def
thf(fact_323_logic_Osmaller__interp__applies__cons,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,Delta4: ( c > d ) > set_a,A: a,S: c > d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( smaller_interp_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta ) @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta4 ) )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S @ Delta4 @ A7 ) ) ) ) ).
% logic.smaller_interp_applies_cons
thf(fact_324_logic_Oapplies__eq_Osimps,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,S: c > d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ S )
= ( collect_a
@ ^ [Uu: a] :
? [A4: a] :
( ( Uu = A4 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A4 @ S @ Delta @ A7 ) ) ) ) ) ).
% logic.applies_eq.simps
thf(fact_325_logic_Oapplies__eq_Oelims,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Uu @ Xb @ Xa @ X ) ) ) ) ) ).
% logic.applies_eq.elims
thf(fact_326_logic_Osem__combinable__equiv,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta ) )
= ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 ) ) ) ).
% logic.sem_combinable_equiv
thf(fact_327_logic_Osem__combinable__appliesE,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sem_co7516848414490435095_b_c_d @ Plus2 @ Mult @ Sadd @ One @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta ) )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S @ Delta @ A7 )
=> ( ( ( some_a @ X )
= ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
=> ( ( ( Sadd @ P @ Q )
= One )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ A7 ) ) ) ) ) ) ) ).
% logic.sem_combinable_appliesE
thf(fact_328_logic_Onot__in__fv__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,S4: set_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ S4 )
= ( ! [Sigma: a,S3: c > d,Delta2: ( c > d ) > set_a,S5: c > d] :
( ( equal_outside_c_d @ S3 @ S5 @ S4 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta2 @ A7 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S5 @ Delta2 @ A7 ) ) ) ) ) ) ).
% logic.not_in_fv_def
thf(fact_329_sem__intui__intuitionistic,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( sem_intui_a_c_d @ plus @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta ) )
= ( ! [S3: c > d] : ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S3 @ Delta @ A7 ) ) ) ).
% sem_intui_intuitionistic
thf(fact_330_instantiate__intui__applies,axiom,
! [S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,Sigma4: a,Sigma5: a] :
( ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S @ Delta @ A7 )
=> ( ( pre_larger_a @ plus @ Sigma4 @ Sigma5 )
=> ( ( member_a @ Sigma5 @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ S ) )
=> ( member_a @ Sigma4 @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ S ) ) ) ) ) ).
% instantiate_intui_applies
thf(fact_331_intuitionisticI,axiom,
! [S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ! [A3: a,B3: a] :
( ( ( pre_larger_a @ plus @ A3 @ B3 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B3 @ S @ Delta @ A7 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A3 @ S @ Delta @ A7 ) )
=> ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S @ Delta @ A7 ) ) ).
% intuitionisticI
thf(fact_332_intuitionistic__def,axiom,
! [S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S @ Delta @ A7 )
= ( ! [A4: a,B4: a] :
( ( ( pre_larger_a @ plus @ A4 @ B4 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B4 @ S @ Delta @ A7 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A4 @ S @ Delta @ A7 ) ) ) ) ).
% intuitionistic_def
thf(fact_333_logic_Oindep__interp__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( indep_interp_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 )
= ( ! [X4: a,S3: c > d,Delta2: ( c > d ) > set_a,Delta3: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X4 @ S3 @ Delta2 @ A7 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X4 @ S3 @ Delta3 @ A7 ) ) ) ) ) ).
% logic.indep_interp_def
thf(fact_334_logic_OintuitionisticI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a,B3: a] :
( ( ( pre_larger_a @ Plus2 @ A3 @ B3 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B3 @ S @ Delta @ A7 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A3 @ S @ Delta @ A7 ) )
=> ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S @ Delta @ A7 ) ) ) ).
% logic.intuitionisticI
thf(fact_335_logic_Ointuitionistic__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S @ Delta @ A7 )
= ( ! [A4: a,B4: a] :
( ( ( pre_larger_a @ Plus2 @ A4 @ B4 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B4 @ S @ Delta @ A7 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A4 @ S @ Delta @ A7 ) ) ) ) ) ).
% logic.intuitionistic_def
thf(fact_336_logic_Oinstantiate__intui__applies,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,Sigma4: a,Sigma5: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S @ Delta @ A7 )
=> ( ( pre_larger_a @ Plus2 @ Sigma4 @ Sigma5 )
=> ( ( member_a @ Sigma5 @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ S ) )
=> ( member_a @ Sigma4 @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ S ) ) ) ) ) ) ).
% logic.instantiate_intui_applies
thf(fact_337_logic_Osem__intui__intuitionistic,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sem_intui_a_c_d @ Plus2 @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta ) )
= ( ! [S3: c > d] : ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S3 @ Delta @ A7 ) ) ) ) ).
% logic.sem_intui_intuitionistic
thf(fact_338_combinable__instantiate,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S @ Delta @ A7 )
=> ( ( ( some_a @ X )
= ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ ( mult_b_a_d_c @ ( sadd @ P @ Q ) @ A7 ) ) ) ) ) ) ).
% combinable_instantiate
thf(fact_339_combinable__imp,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B6: assertion_a_b_d_c] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A7 )
=> ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ).
% combinable_imp
thf(fact_340_sat_Osimps_I10_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ pred_a_b_d_c )
= ( member_a @ Sigma5 @ ( Delta @ S ) ) ) ).
% sat.simps(10)
thf(fact_341_combinable__wand,axiom,
! [Delta: ( c > d ) > set_a,B6: assertion_a_b_d_c,A7: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ).
% combinable_wand
thf(fact_342_combinable__and,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( and_a_b_d_c @ A7 @ B6 ) ) ) ) ).
% combinable_and
thf(fact_343_combinable__forall,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,X: c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( forall_c_a_b_d @ X @ A7 ) ) ) ).
% combinable_forall
thf(fact_344_combinable__mult,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,Pi: b] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( mult_b_a_d_c @ Pi @ A7 ) ) ) ).
% combinable_mult
thf(fact_345_combinable__star,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( star_a_b_d_c @ A7 @ B6 ) ) ) ) ).
% combinable_star
thf(fact_346_sat_Osimps_I1_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,P: b,A7: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( mult_b_a_d_c @ P @ A7 ) )
= ( ? [A4: a] :
( ( Sigma5
= ( mult @ P @ A4 ) )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A4 @ S @ Delta @ A7 ) ) ) ) ).
% sat.simps(1)
thf(fact_347_sat__mult,axiom,
! [Sigma5: a,P: b,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ! [A3: a] :
( ( Sigma5
= ( mult @ P @ A3 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A3 @ S @ Delta @ A7 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( mult_b_a_d_c @ P @ A7 ) ) ) ).
% sat_mult
thf(fact_348_sat_Osimps_I7_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( and_a_b_d_c @ A7 @ B6 ) )
= ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A7 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B6 ) ) ) ).
% sat.simps(7)
thf(fact_349_sat_Osimps_I5_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( imp_a_b_d_c @ A7 @ B6 ) )
= ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B6 ) ) ) ).
% sat.simps(5)
thf(fact_350_sat__imp,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B6 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ).
% sat_imp
thf(fact_351_sat_Osimps_I2_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( star_a_b_d_c @ A7 @ B6 ) )
= ( ? [A4: a,B4: a] :
( ( ( some_a @ Sigma5 )
= ( plus @ A4 @ B4 ) )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A4 @ S @ Delta @ A7 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B4 @ S @ Delta @ B6 ) ) ) ) ).
% sat.simps(2)
thf(fact_352_sat_Osimps_I3_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( wand_a_b_d_c @ A7 @ B6 ) )
= ( ! [A4: a,Sigma2: a] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A4 @ S @ Delta @ A7 )
& ( ( some_a @ Sigma2 )
= ( plus @ Sigma5 @ A4 ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S @ Delta @ B6 ) ) ) ) ).
% sat.simps(3)
thf(fact_353_sat__wand,axiom,
! [S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,Sigma5: a,B6: assertion_a_b_d_c] :
( ! [A3: a,Sigma6: a] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A3 @ S @ Delta @ A7 )
& ( ( some_a @ Sigma6 )
= ( plus @ Sigma5 @ A3 ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma6 @ S @ Delta @ B6 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ).
% sat_wand
thf(fact_354_logic_Osat__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,P: b,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a] :
( ( Sigma5
= ( Mult @ P @ A3 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A3 @ S @ Delta @ A7 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( mult_b_a_d_c @ P @ A7 ) ) ) ) ).
% logic.sat_mult
thf(fact_355_logic_Osat_Osimps_I1_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,P: b,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( mult_b_a_d_c @ P @ A7 ) )
= ( ? [A4: a] :
( ( Sigma5
= ( Mult @ P @ A4 ) )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A4 @ S @ Delta @ A7 ) ) ) ) ) ).
% logic.sat.simps(1)
thf(fact_356_logic_Osat_Osimps_I7_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( and_a_b_d_c @ A7 @ B6 ) )
= ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A7 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B6 ) ) ) ) ).
% logic.sat.simps(7)
thf(fact_357_logic_Osat__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B6 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ).
% logic.sat_imp
thf(fact_358_logic_Osat_Osimps_I5_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( imp_a_b_d_c @ A7 @ B6 ) )
= ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B6 ) ) ) ) ).
% logic.sat.simps(5)
thf(fact_359_logic_Osat_Osimps_I10_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ pred_a_b_d_c )
= ( member_a @ Sigma5 @ ( Delta @ S ) ) ) ) ).
% logic.sat.simps(10)
thf(fact_360_logic_Osat_Osimps_I2_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( star_a_b_d_c @ A7 @ B6 ) )
= ( ? [A4: a,B4: a] :
( ( ( some_a @ Sigma5 )
= ( Plus2 @ A4 @ B4 ) )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A4 @ S @ Delta @ A7 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B4 @ S @ Delta @ B6 ) ) ) ) ) ).
% logic.sat.simps(2)
thf(fact_361_logic_Osat_Osimps_I3_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( wand_a_b_d_c @ A7 @ B6 ) )
= ( ! [A4: a,Sigma2: a] :
( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A4 @ S @ Delta @ A7 )
& ( ( some_a @ Sigma2 )
= ( Plus2 @ Sigma5 @ A4 ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S @ Delta @ B6 ) ) ) ) ) ).
% logic.sat.simps(3)
thf(fact_362_logic_Osat__wand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,Sigma5: a,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a,Sigma6: a] :
( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A3 @ S @ Delta @ A7 )
& ( ( some_a @ Sigma6 )
= ( Plus2 @ Sigma5 @ A3 ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma6 @ S @ Delta @ B6 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ) ).
% logic.sat_wand
thf(fact_363_sat_Osimps_I9_J,axiom,
! [Sigma5: a,S: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A7: assert1556940916145061938on_a_a] :
( ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( forall5484998627543102345tion_a @ X @ A7 ) )
= ( ! [V: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma5 @ ( fun_upd_a_option_a @ S @ X @ V ) @ Delta @ A7 ) ) ) ).
% sat.simps(9)
thf(fact_364_sat_Osimps_I9_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,X: c,A7: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( forall_c_a_b_d @ X @ A7 ) )
= ( ! [V: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ ( fun_upd_c_d @ S @ X @ V ) @ Delta @ A7 ) ) ) ).
% sat.simps(9)
thf(fact_365_sat__forall,axiom,
! [Sigma5: a,S: a > option_a,X: a,Delta: ( a > option_a ) > set_a,A7: assert1556940916145061938on_a_a] :
( ! [V2: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma5 @ ( fun_upd_a_option_a @ S @ X @ V2 ) @ Delta @ A7 )
=> ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( forall5484998627543102345tion_a @ X @ A7 ) ) ) ).
% sat_forall
thf(fact_366_sat__forall,axiom,
! [Sigma5: a,S: c > d,X: c,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ! [V2: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ ( fun_upd_c_d @ S @ X @ V2 ) @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( forall_c_a_b_d @ X @ A7 ) ) ) ).
% sat_forall
thf(fact_367_equivalentI,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ! [Sigma3: a,S2: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ B6 ) )
=> ( ! [Sigma3: a,S2: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ B6 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ A7 ) )
=> ( equivalent_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ B6 ) ) ) ).
% equivalentI
thf(fact_368_unambiguous__def,axiom,
! [Delta: ( a > option_a ) > set_a,A7: assert1556940916145061938on_a_a,X: a] :
( ( unambi704529886615442436tion_a @ plus @ mult @ valid @ Delta @ A7 @ X )
= ( ! [Sigma_1: a,Sigma_2: a,V1: option_a,V22: option_a,S3: a > option_a] :
( ( ( pre_compatible_a @ plus @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_1 @ ( fun_upd_a_option_a @ S3 @ X @ V1 ) @ Delta @ A7 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_2 @ ( fun_upd_a_option_a @ S3 @ X @ V22 ) @ Delta @ A7 ) )
=> ( V1 = V22 ) ) ) ) ).
% unambiguous_def
thf(fact_369_unambiguous__def,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,X: c] :
( ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A7 @ X )
= ( ! [Sigma_1: a,Sigma_2: a,V1: d,V22: d,S3: c > d] :
( ( ( pre_compatible_a @ plus @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_1 @ ( fun_upd_c_d @ S3 @ X @ V1 ) @ Delta @ A7 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_2 @ ( fun_upd_c_d @ S3 @ X @ V22 ) @ Delta @ A7 ) )
=> ( V1 = V22 ) ) ) ) ).
% unambiguous_def
thf(fact_370_unambiguousI,axiom,
! [X: a,Delta: ( a > option_a ) > set_a,A7: assert1556940916145061938on_a_a] :
( ! [Sigma_12: a,Sigma_22: a,V12: option_a,V23: option_a,S2: a > option_a] :
( ( ( pre_compatible_a @ plus @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_12 @ ( fun_upd_a_option_a @ S2 @ X @ V12 ) @ Delta @ A7 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_22 @ ( fun_upd_a_option_a @ S2 @ X @ V23 ) @ Delta @ A7 ) )
=> ( V12 = V23 ) )
=> ( unambi704529886615442436tion_a @ plus @ mult @ valid @ Delta @ A7 @ X ) ) ).
% unambiguousI
thf(fact_371_unambiguousI,axiom,
! [X: c,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ! [Sigma_12: a,Sigma_22: a,V12: d,V23: d,S2: c > d] :
( ( ( pre_compatible_a @ plus @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_12 @ ( fun_upd_c_d @ S2 @ X @ V12 ) @ Delta @ A7 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_22 @ ( fun_upd_c_d @ S2 @ X @ V23 ) @ Delta @ A7 ) )
=> ( V12 = V23 ) )
=> ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A7 @ X ) ) ).
% unambiguousI
thf(fact_372_logic_OequivalentI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma3: a,S2: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ B6 ) )
=> ( ! [Sigma3: a,S2: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ B6 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ A7 ) )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ B6 ) ) ) ) ).
% logic.equivalentI
thf(fact_373_logic_Osat_Osimps_I9_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A7: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( forall5484998627543102345tion_a @ X @ A7 ) )
= ( ! [V: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma5 @ ( fun_upd_a_option_a @ S @ X @ V ) @ Delta @ A7 ) ) ) ) ).
% logic.sat.simps(9)
thf(fact_374_logic_Osat_Osimps_I9_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,X: c,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( forall_c_a_b_d @ X @ A7 ) )
= ( ! [V: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ ( fun_upd_c_d @ S @ X @ V ) @ Delta @ A7 ) ) ) ) ).
% logic.sat.simps(9)
thf(fact_375_logic_Osat__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: a > option_a,X: a,Delta: ( a > option_a ) > set_a,A7: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [V2: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma5 @ ( fun_upd_a_option_a @ S @ X @ V2 ) @ Delta @ A7 )
=> ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( forall5484998627543102345tion_a @ X @ A7 ) ) ) ) ).
% logic.sat_forall
thf(fact_376_logic_Osat__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,X: c,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [V2: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ ( fun_upd_c_d @ S @ X @ V2 ) @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( forall_c_a_b_d @ X @ A7 ) ) ) ) ).
% logic.sat_forall
thf(fact_377_combinable__exists,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,X: c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A7 @ X )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( exists_c_a_b_d @ X @ A7 ) ) ) ) ).
% combinable_exists
thf(fact_378_logic_Ounambiguous__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( a > option_a ) > set_a,A7: assert1556940916145061938on_a_a,X: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( unambi704529886615442436tion_a @ Plus2 @ Mult @ Valid @ Delta @ A7 @ X )
= ( ! [Sigma_1: a,Sigma_2: a,V1: option_a,V22: option_a,S3: a > option_a] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_1 @ ( fun_upd_a_option_a @ S3 @ X @ V1 ) @ Delta @ A7 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_2 @ ( fun_upd_a_option_a @ S3 @ X @ V22 ) @ Delta @ A7 ) )
=> ( V1 = V22 ) ) ) ) ) ).
% logic.unambiguous_def
thf(fact_379_logic_Ounambiguous__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,X: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A7 @ X )
= ( ! [Sigma_1: a,Sigma_2: a,V1: d,V22: d,S3: c > d] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_1 @ ( fun_upd_c_d @ S3 @ X @ V1 ) @ Delta @ A7 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_2 @ ( fun_upd_c_d @ S3 @ X @ V22 ) @ Delta @ A7 ) )
=> ( V1 = V22 ) ) ) ) ) ).
% logic.unambiguous_def
thf(fact_380_logic_OunambiguousI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,Delta: ( a > option_a ) > set_a,A7: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma_12: a,Sigma_22: a,V12: option_a,V23: option_a,S2: a > option_a] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_12 @ ( fun_upd_a_option_a @ S2 @ X @ V12 ) @ Delta @ A7 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_22 @ ( fun_upd_a_option_a @ S2 @ X @ V23 ) @ Delta @ A7 ) )
=> ( V12 = V23 ) )
=> ( unambi704529886615442436tion_a @ Plus2 @ Mult @ Valid @ Delta @ A7 @ X ) ) ) ).
% logic.unambiguousI
thf(fact_381_logic_OunambiguousI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: c,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma_12: a,Sigma_22: a,V12: d,V23: d,S2: c > d] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_12 @ ( fun_upd_c_d @ S2 @ X @ V12 ) @ Delta @ A7 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_22 @ ( fun_upd_c_d @ S2 @ X @ V23 ) @ Delta @ A7 ) )
=> ( V12 = V23 ) )
=> ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A7 @ X ) ) ) ).
% logic.unambiguousI
thf(fact_382_combinable__def,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
= ( ! [P3: b,Q3: b] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P3 @ A7 ) @ ( mult_b_a_d_c @ Q3 @ A7 ) ) @ Delta @ ( mult_b_a_d_c @ ( sadd @ P3 @ Q3 ) @ A7 ) ) ) ) ).
% combinable_def
thf(fact_383_logic_Ocombinable__instantiate,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S @ Delta @ A7 )
=> ( ( ( some_a @ X )
= ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ ( mult_b_a_d_c @ ( Sadd @ P @ Q ) @ A7 ) ) ) ) ) ) ) ).
% logic.combinable_instantiate
thf(fact_384_entailsI,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ! [Sigma3: a,S2: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ B6 ) )
=> ( entails_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ B6 ) ) ).
% entailsI
thf(fact_385_entails__def,axiom,
! [A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B6: assertion_a_b_d_c] :
( ( entails_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ B6 )
= ( ! [Sigma: a,S3: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S3 @ Delta @ A7 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S3 @ Delta @ B6 ) ) ) ) ).
% entails_def
thf(fact_386_sat_Osimps_I6_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( or_a_b_d_c @ A7 @ B6 ) )
= ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A7 )
| ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B6 ) ) ) ).
% sat.simps(6)
thf(fact_387_sat_Osimps_I8_J,axiom,
! [Sigma5: a,S: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A7: assert1556940916145061938on_a_a] :
( ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( exists7165000112504185261tion_a @ X @ A7 ) )
= ( ? [V: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma5 @ ( fun_upd_a_option_a @ S @ X @ V ) @ Delta @ A7 ) ) ) ).
% sat.simps(8)
thf(fact_388_sat_Osimps_I8_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,X: c,A7: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( exists_c_a_b_d @ X @ A7 ) )
= ( ? [V: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ ( fun_upd_c_d @ S @ X @ V ) @ Delta @ A7 ) ) ) ).
% sat.simps(8)
thf(fact_389_logic_Osat_Osimps_I6_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( or_a_b_d_c @ A7 @ B6 ) )
= ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A7 )
| ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B6 ) ) ) ) ).
% logic.sat.simps(6)
thf(fact_390_logic_Oentails__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ B6 )
= ( ! [Sigma: a,S3: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta @ B6 ) ) ) ) ) ).
% logic.entails_def
thf(fact_391_logic_OentailsI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma3: a,S2: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ A7 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ B6 ) )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ B6 ) ) ) ).
% logic.entailsI
thf(fact_392_logic_Ocombinable__exists,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,X: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A7 @ X )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( exists_c_a_b_d @ X @ A7 ) ) ) ) ) ).
% logic.combinable_exists
thf(fact_393_logic_Ocombinable__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
= ( ! [P3: b,Q3: b] : ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P3 @ A7 ) @ ( mult_b_a_d_c @ Q3 @ A7 ) ) @ Delta @ ( mult_b_a_d_c @ ( Sadd @ P3 @ Q3 ) @ A7 ) ) ) ) ) ).
% logic.combinable_def
thf(fact_394_logic_Osat_Osimps_I8_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: a > option_a,Delta: ( a > option_a ) > set_a,X: a,A7: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( exists7165000112504185261tion_a @ X @ A7 ) )
= ( ? [V: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma5 @ ( fun_upd_a_option_a @ S @ X @ V ) @ Delta @ A7 ) ) ) ) ).
% logic.sat.simps(8)
thf(fact_395_logic_Osat_Osimps_I8_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,X: c,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( exists_c_a_b_d @ X @ A7 ) )
= ( ? [V: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ ( fun_upd_c_d @ S @ X @ V ) @ Delta @ A7 ) ) ) ) ).
% logic.sat.simps(8)
thf(fact_396_logic_Ocombinable_Ocong,axiom,
combinable_a_b_c_d = combinable_a_b_c_d ).
% logic.combinable.cong
thf(fact_397_logic_Ocombinable__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,Pi: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( mult_b_a_d_c @ Pi @ A7 ) ) ) ) ).
% logic.combinable_mult
thf(fact_398_logic_Ocombinable__star,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( star_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% logic.combinable_star
thf(fact_399_logic_Ocombinable__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,X: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( forall_c_a_b_d @ X @ A7 ) ) ) ) ).
% logic.combinable_forall
thf(fact_400_logic_Ocombinable__and,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( and_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% logic.combinable_and
thf(fact_401_logic_Ocombinable__wand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,B6: assertion_a_b_d_c,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ) ).
% logic.combinable_wand
thf(fact_402_logic_Ocombinable__pure,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 ) ) ) ).
% logic.combinable_pure
thf(fact_403_logic_OcombinableI__old,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a,B3: a,P2: b,Q2: b,X3: a,Sigma3: a,S2: c > d] :
( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A3 @ S2 @ Delta @ A7 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B3 @ S2 @ Delta @ A7 )
& ( ( some_a @ Sigma3 )
= ( Plus2 @ ( Mult @ P2 @ A3 ) @ ( Mult @ Q2 @ B3 ) ) )
& ( Sigma3
= ( Mult @ ( Sadd @ P2 @ Q2 ) @ X3 ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X3 @ S2 @ Delta @ A7 ) )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 ) ) ) ).
% logic.combinableI_old
thf(fact_404_logic_Ocombinable__instantiate__one,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c,A: a,S: c > d,B: a,X: a,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S @ Delta @ A7 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S @ Delta @ A7 )
=> ( ( ( some_a @ X )
= ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
=> ( ( ( Sadd @ P @ Q )
= One )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ A7 ) ) ) ) ) ) ) ).
% logic.combinable_instantiate_one
thf(fact_405_logic_Ocombinable__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B6 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% logic.combinable_imp
thf(fact_406_combinable__wildcard,axiom,
! [Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A7 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( wildcard_a_b_d_c @ A7 ) ) ) ).
