TPTP Problem File: ITP211^1.p
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%------------------------------------------------------------------------------
% File : ITP211^1 : TPTP v9.0.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00607_017823
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0024_Assertions_00607_017823 [Des22]
% Status : Theorem
% Rating : 0.75 v9.0.0, 0.80 v8.2.0, 0.85 v8.1.0
% Syntax : Number of formulae : 9705 (3210 unt;1879 typ; 0 def)
% Number of atoms : 24722 (9168 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 84528 (3128 ~; 291 |;1450 &;67800 @)
% ( 0 <=>;11859 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 160 ( 159 usr)
% Number of type conns : 8578 (8578 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1723 (1720 usr; 85 con; 0-8 aty)
% Number of variables : 22195 ( 759 ^;20824 !; 612 ?;22195 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 14:51:37.354
%------------------------------------------------------------------------------
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comp_P5544053077415136032nteger: ( produc723880050079548563nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc723880050079548563nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
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comp_P1414880308642389412nteger: ( produc2221115339011613079nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc2221115339011613079nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
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% Relevant facts (7796)
thf(fact_0_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_1_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_2_Abs__assn__eqI_I2_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
( ! [H: produc3658429121746597890et_nat] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_3_Abs__assn__eqI_I1_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
( ! [H: produc3658429121746597890et_nat] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_4_empty__iff,axiom,
! [C: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ C @ bot_bo7824918357723582233_nat_o ) ).
% empty_iff
thf(fact_5_empty__iff,axiom,
! [C: product_unit] :
~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).
% empty_iff
thf(fact_6_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_7_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_8_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_9_all__not__in__conv,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ! [X2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X2 @ A ) )
= ( A = bot_bo7824918357723582233_nat_o ) ) ).
% all_not_in_conv
thf(fact_10_all__not__in__conv,axiom,
! [A: set_Product_unit] :
( ( ! [X2: product_unit] :
~ ( member_Product_unit @ X2 @ A ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% all_not_in_conv
thf(fact_11_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_12_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_13_all__not__in__conv,axiom,
! [A: set_int] :
( ( ! [X2: int] :
~ ( member_int @ X2 @ A ) )
= ( A = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_14_Collect__empty__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ( collec939566748876313656_nat_o @ P )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_15_Collect__empty__eq,axiom,
! [P: product_unit > $o] :
( ( ( collect_Product_unit @ P )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: product_unit] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_16_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_17_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_18_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( bot_bo7824918357723582233_nat_o
= ( collec939566748876313656_nat_o @ P ) )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_20_empty__Collect__eq,axiom,
! [P: product_unit > $o] :
( ( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ P ) )
= ( ! [X2: product_unit] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_21_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_22_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_23_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_24_emptyE,axiom,
! [A2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).
% emptyE
thf(fact_25_emptyE,axiom,
! [A2: product_unit] :
~ ( member_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% emptyE
thf(fact_26_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_27_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_28_emptyE,axiom,
! [A2: int] :
~ ( member_int @ A2 @ bot_bot_set_int ) ).
% emptyE
thf(fact_29_equals0D,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( A = bot_bo3957492148770167129t_unit )
=> ~ ( member_Product_unit @ A2 @ A ) ) ).
% equals0D
thf(fact_30_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_31_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_32_equals0D,axiom,
! [A: set_int,A2: int] :
( ( A = bot_bot_set_int )
=> ~ ( member_int @ A2 @ A ) ) ).
% equals0D
thf(fact_33_equals0D,axiom,
! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( A = bot_bo7824918357723582233_nat_o )
=> ~ ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% equals0D
thf(fact_34_equals0I,axiom,
! [A: set_Product_unit] :
( ! [Y2: product_unit] :
~ ( member_Product_unit @ Y2 @ A )
=> ( A = bot_bo3957492148770167129t_unit ) ) ).
% equals0I
thf(fact_35_equals0I,axiom,
! [A: set_o] :
( ! [Y2: $o] :
~ ( member_o @ Y2 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_36_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_37_equals0I,axiom,
! [A: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A )
=> ( A = bot_bot_set_int ) ) ).
% equals0I
thf(fact_38_equals0I,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ! [Y2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( A = bot_bo7824918357723582233_nat_o ) ) ).
% equals0I
thf(fact_39_ex__in__conv,axiom,
! [A: set_Product_unit] :
( ( ? [X2: product_unit] : ( member_Product_unit @ X2 @ A ) )
= ( A != bot_bo3957492148770167129t_unit ) ) ).
% ex_in_conv
thf(fact_40_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_41_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_42_ex__in__conv,axiom,
! [A: set_int] :
( ( ? [X2: int] : ( member_int @ X2 @ A ) )
= ( A != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_43_ex__in__conv,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
= ( A != bot_bo7824918357723582233_nat_o ) ) ).