% combinable_wildcard
thf(fact_407_sat_Osimps_I12_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( wildcard_a_b_d_c @ A7 ) )
= ( ? [A4: a,P3: b] :
( ( Sigma5
= ( mult @ P3 @ A4 ) )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A4 @ S @ Delta @ A7 ) ) ) ) ).
% sat.simps(12)
thf(fact_408_logic_Osat_Osimps_I12_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( wildcard_a_b_d_c @ A7 ) )
= ( ? [A4: a,P3: b] :
( ( Sigma5
= ( Mult @ P3 @ A4 ) )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A4 @ S @ Delta @ A7 ) ) ) ) ) ).
% logic.sat.simps(12)
thf(fact_409_logic_Ocombinable__wildcard,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A7 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( wildcard_a_b_d_c @ A7 ) ) ) ) ).
% logic.combinable_wildcard
thf(fact_410_logic_Ocan__factorize,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Q: b,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ? [R: b] :
( Q
= ( Smult @ R @ P ) ) ) ).
% logic.can_factorize
thf(fact_411_sat_Osimps_I11_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( bounded_a_b_d_c @ A7 ) )
= ( ( valid @ Sigma5 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A7 ) ) ) ).
% sat.simps(11)
thf(fact_412_indep__implies__non__increasing,axiom,
! [A7: assertion_a_b_d_c] :
( ( indep_interp_a_b_d_c @ plus @ mult @ valid @ A7 )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) ) ) ).
% indep_implies_non_increasing
thf(fact_413_mono__interp,axiom,
monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ pred_a_b_d_c ) ).
% mono_interp
thf(fact_414_non__increasing__or,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( or_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% non_increasing_or
thf(fact_415_mono__or,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( or_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% mono_or
thf(fact_416_local_Omono__mult,axiom,
! [A7: assertion_a_b_d_c,Pi: b] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ Pi @ A7 ) ) ) ) ).
% local.mono_mult
thf(fact_417_non__increasing__mult,axiom,
! [A7: assertion_a_b_d_c,Pi: b] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ Pi @ A7 ) ) ) ) ).
% non_increasing_mult
thf(fact_418_mono__instantiate,axiom,
! [A7: assertion_a_b_d_c,X: a,Delta: ( c > d ) > set_a,S: c > d,Delta4: ( c > d ) > set_a] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( member_a @ X @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ S ) )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta4 )
=> ( member_a @ X @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta4 @ S ) ) ) ) ) ).
% mono_instantiate
thf(fact_419_non__increasing__instantiate,axiom,
! [A7: assertion_a_b_d_c,X: a,Delta4: ( c > d ) > set_a,S: c > d,Delta: ( c > d ) > set_a] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( member_a @ X @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta4 @ S ) )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta4 )
=> ( member_a @ X @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 @ Delta @ S ) ) ) ) ) ).
% non_increasing_instantiate
thf(fact_420_mono__wild,axiom,
! [A7: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ A7 ) ) ) ) ).
% mono_wild
thf(fact_421_non__increasing__wild,axiom,
! [A7: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ A7 ) ) ) ) ).
% non_increasing_wild
thf(fact_422_mono__star,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% mono_star
thf(fact_423_non__inc__star,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% non_inc_star
thf(fact_424_mono__forall,axiom,
! [A7: assertion_a_b_d_c,V3: c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( forall_c_a_b_d @ V3 @ A7 ) ) ) ) ).
% mono_forall
thf(fact_425_non__increasing__forall,axiom,
! [A7: assertion_a_b_d_c,V3: c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( forall_c_a_b_d @ V3 @ A7 ) ) ) ) ).
% non_increasing_forall
thf(fact_426_mono__exists,axiom,
! [A7: assertion_a_b_d_c,V3: c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( exists_c_a_b_d @ V3 @ A7 ) ) ) ) ).
% mono_exists
thf(fact_427_non__increasing__exists,axiom,
! [A7: assertion_a_b_d_c,V3: c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( exists_c_a_b_d @ V3 @ A7 ) ) ) ) ).
% non_increasing_exists
thf(fact_428_mono__and,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( and_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% mono_and
thf(fact_429_non__increasing__and,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( and_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% non_increasing_and
thf(fact_430_mono__bounded,axiom,
! [A7: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( bounded_a_b_d_c @ A7 ) ) ) ) ).
% mono_bounded
thf(fact_431_non__increasing__bounded,axiom,
! [A7: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( bounded_a_b_d_c @ A7 ) ) ) ) ).
% non_increasing_bounded
thf(fact_432_non__increasing__wand,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% non_increasing_wand
thf(fact_433_non__increasing__imp,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% non_increasing_imp
thf(fact_434_mono__wand,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% mono_wand
thf(fact_435_mono__imp,axiom,
! [A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ) ).
% mono_imp
thf(fact_436_logic_Omono__bounded,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( bounded_a_b_d_c @ A7 ) ) ) ) ) ).
% logic.mono_bounded
thf(fact_437_logic_Onon__increasing__bounded,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( bounded_a_b_d_c @ A7 ) ) ) ) ) ).
% logic.non_increasing_bounded
thf(fact_438_logic_Omono__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.mono_imp
thf(fact_439_logic_Omono__wand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.mono_wand
thf(fact_440_logic_Onon__increasing__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( imp_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.non_increasing_imp
thf(fact_441_logic_Onon__increasing__wand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wand_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.non_increasing_wand
thf(fact_442_logic_Osat_Osimps_I11_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( bounded_a_b_d_c @ A7 ) )
= ( ( Valid @ Sigma5 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A7 ) ) ) ) ).
% logic.sat.simps(11)
thf(fact_443_logic_Omono__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Pi: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ Pi @ A7 ) ) ) ) ) ).
% logic.mono_mult
thf(fact_444_logic_Onon__increasing__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,Pi: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ Pi @ A7 ) ) ) ) ) ).
% logic.non_increasing_mult
thf(fact_445_logic_Omono__wild,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ A7 ) ) ) ) ) ).
% logic.mono_wild
thf(fact_446_logic_Onon__increasing__wild,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ A7 ) ) ) ) ) ).
% logic.non_increasing_wild
thf(fact_447_logic_Omono__star,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.mono_star
thf(fact_448_logic_Onon__inc__star,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.non_inc_star
thf(fact_449_logic_Omono__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,V3: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( forall_c_a_b_d @ V3 @ A7 ) ) ) ) ) ).
% logic.mono_forall
thf(fact_450_logic_Onon__increasing__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,V3: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( forall_c_a_b_d @ V3 @ A7 ) ) ) ) ) ).
% logic.non_increasing_forall
thf(fact_451_logic_Omono__and,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( and_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.mono_and
thf(fact_452_logic_Onon__increasing__and,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( and_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.non_increasing_and
thf(fact_453_logic_Omono__exists,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,V3: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( exists_c_a_b_d @ V3 @ A7 ) ) ) ) ) ).
% logic.mono_exists
thf(fact_454_logic_Onon__increasing__exists,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,V3: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( exists_c_a_b_d @ V3 @ A7 ) ) ) ) ) ).
% logic.non_increasing_exists
thf(fact_455_logic_Omono__or,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( or_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.mono_or
thf(fact_456_logic_Onon__increasing__or,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,B6: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ B6 ) )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( or_a_b_d_c @ A7 @ B6 ) ) ) ) ) ) ).
% logic.non_increasing_or
thf(fact_457_logic_Onon__increasing__instantiate,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,X: a,Delta4: ( c > d ) > set_a,S: c > d,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( member_a @ X @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta4 @ S ) )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta4 )
=> ( member_a @ X @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ S ) ) ) ) ) ) ).
% logic.non_increasing_instantiate
thf(fact_458_logic_Omono__instantiate,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,X: a,Delta: ( c > d ) > set_a,S: c > d,Delta4: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) )
=> ( ( member_a @ X @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta @ S ) )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta4 )
=> ( member_a @ X @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ Delta4 @ S ) ) ) ) ) ) ).
% logic.mono_instantiate
thf(fact_459_logic_Omono__interp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ pred_a_b_d_c ) ) ) ).
% logic.mono_interp
thf(fact_460_logic_Oindep__implies__non__increasing,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( indep_interp_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 ) ) ) ) ).
% logic.indep_implies_non_increasing
thf(fact_461_mono__sem,axiom,
! [B6: ( c > d ) > a > $o] : ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( sem_c_d_a_b @ B6 ) ) ) ).
% mono_sem
thf(fact_462_non__increasing__sem,axiom,
! [B6: ( c > d ) > a > $o] : ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( sem_c_d_a_b @ B6 ) ) ) ).
% non_increasing_sem
thf(fact_463_sat_Osimps_I4_J,axiom,
! [Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,B: ( c > d ) > a > $o] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ ( sem_c_d_a_b @ B ) )
= ( B @ S @ Sigma5 ) ) ).
% sat.simps(4)
thf(fact_464_empty__upd__none,axiom,
! [X: a] :
( ( fun_upd_a_option_a
@ ^ [X4: a] : none_a
@ X
@ none_a )
= ( ^ [X4: a] : none_a ) ) ).
% empty_upd_none
thf(fact_465_logic_Osat_Osimps_I4_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma5: a,S: c > d,Delta: ( c > d ) > set_a,B: ( c > d ) > a > $o] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ ( sem_c_d_a_b @ B ) )
= ( B @ S @ Sigma5 ) ) ) ).
% logic.sat.simps(4)
thf(fact_466_logic_Omono__sem,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,B6: ( c > d ) > a > $o] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( sem_c_d_a_b @ B6 ) ) ) ) ).
% logic.mono_sem
thf(fact_467_logic_Onon__increasing__sem,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,B6: ( c > d ) > a > $o] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ ( sem_c_d_a_b @ B6 ) ) ) ) ).
% logic.non_increasing_sem
thf(fact_468_map__upd__eqD1,axiom,
! [M: a > option_a,A: a,X: a,N: a > option_a,Y: a] :
( ( ( fun_upd_a_option_a @ M @ A @ ( some_a @ X ) )
= ( fun_upd_a_option_a @ N @ A @ ( some_a @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_469_map__upd__triv,axiom,
! [T2: a > option_a,K: a,X: a] :
( ( ( T2 @ K )
= ( some_a @ X ) )
=> ( ( fun_upd_a_option_a @ T2 @ K @ ( some_a @ X ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_470_map__upd__Some__unfold,axiom,
! [M: a > option_a,A: a,B: a,X: a,Y: a] :
( ( ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) @ X )
= ( some_a @ Y ) )
= ( ( ( X = A )
& ( B = Y ) )
| ( ( X != A )
& ( ( M @ X )
= ( some_a @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_471_map__upd__nonempty,axiom,
! [T2: a > option_a,K: a,X: a] :
( ( fun_upd_a_option_a @ T2 @ K @ ( some_a @ X ) )
!= ( ^ [X4: a] : none_a ) ) ).
% map_upd_nonempty
thf(fact_472_Bex__def,axiom,
( bex_Product_prod_a_a
= ( ^ [A8: set_Product_prod_a_a,P6: product_prod_a_a > $o] :
? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_473_Bex__def,axiom,
( bex_option_a
= ( ^ [A8: set_option_a,P6: option_a > $o] :
? [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_474_Bex__def,axiom,
( bex_set_a
= ( ^ [A8: set_set_a,P6: set_a > $o] :
? [X4: set_a] :
( ( member_set_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_475_Bex__def,axiom,
( bex_a
= ( ^ [A8: set_a,P6: a > $o] :
? [X4: a] :
( ( member_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_476_Ball__def,axiom,
( ball_P843720320142865617od_a_a
= ( ^ [A8: set_Product_prod_a_a,P6: product_prod_a_a > $o] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A8 )
=> ( P6 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_477_Ball__def,axiom,
( ball_option_a
= ( ^ [A8: set_option_a,P6: option_a > $o] :
! [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
=> ( P6 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_478_Ball__def,axiom,
( ball_set_a
= ( ^ [A8: set_set_a,P6: set_a > $o] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A8 )
=> ( P6 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_479_Ball__def,axiom,
( ball_a
= ( ^ [A8: set_a,P6: a > $o] :
! [X4: a] :
( ( member_a @ X4 @ A8 )
=> ( P6 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_480_Bex__def__raw,axiom,
( bex_Product_prod_a_a
= ( ^ [A8: set_Product_prod_a_a,P6: product_prod_a_a > $o] :
? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def_raw
thf(fact_481_Bex__def__raw,axiom,
( bex_option_a
= ( ^ [A8: set_option_a,P6: option_a > $o] :
? [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def_raw
thf(fact_482_Bex__def__raw,axiom,
( bex_set_a
= ( ^ [A8: set_set_a,P6: set_a > $o] :
? [X4: set_a] :
( ( member_set_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def_raw
thf(fact_483_Bex__def__raw,axiom,
( bex_a
= ( ^ [A8: set_a,P6: a > $o] :
? [X4: a] :
( ( member_a @ X4 @ A8 )
& ( P6 @ X4 ) ) ) ) ).
% Bex_def_raw
thf(fact_484_bex__reg__left,axiom,
! [R3: set_Product_prod_a_a,Q4: product_prod_a_a > $o,P4: product_prod_a_a > $o] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ R3 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ? [X8: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X8 @ R3 )
& ( Q4 @ X8 ) )
=> ? [X_1: product_prod_a_a] : ( P4 @ X_1 ) ) ) ).
% bex_reg_left
thf(fact_485_bex__reg__left,axiom,
! [R3: set_option_a,Q4: option_a > $o,P4: option_a > $o] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ R3 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ? [X8: option_a] :
( ( member_option_a @ X8 @ R3 )
& ( Q4 @ X8 ) )
=> ? [X_1: option_a] : ( P4 @ X_1 ) ) ) ).
% bex_reg_left
thf(fact_486_bex__reg__left,axiom,
! [R3: set_set_a,Q4: set_a > $o,P4: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ R3 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ? [X8: set_a] :
( ( member_set_a @ X8 @ R3 )
& ( Q4 @ X8 ) )
=> ? [X_1: set_a] : ( P4 @ X_1 ) ) ) ).
% bex_reg_left
thf(fact_487_bex__reg__left,axiom,
! [R3: set_a,Q4: a > $o,P4: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ R3 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ? [X8: a] :
( ( member_a @ X8 @ R3 )
& ( Q4 @ X8 ) )
=> ? [X_1: a] : ( P4 @ X_1 ) ) ) ).
% bex_reg_left
thf(fact_488_Ball__Collect,axiom,
( ball_option_a
= ( ^ [A8: set_option_a,P6: option_a > $o] : ( ord_le1955136853071979460tion_a @ A8 @ ( collect_option_a @ P6 ) ) ) ) ).
% Ball_Collect
thf(fact_489_Ball__Collect,axiom,
( ball_set_a
= ( ^ [A8: set_set_a,P6: set_a > $o] : ( ord_le3724670747650509150_set_a @ A8 @ ( collect_set_a @ P6 ) ) ) ) ).
% Ball_Collect
thf(fact_490_Ball__Collect,axiom,
( ball_P843720320142865617od_a_a
= ( ^ [A8: set_Product_prod_a_a,P6: product_prod_a_a > $o] : ( ord_le746702958409616551od_a_a @ A8 @ ( collec3336397797384452498od_a_a @ P6 ) ) ) ) ).
% Ball_Collect
thf(fact_491_Ball__Collect,axiom,
( ball_a
= ( ^ [A8: set_a,P6: a > $o] : ( ord_less_eq_set_a @ A8 @ ( collect_a @ P6 ) ) ) ) ).
% Ball_Collect
thf(fact_492_verit__comp__simplify1_I2_J,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_493_verit__comp__simplify1_I2_J,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_494_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_495_verit__comp__simplify1_I2_J,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_496_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_497_not__in__fv__mod,axiom,
! [A7: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma5: a,S: c > d,Sigma4: a,S6: c > d,X: a,Delta: ( c > d ) > set_a] :
( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A7 @ ( modified_a_c_d @ C ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S6 ) ) ) @ C )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ A7 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S6 @ Delta @ A7 ) ) ) ) ).
% not_in_fv_mod
thf(fact_498_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_499_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_500_empty__iff,axiom,
! [C: option_a] :
~ ( member_option_a @ C @ bot_bot_set_option_a ) ).
% empty_iff
thf(fact_501_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_502_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_503_all__not__in__conv,axiom,
! [A7: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A7 ) )
= ( A7 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_504_all__not__in__conv,axiom,
! [A7: set_option_a] :
( ( ! [X4: option_a] :
~ ( member_option_a @ X4 @ A7 ) )
= ( A7 = bot_bot_set_option_a ) ) ).
% all_not_in_conv
thf(fact_505_all__not__in__conv,axiom,
! [A7: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A7 ) )
= ( A7 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_506_all__not__in__conv,axiom,
! [A7: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A7 ) )
= ( A7 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_507_Collect__empty__eq,axiom,
! [P4: option_a > $o] :
( ( ( collect_option_a @ P4 )
= bot_bot_set_option_a )
= ( ! [X4: option_a] :
~ ( P4 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_508_Collect__empty__eq,axiom,
! [P4: set_a > $o] :
( ( ( collect_set_a @ P4 )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
~ ( P4 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_509_Collect__empty__eq,axiom,
! [P4: a > $o] :
( ( ( collect_a @ P4 )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P4 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_510_empty__Collect__eq,axiom,
! [P4: option_a > $o] :
( ( bot_bot_set_option_a
= ( collect_option_a @ P4 ) )
= ( ! [X4: option_a] :
~ ( P4 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_511_empty__Collect__eq,axiom,
! [P4: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P4 ) )
= ( ! [X4: set_a] :
~ ( P4 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_512_empty__Collect__eq,axiom,
! [P4: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P4 ) )
= ( ! [X4: a] :
~ ( P4 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_513_subset__empty,axiom,
! [A7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ bot_bot_set_option_a )
= ( A7 = bot_bot_set_option_a ) ) ).
% subset_empty
thf(fact_514_subset__empty,axiom,
! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ bot_bot_set_set_a )
= ( A7 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_515_subset__empty,axiom,
! [A7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ bot_bo3357376287454694259od_a_a )
= ( A7 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_516_subset__empty,axiom,
! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ bot_bot_set_a )
= ( A7 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_517_empty__subsetI,axiom,
! [A7: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A7 ) ).
% empty_subsetI
thf(fact_518_empty__subsetI,axiom,
! [A7: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A7 ) ).
% empty_subsetI
thf(fact_519_empty__subsetI,axiom,
! [A7: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A7 ) ).
% empty_subsetI
thf(fact_520_empty__subsetI,axiom,
! [A7: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A7 ) ).
% empty_subsetI
thf(fact_521_ball__empty,axiom,
! [P4: a > $o,X8: a] :
( ( member_a @ X8 @ bot_bot_set_a )
=> ( P4 @ X8 ) ) ).
% ball_empty
thf(fact_522_ball__empty,axiom,
! [P4: option_a > $o,X8: option_a] :
( ( member_option_a @ X8 @ bot_bot_set_option_a )
=> ( P4 @ X8 ) ) ).
% ball_empty
thf(fact_523_ball__empty,axiom,
! [P4: set_a > $o,X8: set_a] :
( ( member_set_a @ X8 @ bot_bot_set_set_a )
=> ( P4 @ X8 ) ) ).
% ball_empty
thf(fact_524_bex__empty,axiom,
! [P4: a > $o] :
~ ? [X8: a] :
( ( member_a @ X8 @ bot_bot_set_a )
& ( P4 @ X8 ) ) ).
% bex_empty
thf(fact_525_bex__empty,axiom,
! [P4: option_a > $o] :
~ ? [X8: option_a] :
( ( member_option_a @ X8 @ bot_bot_set_option_a )
& ( P4 @ X8 ) ) ).
% bex_empty
thf(fact_526_bex__empty,axiom,
! [P4: set_a > $o] :
~ ? [X8: set_a] :
( ( member_set_a @ X8 @ bot_bot_set_set_a )
& ( P4 @ X8 ) ) ).
% bex_empty
thf(fact_527_valid__hoare__tripleI,axiom,
! [Delta: ( c > d ) > set_a,P4: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q4: assertion_a_b_d_c] :
( ! [Sigma3: a,S2: c > d] :
( ( ( valid @ Sigma3 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ P4 ) )
=> ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) ) )
=> ( ! [Sigma3: a,S2: c > d,Sigma6: a,S7: c > d] :
( ( ( valid @ Sigma3 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ P4 ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S7 ) ) ) @ C )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma6 @ S7 @ Delta @ Q4 ) ) )
=> ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P4 @ C @ Q4 @ Delta ) ) ) ).
% valid_hoare_tripleI
thf(fact_528_valid__hoare__triple__def,axiom,
! [P4: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q4: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P4 @ C @ Q4 @ Delta )
= ( ! [Sigma: a,S3: c > d] :
( ( ( valid @ Sigma )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S3 @ Delta @ P4 ) )
=> ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma @ S3 ) )
& ! [Sigma2: a,S5: c > d] :
( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S5 ) ) ) @ C )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S5 @ Delta @ Q4 ) ) ) ) ) ) ).
% valid_hoare_triple_def
thf(fact_529_pred__equals__eq2,axiom,
! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
( ( ( ^ [X4: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 ) )
= ( ^ [X4: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S4 ) ) )
= ( R3 = S4 ) ) ).
% pred_equals_eq2
thf(fact_530_pred__subset__eq2,axiom,
! [R3: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
( ( ord_less_eq_a_a_o
@ ^ [X4: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
@ ^ [X4: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S4 ) )
= ( ord_le746702958409616551od_a_a @ R3 @ S4 ) ) ).
% pred_subset_eq2
thf(fact_531_empty__def,axiom,
( bot_bot_set_option_a
= ( collect_option_a
@ ^ [X4: option_a] : $false ) ) ).
% empty_def
thf(fact_532_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X4: set_a] : $false ) ) ).
% empty_def
thf(fact_533_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% empty_def
thf(fact_534_subset__emptyI,axiom,
! [A7: set_option_a] :
( ! [X3: option_a] :
~ ( member_option_a @ X3 @ A7 )
=> ( ord_le1955136853071979460tion_a @ A7 @ bot_bot_set_option_a ) ) ).
% subset_emptyI
thf(fact_535_subset__emptyI,axiom,
! [A7: set_set_a] :
( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A7 )
=> ( ord_le3724670747650509150_set_a @ A7 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_536_subset__emptyI,axiom,
! [A7: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X3 @ A7 )
=> ( ord_le746702958409616551od_a_a @ A7 @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_537_subset__emptyI,axiom,
! [A7: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A7 )
=> ( ord_less_eq_set_a @ A7 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_538_bot_Oextremum__uniqueI,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ bot_bot_set_option_a )
=> ( A = bot_bot_set_option_a ) ) ).
% bot.extremum_uniqueI
thf(fact_539_bot_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
=> ( A = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_540_bot_Oextremum__uniqueI,axiom,
! [A: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ bot_bot_o_set_a )
=> ( A = bot_bot_o_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_541_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_542_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_543_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_544_bot_Oextremum__unique,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A @ bot_bot_set_option_a )
= ( A = bot_bot_set_option_a ) ) ).
% bot.extremum_unique
thf(fact_545_bot_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
= ( A = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_546_bot_Oextremum__unique,axiom,
! [A: $o > set_a] :
( ( ord_less_eq_o_set_a @ A @ bot_bot_o_set_a )
= ( A = bot_bot_o_set_a ) ) ).
% bot.extremum_unique
thf(fact_547_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_548_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_549_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_550_bot_Oextremum,axiom,
! [A: set_option_a] : ( ord_le1955136853071979460tion_a @ bot_bot_set_option_a @ A ) ).
% bot.extremum
thf(fact_551_bot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% bot.extremum
thf(fact_552_bot_Oextremum,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ bot_bot_o_set_a @ A ) ).
% bot.extremum
thf(fact_553_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_554_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_555_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_556_subrelI,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X3: a,Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y5 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y5 ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R2 @ S ) ) ).
% subrelI
thf(fact_557_ssubst__Pair__rhs,axiom,
! [R2: a,S: a,R3: set_Product_prod_a_a,S6: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R2 @ S ) @ R3 )
=> ( ( S6 = S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R2 @ S6 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_558_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_559_emptyE,axiom,
! [A: option_a] :
~ ( member_option_a @ A @ bot_bot_set_option_a ) ).
% emptyE
thf(fact_560_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_561_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_562_equals0D,axiom,
! [A7: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A7 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A7 ) ) ).
% equals0D
thf(fact_563_equals0D,axiom,
! [A7: set_option_a,A: option_a] :
( ( A7 = bot_bot_set_option_a )
=> ~ ( member_option_a @ A @ A7 ) ) ).
% equals0D
thf(fact_564_equals0D,axiom,
! [A7: set_set_a,A: set_a] :
( ( A7 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A7 ) ) ).
% equals0D
thf(fact_565_equals0D,axiom,
! [A7: set_a,A: a] :
( ( A7 = bot_bot_set_a )
=> ~ ( member_a @ A @ A7 ) ) ).
% equals0D
thf(fact_566_equals0I,axiom,
! [A7: set_Product_prod_a_a] :
( ! [Y5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y5 @ A7 )
=> ( A7 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_567_equals0I,axiom,
! [A7: set_option_a] :
( ! [Y5: option_a] :
~ ( member_option_a @ Y5 @ A7 )
=> ( A7 = bot_bot_set_option_a ) ) ).
% equals0I
thf(fact_568_equals0I,axiom,
! [A7: set_set_a] :
( ! [Y5: set_a] :
~ ( member_set_a @ Y5 @ A7 )
=> ( A7 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_569_equals0I,axiom,
! [A7: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A7 )
=> ( A7 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_570_ex__in__conv,axiom,
! [A7: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A7 ) )
= ( A7 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_571_ex__in__conv,axiom,
! [A7: set_option_a] :
( ( ? [X4: option_a] : ( member_option_a @ X4 @ A7 ) )
= ( A7 != bot_bot_set_option_a ) ) ).
% ex_in_conv
thf(fact_572_ex__in__conv,axiom,
! [A7: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A7 ) )
= ( A7 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_573_ex__in__conv,axiom,
! [A7: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A7 ) )
= ( A7 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_574_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_575_logic_Ovalid__hoare__triple__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P4: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q4: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P4 @ C @ Q4 @ Delta )
= ( ! [Sigma: a,S3: c > d] :
( ( ( Valid @ Sigma )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S3 @ Delta @ P4 ) )
=> ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma @ S3 ) )
& ! [Sigma2: a,S5: c > d] :
( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S5 ) ) ) @ C )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S5 @ Delta @ Q4 ) ) ) ) ) ) ) ).
% logic.valid_hoare_triple_def
thf(fact_576_logic_Ovalid__hoare__tripleI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,P4: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q4: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma3: a,S2: c > d] :
( ( ( Valid @ Sigma3 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ P4 ) )
=> ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) ) )
=> ( ! [Sigma3: a,S2: c > d,Sigma6: a,S7: c > d] :
( ( ( Valid @ Sigma3 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ P4 ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S2 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S7 ) ) ) @ C )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma6 @ S7 @ Delta @ Q4 ) ) )
=> ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P4 @ C @ Q4 @ Delta ) ) ) ) ).
% logic.valid_hoare_tripleI
thf(fact_577_logic_Onot__in__fv__mod,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A7: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma5: a,S: c > d,Sigma4: a,S6: c > d,X: a,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A7 @ ( modified_a_c_d @ C ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S6 ) ) ) @ C )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ A7 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S6 @ Delta @ A7 ) ) ) ) ) ).
% logic.not_in_fv_mod
thf(fact_578_image2__def,axiom,
( bNF_Gr1766759448597441700_a_a_a
= ( ^ [A8: set_a,F2: a > a,G2: a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: a] :
( ( Uu
= ( product_Pair_a_a @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member_a @ A4 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_579_image2__def,axiom,
( bNF_Gr1149069696037075021_a_a_a
= ( ^ [A8: set_Product_prod_a_a,F2: product_prod_a_a > a,G2: product_prod_a_a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: product_prod_a_a] :
( ( Uu
= ( product_Pair_a_a @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member1426531477525435216od_a_a @ A4 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_580_image2__def,axiom,
( bNF_Gr4405194352872318698_a_a_a
= ( ^ [A8: set_option_a,F2: option_a > a,G2: option_a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: option_a] :
( ( Uu
= ( product_Pair_a_a @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member_option_a @ A4 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_581_image2__def,axiom,
( bNF_Gr6628399236160294404_a_a_a
= ( ^ [A8: set_set_a,F2: set_a > a,G2: set_a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: set_a] :
( ( Uu
= ( product_Pair_a_a @ ( F2 @ A4 ) @ ( G2 @ A4 ) ) )
& ( member_set_a @ A4 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_582_min__ext__def,axiom,
( min_ext_a
= ( ^ [R4: set_Product_prod_a_a] :
( collec8259436133773553042_set_a
@ ^ [Uu: produc1703568184450464039_set_a] :
? [X9: set_a,Y7: set_a] :
( ( Uu
= ( produc9088192753505129239_set_a @ X9 @ Y7 ) )
& ( X9 != bot_bot_set_a )
& ! [X4: a] :
( ( member_a @ X4 @ Y7 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ X9 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_583_min__ext__def,axiom,
( min_ext_option_a
= ( ^ [R4: set_Pr7585778909603769095tion_a] :
( collec2929565718378674578tion_a
@ ^ [Uu: produc8652252815484796455tion_a] :
? [X9: set_option_a,Y7: set_option_a] :
( ( Uu
= ( produc8179951581375851543tion_a @ X9 @ Y7 ) )
& ( X9 != bot_bot_set_option_a )
& ! [X4: option_a] :
( ( member_option_a @ X4 @ Y7 )
=> ? [Y3: option_a] :
( ( member_option_a @ Y3 @ X9 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_584_min__ext__def,axiom,
( min_ext_set_a
= ( ^ [R4: set_Pr5845495582615845127_set_a] :
( collec4275731697586626962_set_a
@ ^ [Uu: produc818331024236758311_set_a] :
? [X9: set_set_a,Y7: set_set_a] :
( ( Uu
= ( produc4465669641558533911_set_a @ X9 @ Y7 ) )
& ( X9 != bot_bot_set_set_a )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ Y7 )
=> ? [Y3: set_a] :
( ( member_set_a @ Y3 @ X9 )
& ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_585_relImage__def,axiom,
( bNF_Gr5084790043256381924ge_a_a
= ( ^ [R5: set_Product_prod_a_a,F2: a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A12: a,A22: a] :
( ( Uu
= ( product_Pair_a_a @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A12 @ A22 ) @ R5 ) ) ) ) ) ).
% relImage_def
thf(fact_586_relInvImage__def,axiom,
( bNF_Gr5251999782455585624_a_a_a
= ( ^ [A8: set_Product_prod_a_a,R5: set_Product_prod_a_a,F2: product_prod_a_a > a] :
( collec10116633892588882od_a_a
@ ^ [Uu: produc3498347346309940967od_a_a] :
? [A12: product_prod_a_a,A22: product_prod_a_a] :
( ( Uu
= ( produc7886510207707329367od_a_a @ A12 @ A22 ) )
& ( member1426531477525435216od_a_a @ A12 @ A8 )
& ( member1426531477525435216od_a_a @ A22 @ A8 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) @ R5 ) ) ) ) ) ).
% relInvImage_def
thf(fact_587_relInvImage__def,axiom,
( bNF_Gr701560671560476539on_a_a
= ( ^ [A8: set_option_a,R5: set_Product_prod_a_a,F2: option_a > a] :
( collec4135126896892755346tion_a
@ ^ [Uu: produc3509355604313844263tion_a] :
? [A12: option_a,A22: option_a] :
( ( Uu
= ( produc9011544418120257559tion_a @ A12 @ A22 ) )
& ( member_option_a @ A12 @ A8 )
& ( member_option_a @ A22 @ A8 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) @ R5 ) ) ) ) ) ).
% relInvImage_def
thf(fact_588_relInvImage__def,axiom,
( bNF_Gr9028471054304145889et_a_a
= ( ^ [A8: set_set_a,R5: set_Product_prod_a_a,F2: set_a > a] :
( collec8259436133773553042_set_a
@ ^ [Uu: produc1703568184450464039_set_a] :
? [A12: set_a,A22: set_a] :
( ( Uu
= ( produc9088192753505129239_set_a @ A12 @ A22 ) )
& ( member_set_a @ A12 @ A8 )
& ( member_set_a @ A22 @ A8 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) @ R5 ) ) ) ) ) ).
% relInvImage_def
thf(fact_589_relInvImage__def,axiom,
( bNF_Gr1680063798946433857ge_a_a
= ( ^ [A8: set_a,R5: set_Product_prod_a_a,F2: a > a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A12: a,A22: a] :
( ( Uu
= ( product_Pair_a_a @ A12 @ A22 ) )
& ( member_a @ A12 @ A8 )
& ( member_a @ A22 @ A8 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) @ R5 ) ) ) ) ) ).
% relInvImage_def
thf(fact_590_bot__empty__eq2,axiom,
( bot_bot_a_a_o
= ( ^ [X4: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_591_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_592_bot__empty__eq,axiom,
( bot_bot_option_a_o
= ( ^ [X4: option_a] : ( member_option_a @ X4 @ bot_bot_set_option_a ) ) ) ).
% bot_empty_eq
thf(fact_593_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_594_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_595_bot__set__def,axiom,
( bot_bot_set_option_a
= ( collect_option_a @ bot_bot_option_a_o ) ) ).
% bot_set_def
thf(fact_596_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_597_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_598_relImage__mono,axiom,
! [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a,F: a > a] :
( ( ord_le746702958409616551od_a_a @ R1 @ R22 )
=> ( ord_le746702958409616551od_a_a @ ( bNF_Gr5084790043256381924ge_a_a @ R1 @ F ) @ ( bNF_Gr5084790043256381924ge_a_a @ R22 @ F ) ) ) ).
% relImage_mono
thf(fact_599_relInvImage__mono,axiom,
! [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a,A7: set_a,F: a > a] :
( ( ord_le746702958409616551od_a_a @ R1 @ R22 )
=> ( ord_le746702958409616551od_a_a @ ( bNF_Gr1680063798946433857ge_a_a @ A7 @ R1 @ F ) @ ( bNF_Gr1680063798946433857ge_a_a @ A7 @ R22 @ F ) ) ) ).
% relInvImage_mono
thf(fact_600_image2__eqI,axiom,
! [B: a,F: a > a,X: a,C: a,G: a > a,A7: set_a] :
( ( B
= ( F @ X ) )
=> ( ( C
= ( G @ X ) )
=> ( ( member_a @ X @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr1766759448597441700_a_a_a @ A7 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_601_image2__eqI,axiom,
! [B: a,F: product_prod_a_a > a,X: product_prod_a_a,C: a,G: product_prod_a_a > a,A7: set_Product_prod_a_a] :
( ( B
= ( F @ X ) )
=> ( ( C
= ( G @ X ) )
=> ( ( member1426531477525435216od_a_a @ X @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr1149069696037075021_a_a_a @ A7 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_602_image2__eqI,axiom,
! [B: a,F: option_a > a,X: option_a,C: a,G: option_a > a,A7: set_option_a] :
( ( B
= ( F @ X ) )
=> ( ( C
= ( G @ X ) )
=> ( ( member_option_a @ X @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr4405194352872318698_a_a_a @ A7 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_603_image2__eqI,axiom,
! [B: a,F: set_a > a,X: set_a,C: a,G: set_a > a,A7: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( C
= ( G @ X ) )
=> ( ( member_set_a @ X @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( bNF_Gr6628399236160294404_a_a_a @ A7 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_604_applies__eq_Opelims,axiom,
! [X: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( ( applies_eq_a_b_d_c @ plus @ mult @ valid @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ plus @ mult @ valid @ Uu @ Xb @ Xa @ X ) ) )
=> ~ ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) ) ) ) ) ).
% applies_eq.pelims
thf(fact_605_accp__subset__induct,axiom,
! [D: a > $o,R3: a > a > $o,X: a,P4: a > $o] :
( ( ord_less_eq_a_o @ D @ ( accp_a @ R3 ) )
=> ( ! [X3: a,Z4: a] :
( ( D @ X3 )
=> ( ( R3 @ Z4 @ X3 )
=> ( D @ Z4 ) ) )
=> ( ( D @ X )
=> ( ! [X3: a] :
( ( D @ X3 )
=> ( ! [Z5: a] :
( ( R3 @ Z5 @ X3 )
=> ( P4 @ Z5 ) )
=> ( P4 @ X3 ) ) )
=> ( P4 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_606_accp__subset,axiom,
! [R1: a > a > $o,R22: a > a > $o] :
( ( ord_less_eq_a_a_o @ R1 @ R22 )
=> ( ord_less_eq_a_o @ ( accp_a @ R22 ) @ ( accp_a @ R1 ) ) ) ).
% accp_subset
thf(fact_607_logic_Oapplies__eq_Opelims,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Uu @ Xb @ Xa @ X ) ) )
=> ~ ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) ) ) ) ) ) ).
% logic.applies_eq.pelims
thf(fact_608_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A8: set_a] : ( A8 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_609_Set_Ois__empty__def,axiom,
( is_empty_option_a
= ( ^ [A8: set_option_a] : ( A8 = bot_bot_set_option_a ) ) ) ).
% Set.is_empty_def
thf(fact_610_Set_Ois__empty__def,axiom,
( is_empty_set_a
= ( ^ [A8: set_set_a] : ( A8 = bot_bot_set_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_611_top__apply,axiom,
( top_top_a_o
= ( ^ [X4: a] : top_top_o ) ) ).
% top_apply
thf(fact_612_UNIV__I,axiom,
! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).
% UNIV_I
thf(fact_613_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_614_UNIV__I,axiom,
! [X: option_a] : ( member_option_a @ X @ top_top_set_option_a ) ).
% UNIV_I
thf(fact_615_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_616_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collect_set_a
@ ^ [S3: set_a] : P4 )
= top_top_set_set_a ) )
& ( ~ P4
=> ( ( collect_set_a
@ ^ [S3: set_a] : P4 )
= bot_bot_set_set_a ) ) ) ).
% Collect_const
thf(fact_617_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collect_option_a
@ ^ [S3: option_a] : P4 )
= top_top_set_option_a ) )
& ( ~ P4
=> ( ( collect_option_a
@ ^ [S3: option_a] : P4 )
= bot_bot_set_option_a ) ) ) ).
% Collect_const
thf(fact_618_Collect__const,axiom,
! [P4: $o] :
( ( P4
=> ( ( collect_a
@ ^ [S3: a] : P4 )
= top_top_set_a ) )
& ( ~ P4
=> ( ( collect_a
@ ^ [S3: a] : P4 )
= bot_bot_set_a ) ) ) ).
% Collect_const
thf(fact_619_bex__UNIV,axiom,
! [P4: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ top_top_set_a )
& ( P4 @ X4 ) ) )
= ( ? [X9: a] : ( P4 @ X9 ) ) ) ).
% bex_UNIV
thf(fact_620_bex__UNIV,axiom,
! [P4: option_a > $o] :
( ( ? [X4: option_a] :
( ( member_option_a @ X4 @ top_top_set_option_a )
& ( P4 @ X4 ) ) )
= ( ? [X9: option_a] : ( P4 @ X9 ) ) ) ).
% bex_UNIV
thf(fact_621_empty__not__UNIV,axiom,
bot_bot_set_set_a != top_top_set_set_a ).
% empty_not_UNIV
thf(fact_622_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_623_empty__not__UNIV,axiom,
bot_bot_set_option_a != top_top_set_option_a ).
% empty_not_UNIV
thf(fact_624_UNIV__witness,axiom,
? [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ top_to8063371432257647191od_a_a ) ).
% UNIV_witness
thf(fact_625_UNIV__witness,axiom,
? [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_626_UNIV__witness,axiom,
? [X3: option_a] : ( member_option_a @ X3 @ top_top_set_option_a ) ).
% UNIV_witness
thf(fact_627_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_628_UNIV__eq__I,axiom,
! [A7: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A7 )
=> ( top_to8063371432257647191od_a_a = A7 ) ) ).
% UNIV_eq_I
thf(fact_629_UNIV__eq__I,axiom,
! [A7: set_set_a] :
( ! [X3: set_a] : ( member_set_a @ X3 @ A7 )
=> ( top_top_set_set_a = A7 ) ) ).
% UNIV_eq_I
thf(fact_630_UNIV__eq__I,axiom,
! [A7: set_option_a] :
( ! [X3: option_a] : ( member_option_a @ X3 @ A7 )
=> ( top_top_set_option_a = A7 ) ) ).
% UNIV_eq_I
thf(fact_631_UNIV__eq__I,axiom,
! [A7: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A7 )
=> ( top_top_set_a = A7 ) ) ).
% UNIV_eq_I
thf(fact_632_UNIV__def,axiom,
( top_top_set_set_a
= ( collect_set_a
@ ^ [X4: set_a] : $true ) ) ).
% UNIV_def
thf(fact_633_UNIV__def,axiom,
( top_top_set_option_a
= ( collect_option_a
@ ^ [X4: option_a] : $true ) ) ).
% UNIV_def
thf(fact_634_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X4: a] : $true ) ) ).
% UNIV_def
thf(fact_635_subset__UNIV,axiom,
! [A7: set_option_a] : ( ord_le1955136853071979460tion_a @ A7 @ top_top_set_option_a ) ).
% subset_UNIV
thf(fact_636_subset__UNIV,axiom,
! [A7: set_set_a] : ( ord_le3724670747650509150_set_a @ A7 @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_637_subset__UNIV,axiom,
! [A7: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A7 @ top_to8063371432257647191od_a_a ) ).
% subset_UNIV
thf(fact_638_subset__UNIV,axiom,
! [A7: set_a] : ( ord_less_eq_set_a @ A7 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_639_top_Oextremum__uniqueI,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ top_top_set_option_a @ A )
=> ( A = top_top_set_option_a ) ) ).
% top.extremum_uniqueI
thf(fact_640_top_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
=> ( A = top_top_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_641_top_Oextremum__uniqueI,axiom,
! [A: $o > set_a] :
( ( ord_less_eq_o_set_a @ top_top_o_set_a @ A )
=> ( A = top_top_o_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_642_top_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
=> ( A = top_top_set_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_643_top_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ top_to8063371432257647191od_a_a @ A )
=> ( A = top_to8063371432257647191od_a_a ) ) ).
% top.extremum_uniqueI
thf(fact_644_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_645_top_Oextremum__unique,axiom,
! [A: set_option_a] :
( ( ord_le1955136853071979460tion_a @ top_top_set_option_a @ A )
= ( A = top_top_set_option_a ) ) ).
% top.extremum_unique
thf(fact_646_top_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ top_top_a_o @ A )
= ( A = top_top_a_o ) ) ).
% top.extremum_unique
thf(fact_647_top_Oextremum__unique,axiom,
! [A: $o > set_a] :
( ( ord_less_eq_o_set_a @ top_top_o_set_a @ A )
= ( A = top_top_o_set_a ) ) ).
% top.extremum_unique
thf(fact_648_top_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ top_top_set_set_a @ A )
= ( A = top_top_set_set_a ) ) ).
% top.extremum_unique
thf(fact_649_top_Oextremum__unique,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ top_to8063371432257647191od_a_a @ A )
= ( A = top_to8063371432257647191od_a_a ) ) ).
% top.extremum_unique
thf(fact_650_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_651_top__greatest,axiom,
! [A: set_option_a] : ( ord_le1955136853071979460tion_a @ A @ top_top_set_option_a ) ).
% top_greatest
thf(fact_652_top__greatest,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ A @ top_top_a_o ) ).
% top_greatest
thf(fact_653_top__greatest,axiom,
! [A: $o > set_a] : ( ord_less_eq_o_set_a @ A @ top_top_o_set_a ) ).
% top_greatest
thf(fact_654_top__greatest,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% top_greatest
thf(fact_655_top__greatest,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ top_to8063371432257647191od_a_a ) ).
% top_greatest
thf(fact_656_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_657_graph__def,axiom,
( graph_a_a
= ( ^ [M2: a > option_a] :
( collec3336397797384452498od_a_a
@ ^ [Uu: product_prod_a_a] :
? [A4: a,B4: a] :
( ( Uu
= ( product_Pair_a_a @ A4 @ B4 ) )
& ( ( M2 @ A4 )
= ( some_a @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_658_top__empty__eq2,axiom,
( top_top_a_a_o
= ( ^ [X4: a,Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ top_to8063371432257647191od_a_a ) ) ) ).
% top_empty_eq2
thf(fact_659_top__set__def,axiom,
( top_top_set_set_a
= ( collect_set_a @ top_top_set_a_o ) ) ).
% top_set_def
thf(fact_660_top__set__def,axiom,
( top_top_set_option_a
= ( collect_option_a @ top_top_option_a_o ) ) ).
% top_set_def
thf(fact_661_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_662_top__empty__eq,axiom,
( top_to8687885267596698950_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ top_to8063371432257647191od_a_a ) ) ) ).
% top_empty_eq
thf(fact_663_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_664_top__empty__eq,axiom,
( top_top_option_a_o
= ( ^ [X4: option_a] : ( member_option_a @ X4 @ top_top_set_option_a ) ) ) ).
% top_empty_eq
thf(fact_665_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_666_in__graphI,axiom,
! [M: a > option_a,K: a,V3: a] :
( ( ( M @ K )
= ( some_a @ V3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ M ) ) ) ).
% in_graphI
thf(fact_667_in__graphD,axiom,
! [K: a,V3: a,M: a > option_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ M ) )
=> ( ( M @ K )
= ( some_a @ V3 ) ) ) ).
% in_graphD
thf(fact_668_graph__map__upd,axiom,
! [M: a > option_a,K: a,V3: a] :
( ( graph_a_a @ ( fun_upd_a_option_a @ M @ K @ ( some_a @ V3 ) ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ ( fun_upd_a_option_a @ M @ K @ none_a ) ) ) ) ).
% graph_map_upd
thf(fact_669_ran__map__upd,axiom,
! [M: a > option_a,A: a,B: a] :
( ( ( M @ A )
= none_a )
=> ( ( ran_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) )
= ( insert_a @ B @ ( ran_a_a @ M ) ) ) ) ).
% ran_map_upd
thf(fact_670_map__le__imp__upd__le,axiom,
! [M1: a > option_a,M22: a > option_a,X: a,Y: a] :
( ( map_le_a_a @ M1 @ M22 )
=> ( map_le_a_a @ ( fun_upd_a_option_a @ M1 @ X @ none_a ) @ ( fun_upd_a_option_a @ M22 @ X @ ( some_a @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_671_insert__absorb2,axiom,
! [X: a,A7: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A7 ) )
= ( insert_a @ X @ A7 ) ) ).
% insert_absorb2
thf(fact_672_insert__absorb2,axiom,
! [X: option_a,A7: set_option_a] :
( ( insert_option_a @ X @ ( insert_option_a @ X @ A7 ) )
= ( insert_option_a @ X @ A7 ) ) ).
% insert_absorb2
thf(fact_673_insert__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A7 ) )
= ( ( A = B )
| ( member1426531477525435216od_a_a @ A @ A7 ) ) ) ).
% insert_iff
thf(fact_674_insert__iff,axiom,
! [A: option_a,B: option_a,A7: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B @ A7 ) )
= ( ( A = B )
| ( member_option_a @ A @ A7 ) ) ) ).
% insert_iff
thf(fact_675_insert__iff,axiom,
! [A: set_a,B: set_a,A7: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A7 ) )
= ( ( A = B )
| ( member_set_a @ A @ A7 ) ) ) ).
% insert_iff
thf(fact_676_insert__iff,axiom,
! [A: a,B: a,A7: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A7 ) )
= ( ( A = B )
| ( member_a @ A @ A7 ) ) ) ).
% insert_iff
thf(fact_677_insertCI,axiom,
! [A: product_prod_a_a,B6: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B6 )
=> ( A = B ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_678_insertCI,axiom,
! [A: option_a,B6: set_option_a,B: option_a] :
( ( ~ ( member_option_a @ A @ B6 )
=> ( A = B ) )
=> ( member_option_a @ A @ ( insert_option_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_679_insertCI,axiom,
! [A: set_a,B6: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B6 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_680_insertCI,axiom,
! [A: a,B6: set_a,B: a] :
( ( ~ ( member_a @ A @ B6 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B6 ) ) ) ).