% ex_in_conv
thf(fact_44_set__notEmptyE,axiom,
! [S: set_Product_unit] :
( ( S != bot_bo3957492148770167129t_unit )
=> ~ ! [X3: product_unit] :
~ ( member_Product_unit @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_45_set__notEmptyE,axiom,
! [S: set_o] :
( ( S != bot_bot_set_o )
=> ~ ! [X3: $o] :
~ ( member_o @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_46_set__notEmptyE,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ~ ! [X3: nat] :
~ ( member_nat @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_47_set__notEmptyE,axiom,
! [S: set_int] :
( ( S != bot_bot_set_int )
=> ~ ! [X3: int] :
~ ( member_int @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_48_set__notEmptyE,axiom,
! [S: set_Pr4532377907799695533_nat_o] :
( ( S != bot_bo7824918357723582233_nat_o )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_49_memb__imp__not__empty,axiom,
! [X: product_unit,S: set_Product_unit] :
( ( member_Product_unit @ X @ S )
=> ( S != bot_bo3957492148770167129t_unit ) ) ).
% memb_imp_not_empty
thf(fact_50_memb__imp__not__empty,axiom,
! [X: $o,S: set_o] :
( ( member_o @ X @ S )
=> ( S != bot_bot_set_o ) ) ).
% memb_imp_not_empty
thf(fact_51_memb__imp__not__empty,axiom,
! [X: nat,S: set_nat] :
( ( member_nat @ X @ S )
=> ( S != bot_bot_set_nat ) ) ).
% memb_imp_not_empty
thf(fact_52_memb__imp__not__empty,axiom,
! [X: int,S: set_int] :
( ( member_int @ X @ S )
=> ( S != bot_bot_set_int ) ) ).
% memb_imp_not_empty
thf(fact_53_memb__imp__not__empty,axiom,
! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ S )
=> ( S != bot_bo7824918357723582233_nat_o ) ) ).
% memb_imp_not_empty
thf(fact_54_bot__set__def,axiom,
( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ bot_bo4642748612307482820unit_o ) ) ).
% bot_set_def
thf(fact_55_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_56_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_57_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_58_bot__set__def,axiom,
( bot_bo7824918357723582233_nat_o
= ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).
% bot_set_def
thf(fact_59_mod__false,axiom,
! [H2: produc3658429121746597890et_nat] :
~ ( rep_assn @ bot_bot_assn @ H2 ) ).
% mod_false
thf(fact_60_Set_Ois__empty__def,axiom,
( is_emp5529367068699929823t_unit
= ( ^ [A3: set_Product_unit] : ( A3 = bot_bo3957492148770167129t_unit ) ) ) ).
% Set.is_empty_def
thf(fact_61_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_62_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_63_Set_Ois__empty__def,axiom,
( is_empty_int
= ( ^ [A3: set_int] : ( A3 = bot_bot_set_int ) ) ) ).
% Set.is_empty_def
thf(fact_64_Set_Ois__empty__def,axiom,
( is_emp6650082560925436063_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o] : ( A3 = bot_bo7824918357723582233_nat_o ) ) ) ).
% Set.is_empty_def
thf(fact_65_Collect__empty__eq__bot,axiom,
! [P: product_unit > $o] :
( ( ( collect_Product_unit @ P )
= bot_bo3957492148770167129t_unit )
= ( P = bot_bo4642748612307482820unit_o ) ) ).
% Collect_empty_eq_bot
thf(fact_66_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_67_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_68_Collect__empty__eq__bot,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( P = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_69_Collect__empty__eq__bot,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ( collec939566748876313656_nat_o @ P )
= bot_bo7824918357723582233_nat_o )
= ( P = bot_bo7963750851167320836at_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_70_bot__empty__eq,axiom,
( bot_bo4642748612307482820unit_o
= ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% bot_empty_eq
thf(fact_71_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_72_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_73_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_74_bot__empty__eq,axiom,
( bot_bo7963750851167320836at_o_o
= ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% bot_empty_eq
thf(fact_75_wand__assn__def,axiom,
( wand_assn
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% wand_assn_def
thf(fact_76_one__assn__def,axiom,
( one_one_assn
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_77_Abs__assn__inverse,axiom,
! [Y: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( rep_assn @ ( abs_assn @ Y ) )
= Y ) ) ).
% Abs_assn_inverse
thf(fact_78_one__assn__raw_Osimps,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( As = bot_bot_set_nat ) ) ).
% one_assn_raw.simps
thf(fact_79_one__assn__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( Y
= ( As2 != bot_bot_set_nat ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_80_one__assn__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ( one_assn_raw @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( As2 != bot_bot_set_nat ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_81_one__assn__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( As2 = bot_bot_set_nat ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_82_is__singletonI_H,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y2: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( member_Product_unit @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_sin2160648248035936513t_unit @ A ) ) ) ).
% is_singletonI'
thf(fact_83_is__singletonI_H,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_o @ A ) ) ) ).
% is_singletonI'
thf(fact_84_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_85_is__singletonI_H,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
=> ( ! [X3: int,Y2: int] :
( ( member_int @ X3 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_int @ A ) ) ) ).