% insertCI
thf(fact_681_top1I,axiom,
! [X: a] : ( top_top_a_o @ X ) ).
% top1I
thf(fact_682_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_683_singletonI,axiom,
! [A: option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singletonI
thf(fact_684_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_685_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_686_insert__subset,axiom,
! [X: option_a,A7: set_option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ A7 ) @ B6 )
= ( ( member_option_a @ X @ B6 )
& ( ord_le1955136853071979460tion_a @ A7 @ B6 ) ) ) ).
% insert_subset
thf(fact_687_insert__subset,axiom,
! [X: set_a,A7: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A7 ) @ B6 )
= ( ( member_set_a @ X @ B6 )
& ( ord_le3724670747650509150_set_a @ A7 @ B6 ) ) ) ).
% insert_subset
thf(fact_688_insert__subset,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A7 ) @ B6 )
= ( ( member1426531477525435216od_a_a @ X @ B6 )
& ( ord_le746702958409616551od_a_a @ A7 @ B6 ) ) ) ).
% insert_subset
thf(fact_689_insert__subset,axiom,
! [X: a,A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A7 ) @ B6 )
= ( ( member_a @ X @ B6 )
& ( ord_less_eq_set_a @ A7 @ B6 ) ) ) ).
% insert_subset
thf(fact_690_singleton__conv2,axiom,
! [A: option_a] :
( ( collect_option_a
@ ( ^ [Y4: option_a,Z: option_a] : ( Y4 = Z )
@ A ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singleton_conv2
thf(fact_691_singleton__conv2,axiom,
! [A: set_a] :
( ( collect_set_a
@ ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z )
@ A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv2
thf(fact_692_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y4: a,Z: a] : ( Y4 = Z )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_693_singleton__conv,axiom,
! [A: option_a] :
( ( collect_option_a
@ ^ [X4: option_a] : ( X4 = A ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singleton_conv
thf(fact_694_singleton__conv,axiom,
! [A: set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( X4 = A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv
thf(fact_695_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X4: a] : ( X4 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_696_singleton__insert__inj__eq_H,axiom,
! [A: option_a,A7: set_option_a,B: option_a] :
( ( ( insert_option_a @ A @ A7 )
= ( insert_option_a @ B @ bot_bot_set_option_a ) )
= ( ( A = B )
& ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_697_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A7: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A7 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_698_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A7: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A7 )
= ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_699_singleton__insert__inj__eq_H,axiom,
! [A: a,A7: set_a,B: a] :
( ( ( insert_a @ A @ A7 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A7 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_700_singleton__insert__inj__eq,axiom,
! [B: option_a,A: option_a,A7: set_option_a] :
( ( ( insert_option_a @ B @ bot_bot_set_option_a )
= ( insert_option_a @ A @ A7 ) )
= ( ( A = B )
& ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_701_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A7: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A7 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_702_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_a,A: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A7 ) )
= ( ( A = B )
& ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_703_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A7: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A7 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A7 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_704_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_705_insert__UNIV,axiom,
! [X: option_a] :
( ( insert_option_a @ X @ top_top_set_option_a )
= top_top_set_option_a ) ).
% insert_UNIV
thf(fact_706_Collect__conv__if,axiom,
! [P4: option_a > $o,A: option_a] :
( ( ( P4 @ A )
=> ( ( collect_option_a
@ ^ [X4: option_a] :
( ( X4 = A )
& ( P4 @ X4 ) ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collect_option_a
@ ^ [X4: option_a] :
( ( X4 = A )
& ( P4 @ X4 ) ) )
= bot_bot_set_option_a ) ) ) ).
% Collect_conv_if
thf(fact_707_Collect__conv__if,axiom,
! [P4: set_a > $o,A: set_a] :
( ( ( P4 @ A )
=> ( ( collect_set_a
@ ^ [X4: set_a] :
( ( X4 = A )
& ( P4 @ X4 ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collect_set_a
@ ^ [X4: set_a] :
( ( X4 = A )
& ( P4 @ X4 ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if
thf(fact_708_Collect__conv__if,axiom,
! [P4: a > $o,A: a] :
( ( ( P4 @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A )
& ( P4 @ X4 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A )
& ( P4 @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_709_Collect__conv__if2,axiom,
! [P4: option_a > $o,A: option_a] :
( ( ( P4 @ A )
=> ( ( collect_option_a
@ ^ [X4: option_a] :
( ( A = X4 )
& ( P4 @ X4 ) ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collect_option_a
@ ^ [X4: option_a] :
( ( A = X4 )
& ( P4 @ X4 ) ) )
= bot_bot_set_option_a ) ) ) ).
% Collect_conv_if2
thf(fact_710_Collect__conv__if2,axiom,
! [P4: set_a > $o,A: set_a] :
( ( ( P4 @ A )
=> ( ( collect_set_a
@ ^ [X4: set_a] :
( ( A = X4 )
& ( P4 @ X4 ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collect_set_a
@ ^ [X4: set_a] :
( ( A = X4 )
& ( P4 @ X4 ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_711_Collect__conv__if2,axiom,
! [P4: a > $o,A: a] :
( ( ( P4 @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A = X4 )
& ( P4 @ X4 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P4 @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A = X4 )
& ( P4 @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_712_singletonD,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_713_singletonD,axiom,
! [B: option_a,A: option_a] :
( ( member_option_a @ B @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_714_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_715_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_716_singleton__iff,axiom,
! [B: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_717_singleton__iff,axiom,
! [B: option_a,A: option_a] :
( ( member_option_a @ B @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_718_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_719_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_720_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_721_doubleton__eq__iff,axiom,
! [A: option_a,B: option_a,C: option_a,D2: option_a] :
( ( ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) )
= ( insert_option_a @ C @ ( insert_option_a @ D2 @ bot_bot_set_option_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_722_doubleton__eq__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D2: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D2 @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_723_insert__not__empty,axiom,
! [A: a,A7: set_a] :
( ( insert_a @ A @ A7 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_724_insert__not__empty,axiom,
! [A: option_a,A7: set_option_a] :
( ( insert_option_a @ A @ A7 )
!= bot_bot_set_option_a ) ).
% insert_not_empty
thf(fact_725_insert__not__empty,axiom,
! [A: set_a,A7: set_set_a] :
( ( insert_set_a @ A @ A7 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_726_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_727_singleton__inject,axiom,
! [A: option_a,B: option_a] :
( ( ( insert_option_a @ A @ bot_bot_set_option_a )
= ( insert_option_a @ B @ bot_bot_set_option_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_728_singleton__inject,axiom,
! [A: set_a,B: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_729_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A7 )
=> ? [B8: set_Product_prod_a_a] :
( ( A7
= ( insert4534936382041156343od_a_a @ A @ B8 ) )
& ~ ( member1426531477525435216od_a_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_730_mk__disjoint__insert,axiom,
! [A: option_a,A7: set_option_a] :
( ( member_option_a @ A @ A7 )
=> ? [B8: set_option_a] :
( ( A7
= ( insert_option_a @ A @ B8 ) )
& ~ ( member_option_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_731_mk__disjoint__insert,axiom,
! [A: set_a,A7: set_set_a] :
( ( member_set_a @ A @ A7 )
=> ? [B8: set_set_a] :
( ( A7
= ( insert_set_a @ A @ B8 ) )
& ~ ( member_set_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_732_mk__disjoint__insert,axiom,
! [A: a,A7: set_a] :
( ( member_a @ A @ A7 )
=> ? [B8: set_a] :
( ( A7
= ( insert_a @ A @ B8 ) )
& ~ ( member_a @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_733_insert__commute,axiom,
! [X: a,Y: a,A7: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A7 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A7 ) ) ) ).
% insert_commute
thf(fact_734_insert__commute,axiom,
! [X: option_a,Y: option_a,A7: set_option_a] :
( ( insert_option_a @ X @ ( insert_option_a @ Y @ A7 ) )
= ( insert_option_a @ Y @ ( insert_option_a @ X @ A7 ) ) ) ).
% insert_commute
thf(fact_735_insert__eq__iff,axiom,
! [A: product_prod_a_a,A7: set_Product_prod_a_a,B: product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A7 )
=> ( ~ ( member1426531477525435216od_a_a @ B @ B6 )
=> ( ( ( insert4534936382041156343od_a_a @ A @ A7 )
= ( insert4534936382041156343od_a_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A7 = B6 ) )
& ( ( A != B )
=> ? [C4: set_Product_prod_a_a] :
( ( A7
= ( insert4534936382041156343od_a_a @ B @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ B @ C4 )
& ( B6
= ( insert4534936382041156343od_a_a @ A @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_736_insert__eq__iff,axiom,
! [A: option_a,A7: set_option_a,B: option_a,B6: set_option_a] :
( ~ ( member_option_a @ A @ A7 )
=> ( ~ ( member_option_a @ B @ B6 )
=> ( ( ( insert_option_a @ A @ A7 )
= ( insert_option_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A7 = B6 ) )
& ( ( A != B )
=> ? [C4: set_option_a] :
( ( A7
= ( insert_option_a @ B @ C4 ) )
& ~ ( member_option_a @ B @ C4 )
& ( B6
= ( insert_option_a @ A @ C4 ) )
& ~ ( member_option_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_737_insert__eq__iff,axiom,
! [A: set_a,A7: set_set_a,B: set_a,B6: set_set_a] :
( ~ ( member_set_a @ A @ A7 )
=> ( ~ ( member_set_a @ B @ B6 )
=> ( ( ( insert_set_a @ A @ A7 )
= ( insert_set_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A7 = B6 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A7
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B6
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_738_insert__eq__iff,axiom,
! [A: a,A7: set_a,B: a,B6: set_a] :
( ~ ( member_a @ A @ A7 )
=> ( ~ ( member_a @ B @ B6 )
=> ( ( ( insert_a @ A @ A7 )
= ( insert_a @ B @ B6 ) )
= ( ( ( A = B )
=> ( A7 = B6 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A7
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B6
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_739_insert__absorb,axiom,
! [A: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A7 )
=> ( ( insert4534936382041156343od_a_a @ A @ A7 )
= A7 ) ) ).
% insert_absorb
thf(fact_740_insert__absorb,axiom,
! [A: option_a,A7: set_option_a] :
( ( member_option_a @ A @ A7 )
=> ( ( insert_option_a @ A @ A7 )
= A7 ) ) ).
% insert_absorb
thf(fact_741_insert__absorb,axiom,
! [A: set_a,A7: set_set_a] :
( ( member_set_a @ A @ A7 )
=> ( ( insert_set_a @ A @ A7 )
= A7 ) ) ).
% insert_absorb
thf(fact_742_insert__absorb,axiom,
! [A: a,A7: set_a] :
( ( member_a @ A @ A7 )
=> ( ( insert_a @ A @ A7 )
= A7 ) ) ).
% insert_absorb
thf(fact_743_insert__ident,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( ( insert4534936382041156343od_a_a @ X @ A7 )
= ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( A7 = B6 ) ) ) ) ).
% insert_ident
thf(fact_744_insert__ident,axiom,
! [X: option_a,A7: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ X @ A7 )
=> ( ~ ( member_option_a @ X @ B6 )
=> ( ( ( insert_option_a @ X @ A7 )
= ( insert_option_a @ X @ B6 ) )
= ( A7 = B6 ) ) ) ) ).
% insert_ident
thf(fact_745_insert__ident,axiom,
! [X: set_a,A7: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A7 )
=> ( ~ ( member_set_a @ X @ B6 )
=> ( ( ( insert_set_a @ X @ A7 )
= ( insert_set_a @ X @ B6 ) )
= ( A7 = B6 ) ) ) ) ).
% insert_ident
thf(fact_746_insert__ident,axiom,
! [X: a,A7: set_a,B6: set_a] :
( ~ ( member_a @ X @ A7 )
=> ( ~ ( member_a @ X @ B6 )
=> ( ( ( insert_a @ X @ A7 )
= ( insert_a @ X @ B6 ) )
= ( A7 = B6 ) ) ) ) ).
% insert_ident
thf(fact_747_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A7 )
=> ~ ! [B8: set_Product_prod_a_a] :
( ( A7
= ( insert4534936382041156343od_a_a @ X @ B8 ) )
=> ( member1426531477525435216od_a_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_748_Set_Oset__insert,axiom,
! [X: option_a,A7: set_option_a] :
( ( member_option_a @ X @ A7 )
=> ~ ! [B8: set_option_a] :
( ( A7
= ( insert_option_a @ X @ B8 ) )
=> ( member_option_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_749_Set_Oset__insert,axiom,
! [X: set_a,A7: set_set_a] :
( ( member_set_a @ X @ A7 )
=> ~ ! [B8: set_set_a] :
( ( A7
= ( insert_set_a @ X @ B8 ) )
=> ( member_set_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_750_Set_Oset__insert,axiom,
! [X: a,A7: set_a] :
( ( member_a @ X @ A7 )
=> ~ ! [B8: set_a] :
( ( A7
= ( insert_a @ X @ B8 ) )
=> ( member_a @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_751_insertI2,axiom,
! [A: product_prod_a_a,B6: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ B6 )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_752_insertI2,axiom,
! [A: option_a,B6: set_option_a,B: option_a] :
( ( member_option_a @ A @ B6 )
=> ( member_option_a @ A @ ( insert_option_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_753_insertI2,axiom,
! [A: set_a,B6: set_set_a,B: set_a] :
( ( member_set_a @ A @ B6 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_754_insertI2,axiom,
! [A: a,B6: set_a,B: a] :
( ( member_a @ A @ B6 )
=> ( member_a @ A @ ( insert_a @ B @ B6 ) ) ) ).
% insertI2
thf(fact_755_insertI1,axiom,
! [A: product_prod_a_a,B6: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B6 ) ) ).
% insertI1
thf(fact_756_insertI1,axiom,
! [A: option_a,B6: set_option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ B6 ) ) ).
% insertI1
thf(fact_757_insertI1,axiom,
! [A: set_a,B6: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B6 ) ) ).
% insertI1
thf(fact_758_insertI1,axiom,
! [A: a,B6: set_a] : ( member_a @ A @ ( insert_a @ A @ B6 ) ) ).
% insertI1
thf(fact_759_insertE,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B @ A7 ) )
=> ( ( A != B )
=> ( member1426531477525435216od_a_a @ A @ A7 ) ) ) ).
% insertE
thf(fact_760_insertE,axiom,
! [A: option_a,B: option_a,A7: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B @ A7 ) )
=> ( ( A != B )
=> ( member_option_a @ A @ A7 ) ) ) ).
% insertE
thf(fact_761_insertE,axiom,
! [A: set_a,B: set_a,A7: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A7 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A7 ) ) ) ).
% insertE
thf(fact_762_insertE,axiom,
! [A: a,B: a,A7: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A7 ) )
=> ( ( A != B )
=> ( member_a @ A @ A7 ) ) ) ).
% insertE
thf(fact_763_insert__Collect,axiom,
! [A: set_a,P4: set_a > $o] :
( ( insert_set_a @ A @ ( collect_set_a @ P4 ) )
= ( collect_set_a
@ ^ [U: set_a] :
( ( U != A )
=> ( P4 @ U ) ) ) ) ).
% insert_Collect
thf(fact_764_insert__Collect,axiom,
! [A: option_a,P4: option_a > $o] :
( ( insert_option_a @ A @ ( collect_option_a @ P4 ) )
= ( collect_option_a
@ ^ [U: option_a] :
( ( U != A )
=> ( P4 @ U ) ) ) ) ).
% insert_Collect
thf(fact_765_insert__Collect,axiom,
! [A: a,P4: a > $o] :
( ( insert_a @ A @ ( collect_a @ P4 ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P4 @ U ) ) ) ) ).
% insert_Collect
thf(fact_766_insert__compr,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A4: product_prod_a_a,B7: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( X4 = A4 )
| ( member1426531477525435216od_a_a @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_767_insert__compr,axiom,
( insert_set_a
= ( ^ [A4: set_a,B7: set_set_a] :
( collect_set_a
@ ^ [X4: set_a] :
( ( X4 = A4 )
| ( member_set_a @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_768_insert__compr,axiom,
( insert_option_a
= ( ^ [A4: option_a,B7: set_option_a] :
( collect_option_a
@ ^ [X4: option_a] :
( ( X4 = A4 )
| ( member_option_a @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_769_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B7: set_a] :
( collect_a
@ ^ [X4: a] :
( ( X4 = A4 )
| ( member_a @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_770_insert__subsetI,axiom,
! [X: option_a,A7: set_option_a,X7: set_option_a] :
( ( member_option_a @ X @ A7 )
=> ( ( ord_le1955136853071979460tion_a @ X7 @ A7 )
=> ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ X7 ) @ A7 ) ) ) ).
% insert_subsetI
thf(fact_771_insert__subsetI,axiom,
! [X: set_a,A7: set_set_a,X7: set_set_a] :
( ( member_set_a @ X @ A7 )
=> ( ( ord_le3724670747650509150_set_a @ X7 @ A7 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X7 ) @ A7 ) ) ) ).
% insert_subsetI
thf(fact_772_insert__subsetI,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a,X7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ( ord_le746702958409616551od_a_a @ X7 @ A7 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ X7 ) @ A7 ) ) ) ).
% insert_subsetI
thf(fact_773_insert__subsetI,axiom,
! [X: a,A7: set_a,X7: set_a] :
( ( member_a @ X @ A7 )
=> ( ( ord_less_eq_set_a @ X7 @ A7 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X7 ) @ A7 ) ) ) ).
% insert_subsetI
thf(fact_774_subset__insertI2,axiom,
! [A7: set_option_a,B6: set_option_a,B: option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ B6 )
=> ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_775_subset__insertI2,axiom,
! [A7: set_set_a,B6: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_776_subset__insertI2,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_777_subset__insertI2,axiom,
! [A7: set_a,B6: set_a,B: a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ord_less_eq_set_a @ A7 @ ( insert_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_778_subset__insertI,axiom,
! [B6: set_option_a,A: option_a] : ( ord_le1955136853071979460tion_a @ B6 @ ( insert_option_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_779_subset__insertI,axiom,
! [B6: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B6 @ ( insert_set_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_780_subset__insertI,axiom,
! [B6: set_Product_prod_a_a,A: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B6 @ ( insert4534936382041156343od_a_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_781_subset__insertI,axiom,
! [B6: set_a,A: a] : ( ord_less_eq_set_a @ B6 @ ( insert_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_782_subset__insert,axiom,
! [X: option_a,A7: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ X @ A7 )
=> ( ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ X @ B6 ) )
= ( ord_le1955136853071979460tion_a @ A7 @ B6 ) ) ) ).
% subset_insert
thf(fact_783_subset__insert,axiom,
! [X: set_a,A7: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A7 )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ X @ B6 ) )
= ( ord_le3724670747650509150_set_a @ A7 @ B6 ) ) ) ).
% subset_insert
thf(fact_784_subset__insert,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( ord_le746702958409616551od_a_a @ A7 @ B6 ) ) ) ).
% subset_insert
thf(fact_785_subset__insert,axiom,
! [X: a,A7: set_a,B6: set_a] :
( ~ ( member_a @ X @ A7 )
=> ( ( ord_less_eq_set_a @ A7 @ ( insert_a @ X @ B6 ) )
= ( ord_less_eq_set_a @ A7 @ B6 ) ) ) ).
% subset_insert
thf(fact_786_insert__mono,axiom,
! [C3: set_option_a,D: set_option_a,A: option_a] :
( ( ord_le1955136853071979460tion_a @ C3 @ D )
=> ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ C3 ) @ ( insert_option_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_787_insert__mono,axiom,
! [C3: set_set_a,D: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C3 ) @ ( insert_set_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_788_insert__mono,axiom,
! [C3: set_Product_prod_a_a,D: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C3 @ D )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A @ C3 ) @ ( insert4534936382041156343od_a_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_789_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_790_subset__singleton__iff,axiom,
! [X7: set_option_a,A: option_a] :
( ( ord_le1955136853071979460tion_a @ X7 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( ( X7 = bot_bot_set_option_a )
| ( X7
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_791_subset__singleton__iff,axiom,
! [X7: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X7 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X7 = bot_bot_set_set_a )
| ( X7
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_792_subset__singleton__iff,axiom,
! [X7: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X7 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( ( X7 = bot_bo3357376287454694259od_a_a )
| ( X7
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_793_subset__singleton__iff,axiom,
! [X7: set_a,A: a] :
( ( ord_less_eq_set_a @ X7 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X7 = bot_bot_set_a )
| ( X7
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_794_subset__singletonD,axiom,
! [A7: set_option_a,X: option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
=> ( ( A7 = bot_bot_set_option_a )
| ( A7
= ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% subset_singletonD
thf(fact_795_subset__singletonD,axiom,
! [A7: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A7 = bot_bot_set_set_a )
| ( A7
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_796_subset__singletonD,axiom,
! [A7: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A7 = bot_bo3357376287454694259od_a_a )
| ( A7
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_797_subset__singletonD,axiom,
! [A7: set_a,X: a] :
( ( ord_less_eq_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A7 = bot_bot_set_a )
| ( A7
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_798_upd__None__map__le,axiom,
! [F: a > option_a,X: a] : ( map_le_a_a @ ( fun_upd_a_option_a @ F @ X @ none_a ) @ F ) ).
% upd_None_map_le
thf(fact_799_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_800_the__elem__eq,axiom,
! [X: option_a] :
( ( the_elem_option_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= X ) ).
% the_elem_eq
thf(fact_801_the__elem__eq,axiom,
! [X: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_802_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_803_is__singletonI,axiom,
! [X: option_a] : ( is_sin3348965821858909752tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ).
% is_singletonI
thf(fact_804_is__singletonI,axiom,
! [X: set_a] : ( is_singleton_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ).
% is_singletonI
thf(fact_805_refl__on__singleton,axiom,
! [X: a] : ( refl_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_806_refl__on__singleton,axiom,
! [X: option_a] : ( refl_on_option_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X @ X ) @ bot_bo235252021745139059tion_a ) ) ).
% refl_on_singleton
thf(fact_807_refl__on__singleton,axiom,
! [X: set_a] : ( refl_on_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) @ ( insert5612341225869342455_set_a @ ( produc9088192753505129239_set_a @ X @ X ) @ bot_bo5799363139946352499_set_a ) ) ).
% refl_on_singleton
thf(fact_808_refl__onD2,axiom,
! [A7: set_Product_prod_a_a,R2: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A7 @ R2 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R2 )
=> ( member1426531477525435216od_a_a @ Y @ A7 ) ) ) ).
% refl_onD2
thf(fact_809_refl__onD2,axiom,
! [A7: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( refl_on_option_a @ A7 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( member_option_a @ Y @ A7 ) ) ) ).
% refl_onD2
thf(fact_810_refl__onD2,axiom,
! [A7: set_set_a,R2: set_Pr5845495582615845127_set_a,X: set_a,Y: set_a] :
( ( refl_on_set_a @ A7 @ R2 )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X @ Y ) @ R2 )
=> ( member_set_a @ Y @ A7 ) ) ) ).
% refl_onD2
thf(fact_811_refl__onD2,axiom,
! [A7: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A7 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ Y @ A7 ) ) ) ).
% refl_onD2
thf(fact_812_refl__onD1,axiom,
! [A7: set_Product_prod_a_a,R2: set_Pr8600417178894128327od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A7 @ R2 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R2 )
=> ( member1426531477525435216od_a_a @ X @ A7 ) ) ) ).
% refl_onD1
thf(fact_813_refl__onD1,axiom,
! [A7: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( refl_on_option_a @ A7 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( member_option_a @ X @ A7 ) ) ) ).
% refl_onD1
thf(fact_814_refl__onD1,axiom,
! [A7: set_set_a,R2: set_Pr5845495582615845127_set_a,X: set_a,Y: set_a] :
( ( refl_on_set_a @ A7 @ R2 )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X @ Y ) @ R2 )
=> ( member_set_a @ X @ A7 ) ) ) ).
% refl_onD1
thf(fact_815_refl__onD1,axiom,
! [A7: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A7 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ X @ A7 ) ) ) ).
% refl_onD1
thf(fact_816_refl__onD,axiom,
! [A7: set_Product_prod_a_a,R2: set_Pr8600417178894128327od_a_a,A: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A7 @ R2 )
=> ( ( member1426531477525435216od_a_a @ A @ A7 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_817_refl__onD,axiom,
! [A7: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( refl_on_option_a @ A7 @ R2 )
=> ( ( member_option_a @ A @ A7 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_818_refl__onD,axiom,
! [A7: set_set_a,R2: set_Pr5845495582615845127_set_a,A: set_a] :
( ( refl_on_set_a @ A7 @ R2 )
=> ( ( member_set_a @ A @ A7 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_819_refl__onD,axiom,
! [A7: set_a,R2: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ A7 @ R2 )
=> ( ( member_a @ A @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_820_reflI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 )
=> ( refl_on_a @ top_top_set_a @ R2 ) ) ).
% reflI
thf(fact_821_reflI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ X3 ) @ R2 )
=> ( refl_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% reflI
thf(fact_822_reflD,axiom,
! [R2: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ top_top_set_a @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ).
% reflD
thf(fact_823_reflD,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( refl_on_option_a @ top_top_set_option_a @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ).
% reflD
thf(fact_824_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
( A8
= ( insert_a @ ( the_elem_a @ A8 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_825_is__singleton__the__elem,axiom,
( is_sin3348965821858909752tion_a
= ( ^ [A8: set_option_a] :
( A8
= ( insert_option_a @ ( the_elem_option_a @ A8 ) @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_826_is__singleton__the__elem,axiom,
( is_singleton_set_a
= ( ^ [A8: set_set_a] :
( A8
= ( insert_set_a @ ( the_elem_set_a @ A8 ) @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_827_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% refl_on_empty
thf(fact_828_refl__on__empty,axiom,
refl_on_option_a @ bot_bot_set_option_a @ bot_bo235252021745139059tion_a ).
% refl_on_empty
thf(fact_829_refl__on__empty,axiom,
refl_on_set_a @ bot_bot_set_set_a @ bot_bo5799363139946352499_set_a ).
% refl_on_empty
thf(fact_830_is__singletonI_H,axiom,
! [A7: set_Product_prod_a_a] :
( ( A7 != bot_bo3357376287454694259od_a_a )
=> ( ! [X3: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A7 )
=> ( ( member1426531477525435216od_a_a @ Y5 @ A7 )
=> ( X3 = Y5 ) ) )
=> ( is_sin3171834905898671131od_a_a @ A7 ) ) ) ).
% is_singletonI'
thf(fact_831_is__singletonI_H,axiom,
! [A7: set_option_a] :
( ( A7 != bot_bot_set_option_a )
=> ( ! [X3: option_a,Y5: option_a] :
( ( member_option_a @ X3 @ A7 )
=> ( ( member_option_a @ Y5 @ A7 )
=> ( X3 = Y5 ) ) )
=> ( is_sin3348965821858909752tion_a @ A7 ) ) ) ).
% is_singletonI'
thf(fact_832_is__singletonI_H,axiom,
! [A7: set_set_a] :
( ( A7 != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y5: set_a] :
( ( member_set_a @ X3 @ A7 )
=> ( ( member_set_a @ Y5 @ A7 )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_set_a @ A7 ) ) ) ).
% is_singletonI'
thf(fact_833_is__singletonI_H,axiom,
! [A7: set_a] :
( ( A7 != bot_bot_set_a )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ A7 )
=> ( ( member_a @ Y5 @ A7 )
=> ( X3 = Y5 ) ) )
=> ( is_singleton_a @ A7 ) ) ) ).
% is_singletonI'
thf(fact_834_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
? [X4: a] :
( A8
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_835_is__singleton__def,axiom,
( is_sin3348965821858909752tion_a
= ( ^ [A8: set_option_a] :
? [X4: option_a] :
( A8
= ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_def
thf(fact_836_is__singleton__def,axiom,
( is_singleton_set_a
= ( ^ [A8: set_set_a] :
? [X4: set_a] :
( A8
= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_837_is__singletonE,axiom,
! [A7: set_a] :
( ( is_singleton_a @ A7 )
=> ~ ! [X3: a] :
( A7
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_838_is__singletonE,axiom,
! [A7: set_option_a] :
( ( is_sin3348965821858909752tion_a @ A7 )
=> ~ ! [X3: option_a] :
( A7
!= ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ).
% is_singletonE
thf(fact_839_is__singletonE,axiom,
! [A7: set_set_a] :
( ( is_singleton_set_a @ A7 )
=> ~ ! [X3: set_a] :
( A7
!= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% is_singletonE
thf(fact_840_dom__eq__singleton__conv,axiom,
! [F: a > option_a,X: a] :
( ( ( dom_a_a @ F )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ? [V: a] :
( F
= ( fun_upd_a_option_a
@ ^ [X4: a] : none_a
@ X
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_841_dom__eq__singleton__conv,axiom,
! [F: option_a > option_a,X: option_a] :
( ( ( dom_option_a_a @ F )
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
= ( ? [V: a] :
( F
= ( fun_up1079276522633388797tion_a
@ ^ [X4: option_a] : none_a
@ X
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_842_dom__eq__singleton__conv,axiom,
! [F: set_a > option_a,X: set_a] :
( ( ( dom_set_a_a @ F )
= ( insert_set_a @ X @ bot_bot_set_set_a ) )
= ( ? [V: a] :
( F
= ( fun_up3663993102702442083tion_a
@ ^ [X4: set_a] : none_a
@ X
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_843_subset__singleton__iff__Uniq,axiom,
! [A7: set_option_a] :
( ( ? [A4: option_a] : ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ A4 @ bot_bot_set_option_a ) ) )
= ( uniq_option_a
@ ^ [X4: option_a] : ( member_option_a @ X4 @ A7 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_844_subset__singleton__iff__Uniq,axiom,
! [A7: set_set_a] :
( ( ? [A4: set_a] : ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) )
= ( uniq_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A7 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_845_subset__singleton__iff__Uniq,axiom,
! [A7: set_Product_prod_a_a] :
( ( ? [A4: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ A4 @ bot_bo3357376287454694259od_a_a ) ) )
= ( uniq_P5168762820937118380od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A7 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_846_subset__singleton__iff__Uniq,axiom,
! [A7: set_a] :
( ( ? [A4: a] : ( ord_less_eq_set_a @ A7 @ ( insert_a @ A4 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X4: a] : ( member_a @ X4 @ A7 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_847_these__insert__Some,axiom,
! [X: option_a,A7: set_option_option_a] :
( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X ) @ A7 ) )
= ( insert_option_a @ X @ ( these_option_a @ A7 ) ) ) ).
% these_insert_Some
thf(fact_848_these__insert__Some,axiom,
! [X: a,A7: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X ) @ A7 ) )
= ( insert_a @ X @ ( these_a @ A7 ) ) ) ).
% these_insert_Some
thf(fact_849_these__empty,axiom,
( ( these_option_a @ bot_bo4163488203964334806tion_a )
= bot_bot_set_option_a ) ).
% these_empty
thf(fact_850_these__empty,axiom,
( ( these_set_a @ bot_bo6591985476404692464_set_a )
= bot_bot_set_set_a ) ).
% these_empty
thf(fact_851_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_852_dom__eq__empty__conv,axiom,
! [F: a > option_a] :
( ( ( dom_a_a @ F )
= bot_bot_set_a )
= ( F
= ( ^ [X4: a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_853_dom__eq__empty__conv,axiom,
! [F: option_a > option_a] :
( ( ( dom_option_a_a @ F )
= bot_bot_set_option_a )
= ( F
= ( ^ [X4: option_a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_854_dom__eq__empty__conv,axiom,
! [F: set_a > option_a] :
( ( ( dom_set_a_a @ F )
= bot_bot_set_set_a )
= ( F
= ( ^ [X4: set_a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_855_fun__upd__None__if__notin__dom,axiom,
! [K: product_prod_a_a,M: product_prod_a_a > option_a] :
( ~ ( member1426531477525435216od_a_a @ K @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
=> ( ( fun_up8298456451713467738tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_856_fun__upd__None__if__notin__dom,axiom,
! [K: option_a,M: option_a > option_a] :
( ~ ( member_option_a @ K @ ( dom_option_a_a @ M ) )
=> ( ( fun_up1079276522633388797tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_857_fun__upd__None__if__notin__dom,axiom,
! [K: set_a,M: set_a > option_a] :
( ~ ( member_set_a @ K @ ( dom_set_a_a @ M ) )
=> ( ( fun_up3663993102702442083tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_858_fun__upd__None__if__notin__dom,axiom,
! [K: a,M: a > option_a] :
( ~ ( member_a @ K @ ( dom_a_a @ M ) )
=> ( ( fun_upd_a_option_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_859_dom__const,axiom,
! [F: a > a] :
( ( dom_a_a
@ ^ [X4: a] : ( some_a @ ( F @ X4 ) ) )
= top_top_set_a ) ).
% dom_const
thf(fact_860_dom__const,axiom,
! [F: option_a > a] :
( ( dom_option_a_a
@ ^ [X4: option_a] : ( some_a @ ( F @ X4 ) ) )
= top_top_set_option_a ) ).
% dom_const
thf(fact_861_dom__empty,axiom,
( ( dom_a_a
@ ^ [X4: a] : none_a )
= bot_bot_set_a ) ).
% dom_empty
thf(fact_862_dom__empty,axiom,
( ( dom_option_a_a
@ ^ [X4: option_a] : none_a )
= bot_bot_set_option_a ) ).
% dom_empty
thf(fact_863_dom__empty,axiom,
( ( dom_set_a_a
@ ^ [X4: set_a] : none_a )
= bot_bot_set_set_a ) ).
% dom_empty
thf(fact_864_these__insert__None,axiom,
! [A7: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A7 ) )
= ( these_a @ A7 ) ) ).
% these_insert_None
thf(fact_865_dom__def,axiom,
( dom_set_a_a
= ( ^ [M2: set_a > option_a] :
( collect_set_a
@ ^ [A4: set_a] :
( ( M2 @ A4 )
!= none_a ) ) ) ) ).
% dom_def
thf(fact_866_dom__def,axiom,
( dom_option_a_a
= ( ^ [M2: option_a > option_a] :
( collect_option_a
@ ^ [A4: option_a] :
( ( M2 @ A4 )
!= none_a ) ) ) ) ).
% dom_def
thf(fact_867_dom__def,axiom,
( dom_a_a
= ( ^ [M2: a > option_a] :
( collect_a
@ ^ [A4: a] :
( ( M2 @ A4 )
!= none_a ) ) ) ) ).
% dom_def
thf(fact_868_domIff,axiom,
! [A: product_prod_a_a,M: product_prod_a_a > option_a] :
( ( member1426531477525435216od_a_a @ A @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_869_domIff,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_870_domIff,axiom,
! [A: set_a,M: set_a > option_a] :
( ( member_set_a @ A @ ( dom_set_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_871_domIff,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_872_domI,axiom,
! [M: product_prod_a_a > option_a,A: product_prod_a_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member1426531477525435216od_a_a @ A @ ( dom_Pr6378187988305063785_a_a_a @ M ) ) ) ).
% domI
thf(fact_873_domI,axiom,
! [M: option_a > option_a,A: option_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_option_a @ A @ ( dom_option_a_a @ M ) ) ) ).
% domI
thf(fact_874_domI,axiom,
! [M: set_a > option_a,A: set_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_set_a @ A @ ( dom_set_a_a @ M ) ) ) ).
% domI
thf(fact_875_domI,axiom,
! [M: a > option_a,A: a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_a @ A @ ( dom_a_a @ M ) ) ) ).
% domI
thf(fact_876_domD,axiom,
! [A: product_prod_a_a,M: product_prod_a_a > option_a] :
( ( member1426531477525435216od_a_a @ A @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
=> ? [B3: a] :
( ( M @ A )
= ( some_a @ B3 ) ) ) ).
% domD
thf(fact_877_domD,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
=> ? [B3: a] :
( ( M @ A )
= ( some_a @ B3 ) ) ) ).
% domD
thf(fact_878_domD,axiom,
! [A: set_a,M: set_a > option_a] :
( ( member_set_a @ A @ ( dom_set_a_a @ M ) )
=> ? [B3: a] :
( ( M @ A )
= ( some_a @ B3 ) ) ) ).
% domD
thf(fact_879_domD,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
=> ? [B3: a] :
( ( M @ A )
= ( some_a @ B3 ) ) ) ).
% domD
thf(fact_880_insert__dom,axiom,
! [F: a > option_a,X: a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_a @ X @ ( dom_a_a @ F ) )
= ( dom_a_a @ F ) ) ) ).
% insert_dom
thf(fact_881_insert__dom,axiom,
! [F: option_a > option_a,X: option_a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_option_a @ X @ ( dom_option_a_a @ F ) )
= ( dom_option_a_a @ F ) ) ) ).
% insert_dom
thf(fact_882_in__these__eq,axiom,
! [X: product_prod_a_a,A7: set_op7160277562814721357od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( these_5100388957577570148od_a_a @ A7 ) )
= ( member8183384484874023062od_a_a @ ( some_P3592067295195376908od_a_a @ X ) @ A7 ) ) ).
% in_these_eq
thf(fact_883_in__these__eq,axiom,
! [X: option_a,A7: set_option_option_a] :
( ( member_option_a @ X @ ( these_option_a @ A7 ) )
= ( member5113800082084363315tion_a @ ( some_option_a @ X ) @ A7 ) ) ).
% in_these_eq
thf(fact_884_in__these__eq,axiom,
! [X: set_a,A7: set_option_set_a] :
( ( member_set_a @ X @ ( these_set_a @ A7 ) )
= ( member_option_set_a @ ( some_set_a @ X ) @ A7 ) ) ).
% in_these_eq
thf(fact_885_in__these__eq,axiom,
! [X: a,A7: set_option_a] :
( ( member_a @ X @ ( these_a @ A7 ) )
= ( member_option_a @ ( some_a @ X ) @ A7 ) ) ).
% in_these_eq
thf(fact_886_map__le__implies__dom__le,axiom,
! [F: a > option_a,G: a > option_a] :
( ( map_le_a_a @ F @ G )
=> ( ord_less_eq_set_a @ ( dom_a_a @ F ) @ ( dom_a_a @ G ) ) ) ).
% map_le_implies_dom_le
thf(fact_887_these__not__empty__eq,axiom,
! [B6: set_option_option_a] :
( ( ( these_option_a @ B6 )
!= bot_bot_set_option_a )
= ( ( B6 != bot_bo4163488203964334806tion_a )
& ( B6
!= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_888_these__not__empty__eq,axiom,
! [B6: set_option_set_a] :
( ( ( these_set_a @ B6 )
!= bot_bot_set_set_a )
= ( ( B6 != bot_bo6591985476404692464_set_a )
& ( B6
!= ( insert_option_set_a @ none_set_a @ bot_bo6591985476404692464_set_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_889_these__not__empty__eq,axiom,
! [B6: set_option_a] :
( ( ( these_a @ B6 )
!= bot_bot_set_a )
= ( ( B6 != bot_bot_set_option_a )
& ( B6
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_890_these__empty__eq,axiom,
! [B6: set_option_option_a] :
( ( ( these_option_a @ B6 )
= bot_bot_set_option_a )
= ( ( B6 = bot_bo4163488203964334806tion_a )
| ( B6
= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_empty_eq
thf(fact_891_these__empty__eq,axiom,
! [B6: set_option_set_a] :
( ( ( these_set_a @ B6 )
= bot_bot_set_set_a )
= ( ( B6 = bot_bo6591985476404692464_set_a )
| ( B6
= ( insert_option_set_a @ none_set_a @ bot_bo6591985476404692464_set_a ) ) ) ) ).
% these_empty_eq
thf(fact_892_these__empty__eq,axiom,
! [B6: set_option_a] :
( ( ( these_a @ B6 )
= bot_bot_set_a )
= ( ( B6 = bot_bot_set_option_a )
| ( B6
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_893_dom__fun__upd,axiom,
! [Y: option_a,F: option_a > option_a,X: option_a] :
( ( ( Y = none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( insert_option_a @ X @ ( dom_option_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_894_dom__fun__upd,axiom,
! [Y: option_a,F: set_a > option_a,X: set_a] :
( ( ( Y = none_a )
=> ( ( dom_set_a_a @ ( fun_up3663993102702442083tion_a @ F @ X @ Y ) )
= ( minus_5736297505244876581_set_a @ ( dom_set_a_a @ F ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_set_a_a @ ( fun_up3663993102702442083tion_a @ F @ X @ Y ) )
= ( insert_set_a @ X @ ( dom_set_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_895_dom__fun__upd,axiom,
! [Y: option_a,F: a > option_a,X: a] :
( ( ( Y = none_a )
=> ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X @ Y ) )
= ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X @ Y ) )
= ( insert_a @ X @ ( dom_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_896_alt__ex1E_H,axiom,
! [P4: a > $o] :
( ? [X8: a] :
( ( P4 @ X8 )
& ! [Y5: a] :
( ( P4 @ Y5 )
=> ( Y5 = X8 ) ) )
=> ~ ( ? [X_1: a] : ( P4 @ X_1 )
=> ~ ( uniq_a @ P4 ) ) ) ).
% alt_ex1E'
thf(fact_897_ex1__iff__ex__Uniq,axiom,
( ex1_a
= ( ^ [P6: a > $o] :
( ? [X9: a] : ( P6 @ X9 )
& ( uniq_a @ P6 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_898_DiffI,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A7 )
=> ( ~ ( member1426531477525435216od_a_a @ C @ B6 )
=> ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) ) ) ) ).
% DiffI
thf(fact_899_DiffI,axiom,
! [C: option_a,A7: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ A7 )
=> ( ~ ( member_option_a @ C @ B6 )
=> ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) ) ) ) ).
% DiffI
thf(fact_900_DiffI,axiom,
! [C: set_a,A7: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ A7 )
=> ( ~ ( member_set_a @ C @ B6 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) ) ) ) ).
% DiffI
thf(fact_901_DiffI,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ A7 )
=> ( ~ ( member_a @ C @ B6 )
=> ( member_a @ C @ ( minus_minus_set_a @ A7 @ B6 ) ) ) ) ).
% DiffI
thf(fact_902_Diff__iff,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) )
= ( ( member1426531477525435216od_a_a @ C @ A7 )
& ~ ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_903_Diff__iff,axiom,
! [C: option_a,A7: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) )
= ( ( member_option_a @ C @ A7 )
& ~ ( member_option_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_904_Diff__iff,axiom,
! [C: set_a,A7: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) )
= ( ( member_set_a @ C @ A7 )
& ~ ( member_set_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_905_Diff__iff,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A7 @ B6 ) )
= ( ( member_a @ C @ A7 )
& ~ ( member_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_906_Diff__idemp,axiom,
! [A7: set_a,B6: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A7 @ B6 ) @ B6 )
= ( minus_minus_set_a @ A7 @ B6 ) ) ).
% Diff_idemp
thf(fact_907_Diff__cancel,axiom,
! [A7: set_option_a] :
( ( minus_1574173051537231627tion_a @ A7 @ A7 )
= bot_bot_set_option_a ) ).
% Diff_cancel
thf(fact_908_Diff__cancel,axiom,
! [A7: set_set_a] :
( ( minus_5736297505244876581_set_a @ A7 @ A7 )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_909_Diff__cancel,axiom,
! [A7: set_a] :
( ( minus_minus_set_a @ A7 @ A7 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_910_empty__Diff,axiom,
! [A7: set_option_a] :
( ( minus_1574173051537231627tion_a @ bot_bot_set_option_a @ A7 )
= bot_bot_set_option_a ) ).
% empty_Diff
thf(fact_911_empty__Diff,axiom,
! [A7: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A7 )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_912_empty__Diff,axiom,
! [A7: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A7 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_913_Diff__empty,axiom,
! [A7: set_option_a] :
( ( minus_1574173051537231627tion_a @ A7 @ bot_bot_set_option_a )
= A7 ) ).
% Diff_empty
thf(fact_914_Diff__empty,axiom,
! [A7: set_set_a] :
( ( minus_5736297505244876581_set_a @ A7 @ bot_bot_set_set_a )
= A7 ) ).
% Diff_empty
thf(fact_915_Diff__empty,axiom,
! [A7: set_a] :
( ( minus_minus_set_a @ A7 @ bot_bot_set_a )
= A7 ) ).
% Diff_empty
thf(fact_916_Diff__insert0,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ( minus_6817036919807184750od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( minus_6817036919807184750od_a_a @ A7 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_917_Diff__insert0,axiom,
! [X: option_a,A7: set_option_a,B6: set_option_a] :
( ~ ( member_option_a @ X @ A7 )
=> ( ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ X @ B6 ) )
= ( minus_1574173051537231627tion_a @ A7 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_918_Diff__insert0,axiom,
! [X: set_a,A7: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A7 )
=> ( ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ B6 ) )
= ( minus_5736297505244876581_set_a @ A7 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_919_Diff__insert0,axiom,
! [X: a,A7: set_a,B6: set_a] :
( ~ ( member_a @ X @ A7 )
=> ( ( minus_minus_set_a @ A7 @ ( insert_a @ X @ B6 ) )
= ( minus_minus_set_a @ A7 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_920_insert__Diff1,axiom,
! [X: product_prod_a_a,B6: set_Product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A7 ) @ B6 )
= ( minus_6817036919807184750od_a_a @ A7 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_921_insert__Diff1,axiom,
! [X: option_a,B6: set_option_a,A7: set_option_a] :
( ( member_option_a @ X @ B6 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A7 ) @ B6 )
= ( minus_1574173051537231627tion_a @ A7 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_922_insert__Diff1,axiom,
! [X: set_a,B6: set_set_a,A7: set_set_a] :
( ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A7 ) @ B6 )
= ( minus_5736297505244876581_set_a @ A7 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_923_insert__Diff1,axiom,
! [X: a,B6: set_a,A7: set_a] :
( ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A7 ) @ B6 )
= ( minus_minus_set_a @ A7 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_924_Diff__UNIV,axiom,
! [A7: set_set_a] :
( ( minus_5736297505244876581_set_a @ A7 @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% Diff_UNIV
thf(fact_925_Diff__UNIV,axiom,
! [A7: set_option_a] :
( ( minus_1574173051537231627tion_a @ A7 @ top_top_set_option_a )
= bot_bot_set_option_a ) ).
% Diff_UNIV
thf(fact_926_Diff__UNIV,axiom,
! [A7: set_a] :
( ( minus_minus_set_a @ A7 @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_927_Diff__eq__empty__iff,axiom,
! [A7: set_option_a,B6: set_option_a] :
( ( ( minus_1574173051537231627tion_a @ A7 @ B6 )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ A7 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_928_Diff__eq__empty__iff,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A7 @ B6 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A7 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_929_Diff__eq__empty__iff,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ( minus_6817036919807184750od_a_a @ A7 @ B6 )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ A7 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_930_Diff__eq__empty__iff,axiom,
! [A7: set_a,B6: set_a] :
( ( ( minus_minus_set_a @ A7 @ B6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A7 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_931_insert__Diff__single,axiom,
! [A: option_a,A7: set_option_a] :
( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( insert_option_a @ A @ A7 ) ) ).
% insert_Diff_single
thf(fact_932_insert__Diff__single,axiom,
! [A: set_a,A7: set_set_a] :
( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( insert_set_a @ A @ A7 ) ) ).
% insert_Diff_single
thf(fact_933_insert__Diff__single,axiom,
! [A: a,A7: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A7 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A7 ) ) ).
% insert_Diff_single
thf(fact_934_insert__Diff__if,axiom,
! [X: product_prod_a_a,B6: set_Product_prod_a_a,A7: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A7 ) @ B6 )
= ( minus_6817036919807184750od_a_a @ A7 @ B6 ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ B6 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A7 ) @ B6 )
= ( insert4534936382041156343od_a_a @ X @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_935_insert__Diff__if,axiom,
! [X: option_a,B6: set_option_a,A7: set_option_a] :
( ( ( member_option_a @ X @ B6 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A7 ) @ B6 )
= ( minus_1574173051537231627tion_a @ A7 @ B6 ) ) )
& ( ~ ( member_option_a @ X @ B6 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A7 ) @ B6 )
= ( insert_option_a @ X @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_936_insert__Diff__if,axiom,
! [X: set_a,B6: set_set_a,A7: set_set_a] :
( ( ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A7 ) @ B6 )
= ( minus_5736297505244876581_set_a @ A7 @ B6 ) ) )
& ( ~ ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A7 ) @ B6 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_937_insert__Diff__if,axiom,
! [X: a,B6: set_a,A7: set_a] :
( ( ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A7 ) @ B6 )
= ( minus_minus_set_a @ A7 @ B6 ) ) )
& ( ~ ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A7 ) @ B6 )
= ( insert_a @ X @ ( minus_minus_set_a @ A7 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_938_Diff__mono,axiom,
! [A7: set_set_a,C3: set_set_a,D: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B6 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) @ ( minus_5736297505244876581_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_939_Diff__mono,axiom,
! [A7: set_Product_prod_a_a,C3: set_Product_prod_a_a,D: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ C3 )
=> ( ( ord_le746702958409616551od_a_a @ D @ B6 )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) @ ( minus_6817036919807184750od_a_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_940_Diff__mono,axiom,
! [A7: set_a,C3: set_a,D: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B6 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A7 @ B6 ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_941_Diff__subset,axiom,
! [A7: set_set_a,B6: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) @ A7 ) ).
% Diff_subset
thf(fact_942_Diff__subset,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) @ A7 ) ).
% Diff_subset
thf(fact_943_Diff__subset,axiom,
! [A7: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A7 @ B6 ) @ A7 ) ).
% Diff_subset
thf(fact_944_double__diff,axiom,
! [A7: set_set_a,B6: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ C3 )
=> ( ( minus_5736297505244876581_set_a @ B6 @ ( minus_5736297505244876581_set_a @ C3 @ A7 ) )
= A7 ) ) ) ).
% double_diff
thf(fact_945_double__diff,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a,C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ( ord_le746702958409616551od_a_a @ B6 @ C3 )
=> ( ( minus_6817036919807184750od_a_a @ B6 @ ( minus_6817036919807184750od_a_a @ C3 @ A7 ) )
= A7 ) ) ) ).
% double_diff
thf(fact_946_double__diff,axiom,
! [A7: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( minus_minus_set_a @ B6 @ ( minus_minus_set_a @ C3 @ A7 ) )
= A7 ) ) ) ).
% double_diff
thf(fact_947_set__diff__eq,axiom,
( minus_6817036919807184750od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A8 )
& ~ ( member1426531477525435216od_a_a @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_948_set__diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
( collect_set_a
@ ^ [X4: set_a] :
( ( member_set_a @ X4 @ A8 )
& ~ ( member_set_a @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_949_set__diff__eq,axiom,
( minus_1574173051537231627tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ A8 )
& ~ ( member_option_a @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_950_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A8: set_a,B7: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A8 )
& ~ ( member_a @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_951_DiffE,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A7 )
=> ( member1426531477525435216od_a_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_952_DiffE,axiom,
! [C: option_a,A7: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) )
=> ~ ( ( member_option_a @ C @ A7 )
=> ( member_option_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_953_DiffE,axiom,
! [C: set_a,A7: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) )
=> ~ ( ( member_set_a @ C @ A7 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_954_DiffE,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A7 @ B6 ) )
=> ~ ( ( member_a @ C @ A7 )
=> ( member_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_955_DiffD1,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) )
=> ( member1426531477525435216od_a_a @ C @ A7 ) ) ).
% DiffD1
thf(fact_956_DiffD1,axiom,
! [C: option_a,A7: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) )
=> ( member_option_a @ C @ A7 ) ) ).
% DiffD1
thf(fact_957_DiffD1,axiom,
! [C: set_a,A7: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) )
=> ( member_set_a @ C @ A7 ) ) ).
% DiffD1
thf(fact_958_DiffD1,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A7 @ B6 ) )
=> ( member_a @ C @ A7 ) ) ).
% DiffD1
thf(fact_959_DiffD2,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A7 @ B6 ) )
=> ~ ( member1426531477525435216od_a_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_960_DiffD2,axiom,
! [C: option_a,A7: set_option_a,B6: set_option_a] :
( ( member_option_a @ C @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) )
=> ~ ( member_option_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_961_DiffD2,axiom,
! [C: set_a,A7: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) )
=> ~ ( member_set_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_962_DiffD2,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A7 @ B6 ) )
=> ~ ( member_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_963_Diff__insert__absorb,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ( minus_6817036919807184750od_a_a @ ( insert4534936382041156343od_a_a @ X @ A7 ) @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
= A7 ) ) ).
% Diff_insert_absorb
thf(fact_964_Diff__insert__absorb,axiom,
! [X: option_a,A7: set_option_a] :
( ~ ( member_option_a @ X @ A7 )
=> ( ( minus_1574173051537231627tion_a @ ( insert_option_a @ X @ A7 ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= A7 ) ) ).
% Diff_insert_absorb
thf(fact_965_Diff__insert__absorb,axiom,
! [X: set_a,A7: set_set_a] :
( ~ ( member_set_a @ X @ A7 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A7 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A7 ) ) ).
% Diff_insert_absorb
thf(fact_966_Diff__insert__absorb,axiom,
! [X: a,A7: set_a] :
( ~ ( member_a @ X @ A7 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A7 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A7 ) ) ).
% Diff_insert_absorb
thf(fact_967_Diff__insert2,axiom,
! [A7: set_option_a,A: option_a,B6: set_option_a] :
( ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ A @ B6 ) )
= ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_968_Diff__insert2,axiom,
! [A7: set_set_a,A: set_a,B6: set_set_a] :
( ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A @ B6 ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_969_Diff__insert2,axiom,
! [A7: set_a,A: a,B6: set_a] :
( ( minus_minus_set_a @ A7 @ ( insert_a @ A @ B6 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_970_insert__Diff,axiom,
! [A: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A7 )
=> ( ( insert4534936382041156343od_a_a @ A @ ( minus_6817036919807184750od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) )
= A7 ) ) ).
% insert_Diff
thf(fact_971_insert__Diff,axiom,
! [A: option_a,A7: set_option_a] :
( ( member_option_a @ A @ A7 )
=> ( ( insert_option_a @ A @ ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= A7 ) ) ).
% insert_Diff
thf(fact_972_insert__Diff,axiom,
! [A: set_a,A7: set_set_a] :
( ( member_set_a @ A @ A7 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A7 ) ) ).
% insert_Diff
thf(fact_973_insert__Diff,axiom,
! [A: a,A7: set_a] :
( ( member_a @ A @ A7 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A7 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A7 ) ) ).
% insert_Diff
thf(fact_974_Diff__insert,axiom,
! [A7: set_option_a,A: option_a,B6: set_option_a] :
( ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ A @ B6 ) )
= ( minus_1574173051537231627tion_a @ ( minus_1574173051537231627tion_a @ A7 @ B6 ) @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ).
% Diff_insert
thf(fact_975_Diff__insert,axiom,
! [A7: set_set_a,A: set_a,B6: set_set_a] :
( ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A @ B6 ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A7 @ B6 ) @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ).
% Diff_insert
thf(fact_976_Diff__insert,axiom,
! [A7: set_a,A: a,B6: set_a] :
( ( minus_minus_set_a @ A7 @ ( insert_a @ A @ B6 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A7 @ B6 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_977_subset__Diff__insert,axiom,
! [A7: set_option_a,B6: set_option_a,X: option_a,C3: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( minus_1574173051537231627tion_a @ B6 @ ( insert_option_a @ X @ C3 ) ) )
= ( ( ord_le1955136853071979460tion_a @ A7 @ ( minus_1574173051537231627tion_a @ B6 @ C3 ) )
& ~ ( member_option_a @ X @ A7 ) ) ) ).
% subset_Diff_insert
thf(fact_978_subset__Diff__insert,axiom,
! [A7: set_set_a,B6: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ ( minus_5736297505244876581_set_a @ B6 @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A7 @ ( minus_5736297505244876581_set_a @ B6 @ C3 ) )
& ~ ( member_set_a @ X @ A7 ) ) ) ).
% subset_Diff_insert
thf(fact_979_subset__Diff__insert,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a,X: product_prod_a_a,C3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ ( minus_6817036919807184750od_a_a @ B6 @ ( insert4534936382041156343od_a_a @ X @ C3 ) ) )
= ( ( ord_le746702958409616551od_a_a @ A7 @ ( minus_6817036919807184750od_a_a @ B6 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ X @ A7 ) ) ) ).
% subset_Diff_insert
thf(fact_980_subset__Diff__insert,axiom,
! [A7: set_a,B6: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( minus_minus_set_a @ B6 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A7 @ ( minus_minus_set_a @ B6 @ C3 ) )
& ~ ( member_a @ X @ A7 ) ) ) ).
% subset_Diff_insert
thf(fact_981_subset__insert__iff,axiom,
! [A7: set_option_a,X: option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ X @ B6 ) )
= ( ( ( member_option_a @ X @ A7 )
=> ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B6 ) )
& ( ~ ( member_option_a @ X @ A7 )
=> ( ord_le1955136853071979460tion_a @ A7 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_982_subset__insert__iff,axiom,
! [A7: set_set_a,X: set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ X @ B6 ) )
= ( ( ( member_set_a @ X @ A7 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B6 ) )
& ( ~ ( member_set_a @ X @ A7 )
=> ( ord_le3724670747650509150_set_a @ A7 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_983_subset__insert__iff,axiom,
! [A7: set_Product_prod_a_a,X: product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ B6 ) )
= ( ( ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) @ B6 ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ord_le746702958409616551od_a_a @ A7 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_984_subset__insert__iff,axiom,
! [A7: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( insert_a @ X @ B6 ) )
= ( ( ( member_a @ X @ A7 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) )
& ( ~ ( member_a @ X @ A7 )
=> ( ord_less_eq_set_a @ A7 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_985_Diff__single__insert,axiom,
! [A7: set_option_a,X: option_a,B6: set_option_a] :
( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A7 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) @ B6 )
=> ( ord_le1955136853071979460tion_a @ A7 @ ( insert_option_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_986_Diff__single__insert,axiom,
! [A7: set_set_a,X: set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B6 )
=> ( ord_le3724670747650509150_set_a @ A7 @ ( insert_set_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_987_Diff__single__insert,axiom,
! [A7: set_Product_prod_a_a,X: product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( minus_6817036919807184750od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) @ B6 )
=> ( ord_le746702958409616551od_a_a @ A7 @ ( insert4534936382041156343od_a_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_988_Diff__single__insert,axiom,
! [A7: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 )
=> ( ord_less_eq_set_a @ A7 @ ( insert_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_989_dom__minus,axiom,
! [F: option_a > option_a,X: option_a,A7: set_option_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ A7 ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ A7 ) ) ) ).
% dom_minus
thf(fact_990_dom__minus,axiom,
! [F: a > option_a,X: a,A7: set_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X @ A7 ) )
= ( minus_minus_set_a @ ( dom_a_a @ F ) @ A7 ) ) ) ).
% dom_minus
thf(fact_991_diff__shunt__var,axiom,
! [X: set_option_a,Y: set_option_a] :
( ( ( minus_1574173051537231627tion_a @ X @ Y )
= bot_bot_set_option_a )
= ( ord_le1955136853071979460tion_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_992_diff__shunt__var,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( minus_minus_a_o @ X @ Y )
= bot_bot_a_o )
= ( ord_less_eq_a_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_993_diff__shunt__var,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ( minus_minus_o_set_a @ X @ Y )
= bot_bot_o_set_a )
= ( ord_less_eq_o_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_994_diff__shunt__var,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_995_diff__shunt__var,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( minus_6817036919807184750od_a_a @ X @ Y )
= bot_bo3357376287454694259od_a_a )
= ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_996_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_997_fun__upd__None__restrict,axiom,
! [X: product_prod_a_a,D: set_Product_prod_a_a,M: product_prod_a_a > option_a] :
( ( ( member1426531477525435216od_a_a @ X @ D )
=> ( ( fun_up8298456451713467738tion_a @ ( restri3846233575700461639_a_a_a @ M @ D ) @ X @ none_a )
= ( restri3846233575700461639_a_a_a @ M @ ( minus_6817036919807184750od_a_a @ D @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X @ D )
=> ( ( fun_up8298456451713467738tion_a @ ( restri3846233575700461639_a_a_a @ M @ D ) @ X @ none_a )
= ( restri3846233575700461639_a_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_998_fun__upd__None__restrict,axiom,
! [X: option_a,D: set_option_a,M: option_a > option_a] :
( ( ( member_option_a @ X @ D )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M @ ( minus_1574173051537231627tion_a @ D @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) )
& ( ~ ( member_option_a @ X @ D )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_999_fun__upd__None__restrict,axiom,
! [X: set_a,D: set_set_a,M: set_a > option_a] :
( ( ( member_set_a @ X @ D )
=> ( ( fun_up3663993102702442083tion_a @ ( restrict_map_set_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_set_a_a @ M @ ( minus_5736297505244876581_set_a @ D @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) )
& ( ~ ( member_set_a @ X @ D )
=> ( ( fun_up3663993102702442083tion_a @ ( restrict_map_set_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_set_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1000_fun__upd__None__restrict,axiom,
! [X: a,D: set_a,M: a > option_a] :
( ( ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
& ( ~ ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_1001_restrict__out,axiom,
! [X: product_prod_a_a,A7: set_Product_prod_a_a,M: product_prod_a_a > option_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A7 )
=> ( ( restri3846233575700461639_a_a_a @ M @ A7 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_1002_restrict__out,axiom,
! [X: option_a,A7: set_option_a,M: option_a > option_a] :
( ~ ( member_option_a @ X @ A7 )
=> ( ( restri3984065703976872170on_a_a @ M @ A7 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_1003_restrict__out,axiom,
! [X: set_a,A7: set_set_a,M: set_a > option_a] :
( ~ ( member_set_a @ X @ A7 )
=> ( ( restrict_map_set_a_a @ M @ A7 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_1004_restrict__out,axiom,
! [X: a,A7: set_a,M: a > option_a] :
( ~ ( member_a @ X @ A7 )
=> ( ( restrict_map_a_a @ M @ A7 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_1005_restrict__map__empty,axiom,
! [D: set_a] :
( ( restrict_map_a_a
@ ^ [X4: a] : none_a
@ D )
= ( ^ [X4: a] : none_a ) ) ).
% restrict_map_empty
thf(fact_1006_restrict__map__to__empty,axiom,
! [M: a > option_a] :
( ( restrict_map_a_a @ M @ bot_bot_set_a )
= ( ^ [X4: a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_1007_restrict__map__to__empty,axiom,
! [M: option_a > option_a] :
( ( restri3984065703976872170on_a_a @ M @ bot_bot_set_option_a )
= ( ^ [X4: option_a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_1008_restrict__map__to__empty,axiom,
! [M: set_a > option_a] :
( ( restrict_map_set_a_a @ M @ bot_bot_set_set_a )
= ( ^ [X4: set_a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_1009_minus__set__def,axiom,
( minus_6817036919807184750od_a_a
= ( ^ [A8: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( minus_4793868396798367471_a_a_o
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A8 )
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1010_minus__set__def,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A8: set_set_a,B7: set_set_a] :
( collect_set_a
@ ( minus_minus_set_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A8 )
@ ^ [X4: set_a] : ( member_set_a @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1011_minus__set__def,axiom,
( minus_1574173051537231627tion_a
= ( ^ [A8: set_option_a,B7: set_option_a] :
( collect_option_a
@ ( minus_1105141371419579794on_a_o
@ ^ [X4: option_a] : ( member_option_a @ X4 @ A8 )
@ ^ [X4: option_a] : ( member_option_a @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1012_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A8: set_a,B7: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A8 )
@ ^ [X4: a] : ( member_a @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1013_restrict__map__def,axiom,
( restri3846233575700461639_a_a_a
= ( ^ [M2: product_prod_a_a > option_a,A8: set_Product_prod_a_a,X4: product_prod_a_a] : ( if_option_a @ ( member1426531477525435216od_a_a @ X4 @ A8 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_1014_restrict__map__def,axiom,
( restri3984065703976872170on_a_a
= ( ^ [M2: option_a > option_a,A8: set_option_a,X4: option_a] : ( if_option_a @ ( member_option_a @ X4 @ A8 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_1015_restrict__map__def,axiom,
( restrict_map_set_a_a
= ( ^ [M2: set_a > option_a,A8: set_set_a,X4: set_a] : ( if_option_a @ ( member_set_a @ X4 @ A8 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_1016_restrict__map__def,axiom,
( restrict_map_a_a
= ( ^ [M2: a > option_a,A8: set_a,X4: a] : ( if_option_a @ ( member_a @ X4 @ A8 ) @ ( M2 @ X4 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_1017_ran__restrictD,axiom,
! [Y: a,M: a > option_a,A7: set_a] :
( ( member_a @ Y @ ( ran_a_a @ ( restrict_map_a_a @ M @ A7 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A7 )
& ( ( M @ X3 )
= ( some_a @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_1018_graph__restrictD_I2_J,axiom,
! [K: a,V3: a,M: a > option_a,A7: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K @ V3 ) @ ( graph_a_a @ ( restrict_map_a_a @ M @ A7 ) ) )
=> ( ( M @ K )
= ( some_a @ V3 ) ) ) ).
% graph_restrictD(2)
thf(fact_1019_restrict__upd__same,axiom,
! [M: option_a > option_a,X: option_a,Y: a] :
( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( restri3984065703976872170on_a_a @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% restrict_upd_same
thf(fact_1020_restrict__upd__same,axiom,
! [M: set_a > option_a,X: set_a,Y: a] :
( ( restrict_map_set_a_a @ ( fun_up3663993102702442083tion_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( restrict_map_set_a_a @ M @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% restrict_upd_same
thf(fact_1021_restrict__upd__same,axiom,
! [M: a > option_a,X: a,Y: a] :
( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( restrict_map_a_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% restrict_upd_same
thf(fact_1022_remove__def,axiom,
( remove_option_a
= ( ^ [X4: option_a,A8: set_option_a] : ( minus_1574173051537231627tion_a @ A8 @ ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ) ).
% remove_def
thf(fact_1023_remove__def,axiom,
( remove_set_a
= ( ^ [X4: set_a,A8: set_set_a] : ( minus_5736297505244876581_set_a @ A8 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ).
% remove_def
thf(fact_1024_remove__def,axiom,
( remove_a
= ( ^ [X4: a,A8: set_a] : ( minus_minus_set_a @ A8 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_1025_restrict__complement__singleton__eq,axiom,
! [F: option_a > option_a,X: option_a] :
( ( restri3984065703976872170on_a_a @ F @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( fun_up1079276522633388797tion_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_1026_restrict__complement__singleton__eq,axiom,
! [F: set_a > option_a,X: set_a] :
( ( restrict_map_set_a_a @ F @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( fun_up3663993102702442083tion_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_1027_restrict__complement__singleton__eq,axiom,
! [F: a > option_a,X: a] :
( ( restrict_map_a_a @ F @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( fun_upd_a_option_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_1028_ComplI,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ C @ A7 )
=> ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A7 ) ) ) ).
% ComplI
thf(fact_1029_ComplI,axiom,
! [C: option_a,A7: set_option_a] :
( ~ ( member_option_a @ C @ A7 )
=> ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A7 ) ) ) ).
% ComplI
thf(fact_1030_ComplI,axiom,
! [C: set_a,A7: set_set_a] :
( ~ ( member_set_a @ C @ A7 )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A7 ) ) ) ).
% ComplI
thf(fact_1031_ComplI,axiom,
! [C: a,A7: set_a] :
( ~ ( member_a @ C @ A7 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A7 ) ) ) ).
% ComplI
thf(fact_1032_Compl__iff,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A7 ) )
= ( ~ ( member1426531477525435216od_a_a @ C @ A7 ) ) ) ).
% Compl_iff
thf(fact_1033_Compl__iff,axiom,
! [C: option_a,A7: set_option_a] :
( ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A7 ) )
= ( ~ ( member_option_a @ C @ A7 ) ) ) ).
% Compl_iff
thf(fact_1034_Compl__iff,axiom,
! [C: set_a,A7: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A7 ) )
= ( ~ ( member_set_a @ C @ A7 ) ) ) ).
% Compl_iff
thf(fact_1035_Compl__iff,axiom,
! [C: a,A7: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A7 ) )
= ( ~ ( member_a @ C @ A7 ) ) ) ).
% Compl_iff
thf(fact_1036_Compl__eq__Compl__iff,axiom,
! [A7: set_a,B6: set_a] :
( ( ( uminus_uminus_set_a @ A7 )
= ( uminus_uminus_set_a @ B6 ) )
= ( A7 = B6 ) ) ).
% Compl_eq_Compl_iff
thf(fact_1037_member__remove,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( remove8198300757409973004od_a_a @ Y @ A7 ) )
= ( ( member1426531477525435216od_a_a @ X @ A7 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1038_member__remove,axiom,
! [X: option_a,Y: option_a,A7: set_option_a] :
( ( member_option_a @ X @ ( remove_option_a @ Y @ A7 ) )
= ( ( member_option_a @ X @ A7 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1039_member__remove,axiom,
! [X: set_a,Y: set_a,A7: set_set_a] :
( ( member_set_a @ X @ ( remove_set_a @ Y @ A7 ) )
= ( ( member_set_a @ X @ A7 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1040_member__remove,axiom,
! [X: a,Y: a,A7: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A7 ) )
= ( ( member_a @ X @ A7 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1041_compl__le__compl__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ ( uminus_uminus_a_o @ X ) @ ( uminus_uminus_a_o @ Y ) )
= ( ord_less_eq_a_o @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1042_compl__le__compl__iff,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ ( uminus4708364961709604340_set_a @ X ) @ ( uminus4708364961709604340_set_a @ Y ) )
= ( ord_less_eq_o_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1043_compl__le__compl__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( uminus6103902357914783669_set_a @ Y ) )
= ( ord_le3724670747650509150_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1044_compl__le__compl__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( uminus5530930396987473918od_a_a @ X ) @ ( uminus5530930396987473918od_a_a @ Y ) )
= ( ord_le746702958409616551od_a_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1045_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1046_Compl__anti__mono,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ B6 )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ B6 ) @ ( uminus6103902357914783669_set_a @ A7 ) ) ) ).
% Compl_anti_mono
thf(fact_1047_Compl__anti__mono,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ B6 )
=> ( ord_le746702958409616551od_a_a @ ( uminus5530930396987473918od_a_a @ B6 ) @ ( uminus5530930396987473918od_a_a @ A7 ) ) ) ).
% Compl_anti_mono
thf(fact_1048_Compl__anti__mono,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B6 ) @ ( uminus_uminus_set_a @ A7 ) ) ) ).
% Compl_anti_mono
thf(fact_1049_Compl__subset__Compl__iff,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ A7 ) @ ( uminus6103902357914783669_set_a @ B6 ) )
= ( ord_le3724670747650509150_set_a @ B6 @ A7 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1050_Compl__subset__Compl__iff,axiom,
! [A7: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( uminus5530930396987473918od_a_a @ A7 ) @ ( uminus5530930396987473918od_a_a @ B6 ) )
= ( ord_le746702958409616551od_a_a @ B6 @ A7 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1051_Compl__subset__Compl__iff,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A7 ) @ ( uminus_uminus_set_a @ B6 ) )
= ( ord_less_eq_set_a @ B6 @ A7 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1052_subset__Compl__singleton,axiom,
! [A7: set_option_a,B: option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
= ( ~ ( member_option_a @ B @ A7 ) ) ) ).
% subset_Compl_singleton
thf(fact_1053_subset__Compl__singleton,axiom,
! [A7: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B @ A7 ) ) ) ).
% subset_Compl_singleton
thf(fact_1054_subset__Compl__singleton,axiom,
! [A7: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ ( uminus5530930396987473918od_a_a @ ( insert4534936382041156343od_a_a @ B @ bot_bo3357376287454694259od_a_a ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B @ A7 ) ) ) ).
% subset_Compl_singleton
thf(fact_1055_subset__Compl__singleton,axiom,
! [A7: set_a,B: a] :
( ( ord_less_eq_set_a @ A7 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A7 ) ) ) ).
% subset_Compl_singleton
thf(fact_1056_compl__le__swap2,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ ( uminus_uminus_a_o @ Y ) @ X )
=> ( ord_less_eq_a_o @ ( uminus_uminus_a_o @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1057_compl__le__swap2,axiom,
! [Y: $o > set_a,X: $o > set_a] :
( ( ord_less_eq_o_set_a @ ( uminus4708364961709604340_set_a @ Y ) @ X )
=> ( ord_less_eq_o_set_a @ ( uminus4708364961709604340_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1058_compl__le__swap2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ Y ) @ X )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1059_compl__le__swap2,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( uminus5530930396987473918od_a_a @ Y ) @ X )
=> ( ord_le746702958409616551od_a_a @ ( uminus5530930396987473918od_a_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1060_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1061_compl__le__swap1,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ Y @ ( uminus_uminus_a_o @ X ) )
=> ( ord_less_eq_a_o @ X @ ( uminus_uminus_a_o @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1062_compl__le__swap1,axiom,
! [Y: $o > set_a,X: $o > set_a] :
( ( ord_less_eq_o_set_a @ Y @ ( uminus4708364961709604340_set_a @ X ) )
=> ( ord_less_eq_o_set_a @ X @ ( uminus4708364961709604340_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1063_compl__le__swap1,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ ( uminus6103902357914783669_set_a @ X ) )
=> ( ord_le3724670747650509150_set_a @ X @ ( uminus6103902357914783669_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1064_compl__le__swap1,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ ( uminus5530930396987473918od_a_a @ X ) )
=> ( ord_le746702958409616551od_a_a @ X @ ( uminus5530930396987473918od_a_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1065_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1066_compl__mono,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ord_less_eq_a_o @ ( uminus_uminus_a_o @ Y ) @ ( uminus_uminus_a_o @ X ) ) ) ).
% compl_mono
thf(fact_1067_compl__mono,axiom,
! [X: $o > set_a,Y: $o > set_a] :
( ( ord_less_eq_o_set_a @ X @ Y )
=> ( ord_less_eq_o_set_a @ ( uminus4708364961709604340_set_a @ Y ) @ ( uminus4708364961709604340_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1068_compl__mono,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ Y ) @ ( uminus6103902357914783669_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1069_compl__mono,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ord_le746702958409616551od_a_a @ ( uminus5530930396987473918od_a_a @ Y ) @ ( uminus5530930396987473918od_a_a @ X ) ) ) ).
% compl_mono
thf(fact_1070_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1071_Compl__eq__Diff__UNIV,axiom,
( uminus6205308855922866075tion_a
= ( minus_1574173051537231627tion_a @ top_top_set_option_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1072_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_a
= ( minus_minus_set_a @ top_top_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1073_Collect__neg__eq,axiom,
! [P4: set_a > $o] :
( ( collect_set_a
@ ^ [X4: set_a] :
~ ( P4 @ X4 ) )
= ( uminus6103902357914783669_set_a @ ( collect_set_a @ P4 ) ) ) ).
% Collect_neg_eq
thf(fact_1074_Collect__neg__eq,axiom,
! [P4: option_a > $o] :
( ( collect_option_a
@ ^ [X4: option_a] :
~ ( P4 @ X4 ) )
= ( uminus6205308855922866075tion_a @ ( collect_option_a @ P4 ) ) ) ).
% Collect_neg_eq
thf(fact_1075_Collect__neg__eq,axiom,
! [P4: a > $o] :
( ( collect_a
@ ^ [X4: a] :
~ ( P4 @ X4 ) )
= ( uminus_uminus_set_a @ ( collect_a @ P4 ) ) ) ).
% Collect_neg_eq
thf(fact_1076_Compl__eq,axiom,
( uminus5530930396987473918od_a_a
= ( ^ [A8: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_1077_Compl__eq,axiom,
( uminus6103902357914783669_set_a
= ( ^ [A8: set_set_a] :
( collect_set_a
@ ^ [X4: set_a] :
~ ( member_set_a @ X4 @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_1078_Compl__eq,axiom,
( uminus6205308855922866075tion_a
= ( ^ [A8: set_option_a] :
( collect_option_a
@ ^ [X4: option_a] :
~ ( member_option_a @ X4 @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_1079_Compl__eq,axiom,
( uminus_uminus_set_a
= ( ^ [A8: set_a] :
( collect_a
@ ^ [X4: a] :
~ ( member_a @ X4 @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_1080_ComplD,axiom,
! [C: product_prod_a_a,A7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( uminus5530930396987473918od_a_a @ A7 ) )
=> ~ ( member1426531477525435216od_a_a @ C @ A7 ) ) ).
% ComplD
thf(fact_1081_ComplD,axiom,
! [C: option_a,A7: set_option_a] :
( ( member_option_a @ C @ ( uminus6205308855922866075tion_a @ A7 ) )
=> ~ ( member_option_a @ C @ A7 ) ) ).
% ComplD
thf(fact_1082_ComplD,axiom,
! [C: set_a,A7: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A7 ) )
=> ~ ( member_set_a @ C @ A7 ) ) ).
% ComplD
thf(fact_1083_ComplD,axiom,
! [C: a,A7: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A7 ) )
=> ~ ( member_a @ C @ A7 ) ) ).
% ComplD
thf(fact_1084_double__complement,axiom,
! [A7: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A7 ) )
= A7 ) ).
% double_complement
thf(fact_1085_subset__Compl__self__eq,axiom,
! [A7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( uminus6205308855922866075tion_a @ A7 ) )
= ( A7 = bot_bot_set_option_a ) ) ).
% subset_Compl_self_eq
thf(fact_1086_subset__Compl__self__eq,axiom,
! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ ( uminus6103902357914783669_set_a @ A7 ) )
= ( A7 = bot_bot_set_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1087_subset__Compl__self__eq,axiom,
! [A7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A7 @ ( uminus5530930396987473918od_a_a @ A7 ) )
= ( A7 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_Compl_self_eq
thf(fact_1088_subset__Compl__self__eq,axiom,
! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( uminus_uminus_set_a @ A7 ) )
= ( A7 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1089_Compl__UNIV__eq,axiom,
( ( uminus6103902357914783669_set_a @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% Compl_UNIV_eq
thf(fact_1090_Compl__UNIV__eq,axiom,
( ( uminus6205308855922866075tion_a @ top_top_set_option_a )
= bot_bot_set_option_a ) ).
% Compl_UNIV_eq
thf(fact_1091_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_1092_Compl__empty__eq,axiom,
( ( uminus6103902357914783669_set_a @ bot_bot_set_set_a )
= top_top_set_set_a ) ).
% Compl_empty_eq
thf(fact_1093_Compl__empty__eq,axiom,
( ( uminus6205308855922866075tion_a @ bot_bot_set_option_a )
= top_top_set_option_a ) ).
% Compl_empty_eq
thf(fact_1094_Compl__empty__eq,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% Compl_empty_eq
thf(fact_1095_Compl__insert,axiom,
! [X: option_a,A7: set_option_a] :
( ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ A7 ) )
= ( minus_1574173051537231627tion_a @ ( uminus6205308855922866075tion_a @ A7 ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ).
% Compl_insert
thf(fact_1096_Compl__insert,axiom,
! [X: set_a,A7: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( insert_set_a @ X @ A7 ) )
= ( minus_5736297505244876581_set_a @ ( uminus6103902357914783669_set_a @ A7 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ).
% Compl_insert
thf(fact_1097_Compl__insert,axiom,
! [X: a,A7: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X @ A7 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A7 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_1098_finite__Map__induct,axiom,
! [M: product_prod_a_a > option_a,P4: ( product_prod_a_a > option_a ) > $o] :
( ( finite6544458595007987280od_a_a @ ( dom_Pr6378187988305063785_a_a_a @ M ) )
=> ( ( P4
@ ^ [X4: product_prod_a_a] : none_a )
=> ( ! [K2: product_prod_a_a,V2: a,M3: product_prod_a_a > option_a] :
( ( finite6544458595007987280od_a_a @ ( dom_Pr6378187988305063785_a_a_a @ M3 ) )
=> ( ~ ( member1426531477525435216od_a_a @ K2 @ ( dom_Pr6378187988305063785_a_a_a @ M3 ) )
=> ( ( P4 @ M3 )
=> ( P4 @ ( fun_up8298456451713467738tion_a @ M3 @ K2 @ ( some_a @ V2 ) ) ) ) ) )
=> ( P4 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_1099_finite__Map__induct,axiom,
! [M: set_a > option_a,P4: ( set_a > option_a ) > $o] :
( ( finite_finite_set_a @ ( dom_set_a_a @ M ) )
=> ( ( P4
@ ^ [X4: set_a] : none_a )
=> ( ! [K2: set_a,V2: a,M3: set_a > option_a] :
( ( finite_finite_set_a @ ( dom_set_a_a @ M3 ) )
=> ( ~ ( member_set_a @ K2 @ ( dom_set_a_a @ M3 ) )
=> ( ( P4 @ M3 )
=> ( P4 @ ( fun_up3663993102702442083tion_a @ M3 @ K2 @ ( some_a @ V2 ) ) ) ) ) )
=> ( P4 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_1100_finite__Map__induct,axiom,
! [M: option_a > option_a,P4: ( option_a > option_a ) > $o] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ M ) )
=> ( ( P4
@ ^ [X4: option_a] : none_a )
=> ( ! [K2: option_a,V2: a,M3: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ M3 ) )
=> ( ~ ( member_option_a @ K2 @ ( dom_option_a_a @ M3 ) )
=> ( ( P4 @ M3 )
=> ( P4 @ ( fun_up1079276522633388797tion_a @ M3 @ K2 @ ( some_a @ V2 ) ) ) ) ) )
=> ( P4 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_1101_finite__Map__induct,axiom,
! [M: a > option_a,P4: ( a > option_a ) > $o] :
( ( finite_finite_a @ ( dom_a_a @ M ) )
=> ( ( P4
@ ^ [X4: a] : none_a )
=> ( ! [K2: a,V2: a,M3: a > option_a] :
( ( finite_finite_a @ ( dom_a_a @ M3 ) )
=> ( ~ ( member_a @ K2 @ ( dom_a_a @ M3 ) )
=> ( ( P4 @ M3 )
=> ( P4 @ ( fun_upd_a_option_a @ M3 @ K2 @ ( some_a @ V2 ) ) ) ) ) )
=> ( P4 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_1102_the__elem__def,axiom,
( the_elem_a
= ( ^ [X9: set_a] :
( the_a
@ ^ [X4: a] :
( X9
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1103_the__elem__def,axiom,
( the_elem_option_a
= ( ^ [X9: set_option_a] :
( the_option_a
@ ^ [X4: option_a] :
( X9
= ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1104_the__elem__def,axiom,
( the_elem_set_a
= ( ^ [X9: set_set_a] :
( the_set_a
@ ^ [X4: set_a] :
( X9
= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_1105_the__sym__eq__trivial,axiom,
! [X: a] :
( ( the_a
@ ( ^ [Y4: a,Z: a] : ( Y4 = Z )
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_1106_the__eq__trivial,axiom,
! [A: a] :
( ( the_a
@ ^ [X4: a] : ( X4 = A ) )
= A ) ).
% the_eq_trivial
thf(fact_1107_the__equality,axiom,
! [P4: a > $o,A: a] :
( ( P4 @ A )
=> ( ! [X3: a] :
( ( P4 @ X3 )
=> ( X3 = A ) )
=> ( ( the_a @ P4 )
= A ) ) ) ).
% the_equality
thf(fact_1108_finite__option__UNIV,axiom,
( ( finite3831083272032232269_set_a @ top_to3949272007228979924_set_a )
= ( finite_finite_set_a @ top_top_set_set_a ) ) ).
% finite_option_UNIV
thf(fact_1109_finite__option__UNIV,axiom,
( ( finite8114217219359860531tion_a @ top_to1659475022456381882tion_a )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% finite_option_UNIV
thf(fact_1110_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_1111_uminus__set__def,axiom,
( uminus5530930396987473918od_a_a
= ( ^ [A8: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( uminus1956187339570297439_a_a_o
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1112_uminus__set__def,axiom,
( uminus6103902357914783669_set_a
= ( ^ [A8: set_set_a] :
( collect_set_a
@ ( uminus8300850153153071976et_a_o
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1113_uminus__set__def,axiom,
( uminus6205308855922866075tion_a
= ( ^ [A8: set_option_a] :
( collect_option_a
@ ( uminus6867298174334746882on_a_o
@ ^ [X4: option_a] : ( member_option_a @ X4 @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1114_uminus__set__def,axiom,
( uminus_uminus_set_a
= ( ^ [A8: set_a] :
( collect_a
@ ( uminus_uminus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1115_the1__equality_H,axiom,
! [P4: a > $o,A: a] :
( ( uniq_a @ P4 )
=> ( ( P4 @ A )
=> ( ( the_a @ P4 )
= A ) ) ) ).
% the1_equality'
thf(fact_1116_the1__equality,axiom,
! [P4: a > $o,A: a] :
( ? [X8: a] :
( ( P4 @ X8 )
& ! [Y5: a] :
( ( P4 @ Y5 )
=> ( Y5 = X8 ) ) )
=> ( ( P4 @ A )
=> ( ( the_a @ P4 )
= A ) ) ) ).
% the1_equality
thf(fact_1117_the1I2,axiom,
! [P4: a > $o,Q4: a > $o] :
( ? [X8: a] :
( ( P4 @ X8 )
& ! [Y5: a] :
( ( P4 @ Y5 )
=> ( Y5 = X8 ) ) )
=> ( ! [X3: a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( Q4 @ ( the_a @ P4 ) ) ) ) ).
% the1I2
thf(fact_1118_If__def,axiom,
( if_a
= ( ^ [P6: $o,X4: a,Y3: a] :
( the_a
@ ^ [Z6: a] :
( ( P6
=> ( Z6 = X4 ) )
& ( ~ P6
=> ( Z6 = Y3 ) ) ) ) ) ) ).
% If_def
thf(fact_1119_theI2,axiom,
! [P4: a > $o,A: a,Q4: a > $o] :
( ( P4 @ A )
=> ( ! [X3: a] :
( ( P4 @ X3 )
=> ( X3 = A ) )
=> ( ! [X3: a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( Q4 @ ( the_a @ P4 ) ) ) ) ) ).
% theI2
thf(fact_1120_theI_H,axiom,
! [P4: a > $o] :
( ? [X8: a] :
( ( P4 @ X8 )
& ! [Y5: a] :
( ( P4 @ Y5 )
=> ( Y5 = X8 ) ) )
=> ( P4 @ ( the_a @ P4 ) ) ) ).
% theI'
thf(fact_1121_theI,axiom,
! [P4: a > $o,A: a] :
( ( P4 @ A )
=> ( ! [X3: a] :
( ( P4 @ X3 )
=> ( X3 = A ) )
=> ( P4 @ ( the_a @ P4 ) ) ) ) ).
% theI
thf(fact_1122_finite__set__of__finite__maps,axiom,
! [A7: set_a,B6: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( finite_finite_a @ B6 )
=> ( finite5998080203967203522tion_a
@ ( collect_a_option_a
@ ^ [M2: a > option_a] :
( ( ( dom_a_a @ M2 )
= A7 )
& ( ord_less_eq_set_a @ ( ran_a_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1123_finite__set__of__finite__maps,axiom,
! [A7: set_a,B6: set_option_a] :
( ( finite_finite_a @ A7 )
=> ( ( finite1674126218327898605tion_a @ B6 )
=> ( finite595534299322690568tion_a
@ ( collec6680173830675942470tion_a
@ ^ [M2: a > option_option_a] :
( ( ( dom_a_option_a @ M2 )
= A7 )
& ( ord_le1955136853071979460tion_a @ ( ran_a_option_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1124_finite__set__of__finite__maps,axiom,
! [A7: set_set_a,B6: set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( finite_finite_a @ B6 )
=> ( finite668978520222027618tion_a
@ ( collec5355523187277114528tion_a
@ ^ [M2: set_a > option_a] :
( ( ( dom_set_a_a @ M2 )
= A7 )
& ( ord_less_eq_set_a @ ( ran_set_a_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1125_finite__set__of__finite__maps,axiom,
! [A7: set_option_a,B6: set_a] :
( ( finite1674126218327898605tion_a @ A7 )
=> ( ( finite_finite_a @ B6 )
=> ( finite8942580144290239484tion_a
@ ( collec5803847638788715578tion_a
@ ^ [M2: option_a > option_a] :
( ( ( dom_option_a_a @ M2 )
= A7 )
& ( ord_less_eq_set_a @ ( ran_option_a_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1126_finite__set__of__finite__maps,axiom,
! [A7: set_a,B6: set_set_a] :
( ( finite_finite_a @ A7 )
=> ( ( finite_finite_set_a @ B6 )
=> ( finite201771827693623842_set_a
@ ( collec4888316494748710752_set_a
@ ^ [M2: a > option_set_a] :
( ( ( dom_a_set_a @ M2 )
= A7 )
& ( ord_le3724670747650509150_set_a @ ( ran_a_set_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1127_finite__set__of__finite__maps,axiom,
! [A7: set_set_a,B6: set_option_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( finite1674126218327898605tion_a @ B6 )
=> ( finite8850206357266170536tion_a
@ ( collec79705156120029926tion_a
@ ^ [M2: set_a > option_option_a] :
( ( ( dom_set_a_option_a @ M2 )
= A7 )
& ( ord_le1955136853071979460tion_a @ ( ran_set_a_option_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1128_finite__set__of__finite__maps,axiom,
! [A7: set_option_a,B6: set_option_a] :
( ( finite1674126218327898605tion_a @ A7 )
=> ( ( finite1674126218327898605tion_a @ B6 )
=> ( finite3089212293327181890tion_a
@ ( collec2458836999851688832tion_a
@ ^ [M2: option_a > option_option_a] :
( ( ( dom_op4724496951392727122tion_a @ M2 )
= A7 )
& ( ord_le1955136853071979460tion_a @ ( ran_op6317565877353657455tion_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1129_finite__set__of__finite__maps,axiom,
! [A7: set_set_a,B6: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( finite_finite_set_a @ B6 )
=> ( finite3953205783614742722_set_a
@ ( collec1684453483449566720_set_a
@ ^ [M2: set_a > option_set_a] :
( ( ( dom_set_a_set_a @ M2 )
= A7 )
& ( ord_le3724670747650509150_set_a @ ( ran_set_a_set_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1130_finite__set__of__finite__maps,axiom,
! [A7: set_option_a,B6: set_set_a] :
( ( finite1674126218327898605tion_a @ A7 )
=> ( ( finite_finite_set_a @ B6 )
=> ( finite3474160937502457180_set_a
@ ( collec3927031773211092378_set_a
@ ^ [M2: option_a > option_set_a] :
( ( ( dom_option_a_set_a @ M2 )
= A7 )
& ( ord_le3724670747650509150_set_a @ ( ran_option_a_set_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1131_finite__set__of__finite__maps,axiom,
! [A7: set_a,B6: set_Product_prod_a_a] :
( ( finite_finite_a @ A7 )
=> ( ( finite6544458595007987280od_a_a @ B6 )
=> ( finite1927657904573721579od_a_a
@ ( collec7441720289058034473od_a_a
@ ^ [M2: a > option5210160422955383789od_a_a] :
( ( ( dom_a_1117496481140825723od_a_a @ M2 )
= A7 )
& ( ord_le746702958409616551od_a_a @ ( ran_a_780659196698074264od_a_a @ M2 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1132_finite__map__freshness,axiom,
! [F: set_a > option_a] :
( ( finite_finite_set_a @ ( dom_set_a_a @ F ) )
=> ( ~ ( finite_finite_set_a @ top_top_set_set_a )
=> ? [X3: set_a] :
( ( F @ X3 )
= none_a ) ) ) ).
% finite_map_freshness
thf(fact_1133_finite__map__freshness,axiom,
! [F: a > option_a] :
( ( finite_finite_a @ ( dom_a_a @ F ) )
=> ( ~ ( finite_finite_a @ top_top_set_a )
=> ? [X3: a] :
( ( F @ X3 )
= none_a ) ) ) ).
% finite_map_freshness
thf(fact_1134_finite__map__freshness,axiom,
! [F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ F ) )
=> ( ~ ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ? [X3: option_a] :
( ( F @ X3 )
= none_a ) ) ) ).
% finite_map_freshness
thf(fact_1135_finite__Collect__bex,axiom,
! [A7: set_a,Q4: a > a > $o] :
( ( finite_finite_a @ A7 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
? [Y3: a] :
( ( member_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A7 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y3: a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1136_finite__Collect__bex,axiom,
! [A7: set_a,Q4: set_a > a > $o] :
( ( finite_finite_a @ A7 )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
? [Y3: a] :
( ( member_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A7 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [Y3: set_a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1137_finite__Collect__bex,axiom,
! [A7: set_a,Q4: option_a > a > $o] :
( ( finite_finite_a @ A7 )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
? [Y3: a] :
( ( member_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A7 )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [Y3: option_a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1138_finite__Collect__bex,axiom,
! [A7: set_set_a,Q4: a > set_a > $o] :
( ( finite_finite_set_a @ A7 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
? [Y3: set_a] :
( ( member_set_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A7 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y3: a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1139_finite__Collect__bex,axiom,
! [A7: set_set_a,Q4: set_a > set_a > $o] :
( ( finite_finite_set_a @ A7 )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
? [Y3: set_a] :
( ( member_set_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A7 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [Y3: set_a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1140_finite__Collect__bex,axiom,
! [A7: set_set_a,Q4: option_a > set_a > $o] :
( ( finite_finite_set_a @ A7 )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
? [Y3: set_a] :
( ( member_set_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A7 )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [Y3: option_a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1141_finite__Collect__bex,axiom,
! [A7: set_option_a,Q4: a > option_a > $o] :
( ( finite1674126218327898605tion_a @ A7 )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
? [Y3: option_a] :
( ( member_option_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Y3: a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1142_finite__Collect__bex,axiom,
! [A7: set_option_a,Q4: set_a > option_a > $o] :
( ( finite1674126218327898605tion_a @ A7 )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
? [Y3: option_a] :
( ( member_option_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [Y3: set_a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1143_finite__Collect__bex,axiom,
! [A7: set_option_a,Q4: option_a > option_a > $o] :
( ( finite1674126218327898605tion_a @ A7 )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
? [Y3: option_a] :
( ( member_option_a @ Y3 @ A7 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [Y3: option_a] : ( Q4 @ Y3 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1144_finite__Collect__bounded__ex,axiom,
! [P4: a > $o,Q4: set_a > a > $o] :
( ( finite_finite_a @ ( collect_a @ P4 ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
? [Y3: a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: a] :
( ( P4 @ Y3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1145_finite__Collect__bounded__ex,axiom,
! [P4: a > $o,Q4: option_a > a > $o] :
( ( finite_finite_a @ ( collect_a @ P4 ) )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
? [Y3: a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: a] :
( ( P4 @ Y3 )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1146_finite__Collect__bounded__ex,axiom,
! [P4: set_a > $o,Q4: a > set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P4 ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
? [Y3: set_a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: set_a] :
( ( P4 @ Y3 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1147_finite__Collect__bounded__ex,axiom,
! [P4: set_a > $o,Q4: set_a > set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P4 ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
? [Y3: set_a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: set_a] :
( ( P4 @ Y3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1148_finite__Collect__bounded__ex,axiom,
! [P4: set_a > $o,Q4: option_a > set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P4 ) )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
? [Y3: set_a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: set_a] :
( ( P4 @ Y3 )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1149_finite__Collect__bounded__ex,axiom,
! [P4: option_a > $o,Q4: a > option_a > $o] :
( ( finite1674126218327898605tion_a @ ( collect_option_a @ P4 ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
? [Y3: option_a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: option_a] :
( ( P4 @ Y3 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1150_finite__Collect__bounded__ex,axiom,
! [P4: option_a > $o,Q4: set_a > option_a > $o] :
( ( finite1674126218327898605tion_a @ ( collect_option_a @ P4 ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
? [Y3: option_a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: option_a] :
( ( P4 @ Y3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1151_finite__Collect__bounded__ex,axiom,
! [P4: option_a > $o,Q4: option_a > option_a > $o] :
( ( finite1674126218327898605tion_a @ ( collect_option_a @ P4 ) )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
? [Y3: option_a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: option_a] :
( ( P4 @ Y3 )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1152_finite__Collect__bounded__ex,axiom,
! [P4: a > $o,Q4: a > a > $o] :
( ( finite_finite_a @ ( collect_a @ P4 ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
? [Y3: a] :
( ( P4 @ Y3 )
& ( Q4 @ X4 @ Y3 ) ) ) )
= ( ! [Y3: a] :
( ( P4 @ Y3 )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] : ( Q4 @ X4 @ Y3 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_1153_finite__Collect__disjI,axiom,
! [P4: set_a > $o,Q4: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( P4 @ X4 )
| ( Q4 @ X4 ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P4 ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q4 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1154_finite__Collect__disjI,axiom,
! [P4: option_a > $o,Q4: option_a > $o] :
( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( P4 @ X4 )
| ( Q4 @ X4 ) ) ) )
= ( ( finite1674126218327898605tion_a @ ( collect_option_a @ P4 ) )
& ( finite1674126218327898605tion_a @ ( collect_option_a @ Q4 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1155_finite__Collect__disjI,axiom,
! [P4: a > $o,Q4: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( P4 @ X4 )
| ( Q4 @ X4 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P4 ) )
& ( finite_finite_a @ ( collect_a @ Q4 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1156_finite__Collect__conjI,axiom,
! [P4: set_a > $o,Q4: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P4 ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q4 ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1157_finite__Collect__conjI,axiom,
! [P4: option_a > $o,Q4: option_a > $o] :
( ( ( finite1674126218327898605tion_a @ ( collect_option_a @ P4 ) )
| ( finite1674126218327898605tion_a @ ( collect_option_a @ Q4 ) ) )
=> ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1158_finite__Collect__conjI,axiom,
! [P4: a > $o,Q4: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P4 ) )
| ( finite_finite_a @ ( collect_a @ Q4 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
( ( P4 @ X4 )
& ( Q4 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1159_finite__Collect__not,axiom,
! [P4: set_a > $o] :
( ( finite_finite_set_a @ ( collect_set_a @ P4 ) )
=> ( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X4: set_a] :
~ ( P4 @ X4 ) ) )
= ( finite_finite_set_a @ top_top_set_set_a ) ) ) ).
% finite_Collect_not
thf(fact_1160_finite__Collect__not,axiom,
! [P4: option_a > $o] :
( ( finite1674126218327898605tion_a @ ( collect_option_a @ P4 ) )
=> ( ( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [X4: option_a] :
~ ( P4 @ X4 ) ) )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ) ).
% finite_Collect_not
thf(fact_1161_finite__Collect__not,axiom,
! [P4: a > $o] :
( ( finite_finite_a @ ( collect_a @ P4 ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X4: a] :
~ ( P4 @ X4 ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_1162_finite__Collect__subsets,axiom,
! [A7: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A7 )
=> ( finite8717734299975451184od_a_a
@ ( collec1673347964119250290od_a_a
@ ^ [B7: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B7 @ A7 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1163_finite__Collect__subsets,axiom,
! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B7: set_a] : ( ord_less_eq_set_a @ B7 @ A7 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1164_not__finite__existsD,axiom,
! [P4: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P4 ) )
=> ? [X_1: a] : ( P4 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1165_finite__has__maximal2,axiom,
! [A7: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( member_set_a @ A @ A7 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A7 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1166_finite__has__minimal2,axiom,
! [A7: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( member_set_a @ A @ A7 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A7 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1167_finite__subset,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( finite_finite_a @ B6 )
=> ( finite_finite_a @ A7 ) ) ) ).
% finite_subset
thf(fact_1168_infinite__super,axiom,
! [S4: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S4 @ T3 )
=> ( ~ ( finite_finite_a @ S4 )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_1169_rev__finite__subset,axiom,
! [B6: set_a,A7: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( finite_finite_a @ A7 ) ) ) ).
% rev_finite_subset
thf(fact_1170_finite__image__set2,axiom,
! [P4: a > $o,Q4: a > $o,F: a > a > a] :
( ( finite_finite_a @ ( collect_a @ P4 ) )
=> ( ( finite_finite_a @ ( collect_a @ Q4 ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Uu: a] :
? [X4: a,Y3: a] :
( ( Uu
= ( F @ X4 @ Y3 ) )
& ( P4 @ X4 )
& ( Q4 @ Y3 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_1171_finite__image__set,axiom,
! [P4: a > $o,F: a > a] :
( ( finite_finite_a @ ( collect_a @ P4 ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Uu: a] :
? [X4: a] :
( ( Uu
= ( F @ X4 ) )
& ( P4 @ X4 ) ) ) ) ) ).
% finite_image_set
thf(fact_1172_finite__has__minimal,axiom,
! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A7 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1173_finite__has__maximal,axiom,
! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A7 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1174_finite__subset__induct_H,axiom,
! [F3: set_a,A7: set_a,P4: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A7 )
=> ( ( P4 @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A7 )
=> ( ( ord_less_eq_set_a @ F4 @ A7 )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert_a @ A3 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1175_finite__subset__induct,axiom,
! [F3: set_a,A7: set_a,P4: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A7 )
=> ( ( P4 @ bot_bot_set_a )
=> ( ! [A3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A3 @ A7 )
=> ( ~ ( member_a @ A3 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert_a @ A3 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1176_remove__induct,axiom,
! [P4: set_a > $o,B6: set_a] :
( ( P4 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B6 )
=> ( P4 @ B6 ) )
=> ( ! [A9: set_a] :
( ( finite_finite_a @ A9 )
=> ( ( A9 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A9 @ B6 )
=> ( ! [X8: a] :
( ( member_a @ X8 @ A9 )
=> ( P4 @ ( minus_minus_set_a @ A9 @ ( insert_a @ X8 @ bot_bot_set_a ) ) ) )
=> ( P4 @ A9 ) ) ) ) )
=> ( P4 @ B6 ) ) ) ) ).
% remove_induct
thf(fact_1177_finite__remove__induct,axiom,
! [B6: set_a,P4: set_a > $o] :
( ( finite_finite_a @ B6 )
=> ( ( P4 @ bot_bot_set_a )
=> ( ! [A9: set_a] :
( ( finite_finite_a @ A9 )
=> ( ( A9 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A9 @ B6 )
=> ( ! [X8: a] :
( ( member_a @ X8 @ A9 )
=> ( P4 @ ( minus_minus_set_a @ A9 @ ( insert_a @ X8 @ bot_bot_set_a ) ) ) )
=> ( P4 @ A9 ) ) ) ) )
=> ( P4 @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_1178_max__extp_Ocases,axiom,
! [R3: a > a > $o,A1: set_a,A2: set_a] :
( ( max_extp_a @ R3 @ A1 @ A2 )
=> ~ ( ( finite_finite_a @ A1 )
=> ( ( finite_finite_a @ A2 )
=> ( ( A2
!= ( collect_a @ bot_bot_a_o ) )
=> ~ ! [X8: a] :
( ( member_a @ X8 @ A1 )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ A2 )
& ( R3 @ X8 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_1179_max__extp_Omax__extI,axiom,
! [X7: set_a,Y8: set_a,R3: a > a > $o] :
( ( finite_finite_a @ X7 )
=> ( ( finite_finite_a @ Y8 )
=> ( ( Y8
!= ( collect_a @ bot_bot_a_o ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ X7 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ Y8 )
& ( R3 @ X3 @ Xa2 ) ) )
=> ( max_extp_a @ R3 @ X7 @ Y8 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_1180_max__extp_Osimps,axiom,
( max_extp_a
= ( ^ [R5: a > a > $o,A12: set_a,A22: set_a] :
( ( finite_finite_a @ A12 )
& ( finite_finite_a @ A22 )
& ( A22
!= ( collect_a @ bot_bot_a_o ) )
& ! [X4: a] :
( ( member_a @ X4 @ A12 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ A22 )
& ( R5 @ X4 @ Y3 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_1181_lfp__lfp,axiom,
! [F: set_a > set_a > set_a] :
( ! [X3: set_a,Y5: set_a,W: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ( ord_less_eq_set_a @ W @ Z4 )
=> ( ord_less_eq_set_a @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( comple1558298011288954135_set_a
@ ^ [X4: set_a] : ( comple1558298011288954135_set_a @ ( F @ X4 ) ) )
= ( comple1558298011288954135_set_a
@ ^ [X4: set_a] : ( F @ X4 @ X4 ) ) ) ) ).
% lfp_lfp
thf(fact_1182_lfp__greatest,axiom,
! [F: set_a > set_a,A7: set_a] :
( ! [U2: set_a] :
( ( ord_less_eq_set_a @ ( F @ U2 ) @ U2 )
=> ( ord_less_eq_set_a @ A7 @ U2 ) )
=> ( ord_less_eq_set_a @ A7 @ ( comple1558298011288954135_set_a @ F ) ) ) ).
% lfp_greatest
thf(fact_1183_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X9: $o > set_a,Y7: $o > set_a] :
( ( ord_less_eq_set_a @ ( X9 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_a @ ( X9 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_1184_lfp__mono,axiom,
! [F: set_a > set_a,G: set_a > set_a] :
( ! [Z7: set_a] : ( ord_less_eq_set_a @ ( F @ Z7 ) @ ( G @ Z7 ) )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ ( comple1558298011288954135_set_a @ G ) ) ) ).
% lfp_mono
thf(fact_1185_lfp__lowerbound,axiom,
! [F: set_a > set_a,A7: set_a] :
( ( ord_less_eq_set_a @ ( F @ A7 ) @ A7 )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ A7 ) ) ).
% lfp_lowerbound
thf(fact_1186_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_1187_image__eqI,axiom,
! [B: a,F: a > a,X: a,A7: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A7 )
=> ( member_a @ B @ ( image_a_a @ F @ A7 ) ) ) ) ).
% image_eqI
thf(fact_1188_image__map__upd,axiom,
! [X: a,A7: set_a,M: a > option_a,Y: a] :
( ~ ( member_a @ X @ A7 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Y ) ) @ A7 )
= ( image_a_option_a @ M @ A7 ) ) ) ).
% image_map_upd
thf(fact_1189_these__image__Some__eq,axiom,
! [A7: set_a] :
( ( these_a @ ( image_a_option_a @ some_a @ A7 ) )
= A7 ) ).
% these_image_Some_eq
thf(fact_1190_all__finite__subset__image,axiom,
! [F: a > a,A7: set_a,P4: set_a > $o] :
( ( ! [B7: set_a] :
( ( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F @ A7 ) ) )
=> ( P4 @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ A7 ) )
=> ( P4 @ ( image_a_a @ F @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1191_ex__finite__subset__image,axiom,
! [F: a > a,A7: set_a,P4: set_a > $o] :
( ( ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F @ A7 ) )
& ( P4 @ B7 ) ) )
= ( ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ord_less_eq_set_a @ B7 @ A7 )
& ( P4 @ ( image_a_a @ F @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1192_finite__subset__image,axiom,
! [B6: set_a,F: a > a,A7: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A7 ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A7 )
& ( finite_finite_a @ C5 )
& ( B6
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1193_image__mono,axiom,
! [A7: set_a,B6: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A7 ) @ ( image_a_a @ F @ B6 ) ) ) ).
% image_mono
thf(fact_1194_image__subsetI,axiom,
! [A7: set_a,F: a > a,B6: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ ( F @ X3 ) @ B6 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A7 ) @ B6 ) ) ).
% image_subsetI
thf(fact_1195_subset__imageE,axiom,
! [B6: set_a,F: a > a,A7: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A7 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A7 )
=> ( B6
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1196_subset__image__iff,axiom,
! [B6: set_a,F: a > a,A7: set_a] :
( ( ord_less_eq_set_a @ B6 @ ( image_a_a @ F @ A7 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A7 )
& ( B6
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1197_setcompr__eq__image,axiom,
! [F: a > a,P4: a > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X4: a] :
( ( Uu
= ( F @ X4 ) )
& ( P4 @ X4 ) ) )
= ( image_a_a @ F @ ( collect_a @ P4 ) ) ) ).
% setcompr_eq_image
thf(fact_1198_Setcompr__eq__image,axiom,
! [F: a > a,A7: set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [X4: a] :
( ( Uu
= ( F @ X4 ) )
& ( member_a @ X4 @ A7 ) ) )
= ( image_a_a @ F @ A7 ) ) ).
% Setcompr_eq_image
thf(fact_1199_image__Collect__subsetI,axiom,
! [P4: a > $o,F: a > a,B6: set_a] :
( ! [X3: a] :
( ( P4 @ X3 )
=> ( member_a @ ( F @ X3 ) @ B6 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P4 ) ) @ B6 ) ) ).
% image_Collect_subsetI
thf(fact_1200_imageE,axiom,
! [B: a,F: a > a,A7: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A7 ) )
=> ~ ! [X3: a] :
( ( B
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A7 ) ) ) ).
% imageE
thf(fact_1201_Compr__image__eq,axiom,
! [F: a > a,A7: set_a,P4: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A7 ) )
& ( P4 @ X4 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A7 )
& ( P4 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1202_imageI,axiom,
! [X: a,A7: set_a,F: a > a] :
( ( member_a @ X @ A7 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A7 ) ) ) ).
% imageI
thf(fact_1203_rev__image__eqI,axiom,
! [X: a,A7: set_a,B: a,F: a > a] :
( ( member_a @ X @ A7 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A7 ) ) ) ) ).
% rev_image_eqI
thf(fact_1204_None__notin__image__Some,axiom,
! [A7: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A7 ) ) ).
% None_notin_image_Some
thf(fact_1205_all__subset__image,axiom,
! [F: a > a,A7: set_a,P4: set_a > $o] :
( ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F @ A7 ) )
=> ( P4 @ B7 ) ) )
= ( ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ A7 )
=> ( P4 @ ( image_a_a @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_1206_notin__range__Some,axiom,
! [X: option_a] :
( ( ~ ( member_option_a @ X @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
= ( X = none_a ) ) ).
% notin_range_Some
thf(fact_1207_Some__image__these__eq,axiom,
! [A7: set_option_a] :
( ( image_a_option_a @ some_a @ ( these_a @ A7 ) )
= ( collect_option_a
@ ^ [X4: option_a] :
( ( member_option_a @ X4 @ A7 )
& ( X4 != none_a ) ) ) ) ).
% Some_image_these_eq
thf(fact_1208_finite__range__Some,axiom,
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ some_a @ top_top_set_a ) )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_range_Some
thf(fact_1209_in__image__insert__iff,axiom,
! [B6: set_set_a,X: a,A7: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B6 )
=> ~ ( member_a @ X @ C5 ) )
=> ( ( member_set_a @ A7 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B6 ) )
= ( ( member_a @ X @ A7 )
& ( member_set_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1210_flat__lub__def,axiom,
( partial_flat_lub_a
= ( ^ [B4: a,A8: set_a] :
( if_a @ ( ord_less_eq_set_a @ A8 @ ( insert_a @ B4 @ bot_bot_set_a ) ) @ B4
@ ( the_a
@ ^ [X4: a] : ( member_a @ X4 @ ( minus_minus_set_a @ A8 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_1211_Fpow__mono,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A7 ) @ ( finite_Fpow_a @ B6 ) ) ) ).
% Fpow_mono
thf(fact_1212_refl__onI,axiom,
! [R2: set_Product_prod_a_a,A7: set_a] :
( ( ord_le746702958409616551od_a_a @ R2
@ ( product_Sigma_a_a @ A7
@ ^ [Uu: a] : A7 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 ) )
=> ( refl_on_a @ A7 @ R2 ) ) ) ).
% refl_onI
thf(fact_1213_Fpow__def,axiom,
( finite_Fpow_a
= ( ^ [A8: set_a] :
( collect_set_a
@ ^ [X9: set_a] :
( ( ord_less_eq_set_a @ X9 @ A8 )
& ( finite_finite_a @ X9 ) ) ) ) ) ).
% Fpow_def
thf(fact_1214_Sigma__mono,axiom,
! [A7: set_a,C3: set_a,B6: a > set_a,D: a > set_a] :
( ( ord_less_eq_set_a @ A7 @ C3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( ord_less_eq_set_a @ ( B6 @ X3 ) @ ( D @ X3 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( product_Sigma_a_a @ A7 @ B6 ) @ ( product_Sigma_a_a @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_1215_Times__subset__cancel2,axiom,
! [X: a,C3: set_a,A7: set_a,B6: set_a] :
( ( member_a @ X @ C3 )
=> ( ( ord_le746702958409616551od_a_a
@ ( product_Sigma_a_a @ A7
@ ^ [Uu: a] : C3 )
@ ( product_Sigma_a_a @ B6
@ ^ [Uu: a] : C3 ) )
= ( ord_less_eq_set_a @ A7 @ B6 ) ) ) ).
% Times_subset_cancel2
thf(fact_1216_times__subset__iff,axiom,
! [A7: set_a,C3: set_a,B6: set_a,D: set_a] :
( ( ord_le746702958409616551od_a_a
@ ( product_Sigma_a_a @ A7
@ ^ [Uu: a] : C3 )
@ ( product_Sigma_a_a @ B6
@ ^ [Uu: a] : D ) )
= ( ( A7 = bot_bot_set_a )
| ( C3 = bot_bot_set_a )
| ( ( ord_less_eq_set_a @ A7 @ B6 )
& ( ord_less_eq_set_a @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1217_UnCI,axiom,
! [C: a,B6: set_a,A7: set_a] :
( ( ~ ( member_a @ C @ B6 )
=> ( member_a @ C @ A7 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A7 @ B6 ) ) ) ).
% UnCI
thf(fact_1218_Un__iff,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A7 @ B6 ) )
= ( ( member_a @ C @ A7 )
| ( member_a @ C @ B6 ) ) ) ).
% Un_iff
thf(fact_1219_Un__subset__iff,axiom,
! [A7: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A7 @ B6 ) @ C3 )
= ( ( ord_less_eq_set_a @ A7 @ C3 )
& ( ord_less_eq_set_a @ B6 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_1220_sup__shunt,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= top_top_set_a )
= ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_1221_Collect__imp__eq,axiom,
! [P4: a > $o,Q4: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P4 @ X4 )
=> ( Q4 @ X4 ) ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ ( collect_a @ P4 ) ) @ ( collect_a @ Q4 ) ) ) ).
% Collect_imp_eq
thf(fact_1222_Diff__partition,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( sup_sup_set_a @ A7 @ ( minus_minus_set_a @ B6 @ A7 ) )
= B6 ) ) ).
% Diff_partition
thf(fact_1223_Diff__subset__conv,axiom,
! [A7: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A7 @ B6 ) @ C3 )
= ( ord_less_eq_set_a @ A7 @ ( sup_sup_set_a @ B6 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_1224_Un__mono,axiom,
! [A7: set_a,C3: set_a,B6: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A7 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A7 @ B6 ) @ ( sup_sup_set_a @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_1225_Un__least,axiom,
! [A7: set_a,C3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A7 @ B6 ) @ C3 ) ) ) ).
% Un_least
thf(fact_1226_Un__upper1,axiom,
! [A7: set_a,B6: set_a] : ( ord_less_eq_set_a @ A7 @ ( sup_sup_set_a @ A7 @ B6 ) ) ).
% Un_upper1
thf(fact_1227_Un__upper2,axiom,
! [B6: set_a,A7: set_a] : ( ord_less_eq_set_a @ B6 @ ( sup_sup_set_a @ A7 @ B6 ) ) ).
% Un_upper2
thf(fact_1228_Un__absorb1,axiom,
! [A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( sup_sup_set_a @ A7 @ B6 )
= B6 ) ) ).
% Un_absorb1
thf(fact_1229_Un__absorb2,axiom,
! [B6: set_a,A7: set_a] :
( ( ord_less_eq_set_a @ B6 @ A7 )
=> ( ( sup_sup_set_a @ A7 @ B6 )
= A7 ) ) ).
% Un_absorb2
thf(fact_1230_subset__UnE,axiom,
! [C3: set_a,A7: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A7 @ B6 ) )
=> ~ ! [A10: set_a] :
( ( ord_less_eq_set_a @ A10 @ A7 )
=> ! [B9: set_a] :
( ( ord_less_eq_set_a @ B9 @ B6 )
=> ( C3
!= ( sup_sup_set_a @ A10 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_1231_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A8: set_a,B7: set_a] :
( ( sup_sup_set_a @ A8 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_1232_insert__def,axiom,
( insert_a
= ( ^ [A4: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X4: a] : ( X4 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1233_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A8: set_a,B7: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A8 )
| ( member_a @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_1234_Collect__disj__eq,axiom,
! [P4: a > $o,Q4: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P4 @ X4 )
| ( Q4 @ X4 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q4 ) ) ) ).
% Collect_disj_eq
thf(fact_1235_UnE,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A7 @ B6 ) )
=> ( ~ ( member_a @ C @ A7 )
=> ( member_a @ C @ B6 ) ) ) ).
% UnE
thf(fact_1236_UnI1,axiom,
! [C: a,A7: set_a,B6: set_a] :
( ( member_a @ C @ A7 )
=> ( member_a @ C @ ( sup_sup_set_a @ A7 @ B6 ) ) ) ).
% UnI1
thf(fact_1237_UnI2,axiom,
! [C: a,B6: set_a,A7: set_a] :
( ( member_a @ C @ B6 )
=> ( member_a @ C @ ( sup_sup_set_a @ A7 @ B6 ) ) ) ).
% UnI2
thf(fact_1238_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1239_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1240_sup__Un__eq,axiom,
! [R3: set_a,S4: set_a] :
( ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R3 )
@ ^ [X4: a] : ( member_a @ X4 @ S4 ) )
= ( ^ [X4: a] : ( member_a @ X4 @ ( sup_sup_set_a @ R3 @ S4 ) ) ) ) ).
% sup_Un_eq
thf(fact_1241_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A8: set_a,B7: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A8 )
@ ^ [X4: a] : ( member_a @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_1242_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1243_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1244_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_1245_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1246_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1247_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1248_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_1249_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1250_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1251_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1252_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1253_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1254_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1255_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y5: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_a,Y5: set_a] : ( ord_less_eq_set_a @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_a,Y5: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X3 )
=> ( ( ord_less_eq_set_a @ Z4 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1256_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_1257_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1258_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1259_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1260_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1261_sup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1262_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_1263_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_1264_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_1265_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_1266_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1267_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_1268_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1269_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1270_inj__on__image__mem__iff,axiom,
! [F: a > a,B6: set_a,A: a,A7: set_a] :
( ( inj_on_a_a @ F @ B6 )
=> ( ( member_a @ A @ B6 )
=> ( ( ord_less_eq_set_a @ A7 @ B6 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A7 ) )
= ( member_a @ A @ A7 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1271_inj__image__subset__iff,axiom,
! [F: a > a,A7: set_a,B6: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A7 ) @ ( image_a_a @ F @ B6 ) )
= ( ord_less_eq_set_a @ A7 @ B6 ) ) ) ).
% inj_image_subset_iff
thf(fact_1272_mono__Field,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R2 @ S )
=> ( ord_less_eq_set_a @ ( field_a @ R2 ) @ ( field_a @ S ) ) ) ).
% mono_Field
thf(fact_1273_inj__Some,axiom,
! [A7: set_a] : ( inj_on_a_option_a @ some_a @ A7 ) ).
% inj_Some
thf(fact_1274_FieldI2,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ J @ ( field_a @ R3 ) ) ) ).
% FieldI2
thf(fact_1275_FieldI1,axiom,
! [I: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
=> ( member_a @ I @ ( field_a @ R3 ) ) ) ).
% FieldI1
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
? [Sigma: a,P3: b,Q3: b] :
( ( X4 = Sigma )
& ( ( sadd @ P3 @ Q3 )
= one )
& ( ( some_a @ Sigma )
= ( plus @ ( mult @ P3 @ a2 ) @ ( mult @ Q3 @ b2 ) ) ) ) )
@ ( delta @ s ) ) ).
%------------------------------------------------------------------------------