% is_singletonI'
thf(fact_86_is__singletonI_H,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_sin5180296473474724033_nat_o @ A ) ) ) ).
% is_singletonI'
thf(fact_87_wand__proper,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q2 ) ) ).
% wand_proper
thf(fact_88_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_89_assn__basic__inequalities_I3_J,axiom,
bot_bot_assn != one_one_assn ).
% assn_basic_inequalities(3)
thf(fact_90_bex2I,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
=> ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
=> ( P @ A2 @ B ) )
=> ? [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ S )
& ( P @ A4 @ B2 ) ) ) ) ).
% bex2I
thf(fact_91_bex2I,axiom,
! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
=> ( P @ A2 @ B ) )
=> ? [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ S )
& ( P @ A4 @ B2 ) ) ) ) ).
% bex2I
thf(fact_92_bex2I,axiom,
! [A2: int,B: int,S: set_Pr958786334691620121nt_int,P: int > int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
=> ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
=> ( P @ A2 @ B ) )
=> ? [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ S )
& ( P @ A4 @ B2 ) ) ) ) ).
% bex2I
thf(fact_93_bex2I,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
=> ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
=> ( P @ A2 @ B ) )
=> ? [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ S )
& ( P @ A4 @ B2 ) ) ) ) ).
% bex2I
thf(fact_94_bex2I,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
=> ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
=> ( P @ A2 @ B ) )
=> ? [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ S )
& ( P @ A4 @ B2 ) ) ) ) ).
% bex2I
thf(fact_95_one__assn__raw_Ocases,axiom,
! [X: produc3658429121746597890et_nat] :
~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( X
!= ( produc7507926704131184380et_nat @ H @ As2 ) ) ).
% one_assn_raw.cases
thf(fact_96_sndE,axiom,
! [X: product_prod_nat_nat,A2: nat,B: nat,P: nat > $o] :
( ( X
= ( product_Pair_nat_nat @ A2 @ B ) )
=> ( ( P @ ( product_snd_nat_nat @ X ) )
=> ( P @ B ) ) ) ).
% sndE
thf(fact_97_sndE,axiom,
! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
( ( X
= ( produc5001842942810119800et_nat @ A2 @ B ) )
=> ( ( P @ ( produc4011572625026189258et_nat @ X ) )
=> ( P @ B ) ) ) ).
% sndE
thf(fact_98_sndE,axiom,
! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: produc3925858234332021118et_nat > $o] :
( ( X
= ( produc2245416461498447860et_nat @ A2 @ B ) )
=> ( ( P @ ( produc5374455773327741254et_nat @ X ) )
=> ( P @ B ) ) ) ).
% sndE
thf(fact_99_sndE,axiom,
! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: set_nat > $o] :
( ( X
= ( produc7507926704131184380et_nat @ A2 @ B ) )
=> ( ( P @ ( produc8586169260539613262et_nat @ X ) )
=> ( P @ B ) ) ) ).
% sndE
thf(fact_100_sndE,axiom,
! [X: product_prod_int_int,A2: int,B: int,P: int > $o] :
( ( X
= ( product_Pair_int_int @ A2 @ B ) )
=> ( ( P @ ( product_snd_int_int @ X ) )
=> ( P @ B ) ) ) ).
% sndE
thf(fact_101_Rep__assn,axiom,
! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).
% Rep_assn
thf(fact_102_mem__Collect__eq,axiom,
! [A2: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A2: product_unit,P: product_unit > $o] :
( ( member_Product_unit @ A2 @ ( collect_Product_unit @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( collec939566748876313656_nat_o
@ ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A: set_Product_unit] :
( ( collect_Product_unit
@ ^ [X2: product_unit] : ( member_Product_unit @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_112_Collect__cong,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ! [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collec939566748876313656_nat_o @ P )
= ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% Collect_cong
thf(fact_113_Collect__cong,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_114_Collect__cong,axiom,
! [P: int > $o,Q2: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q2 ) ) ) ).
% Collect_cong
thf(fact_115_Collect__cong,axiom,
! [P: $o > $o,Q2: $o > $o] :
( ! [X3: $o] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q2 ) ) ) ).
% Collect_cong
thf(fact_116_Collect__cong,axiom,
! [P: product_unit > $o,Q2: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_Product_unit @ P )
= ( collect_Product_unit @ Q2 ) ) ) ).
% Collect_cong
thf(fact_117_Rep__assn__cases,axiom,
! [Y: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ~ ! [X3: assn] :
( Y
!= ( rep_assn @ X3 ) ) ) ).
% Rep_assn_cases
thf(fact_118_Rep__assn__induct,axiom,
! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ! [X3: assn] : ( P @ ( rep_assn @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_assn_induct
thf(fact_119_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: produc3658429121746597890et_nat > $o] :
( ( X
= ( abs_assn @ Y2 ) )
=> ~ ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).
% Abs_assn_cases
thf(fact_120_Abs__assn__induct,axiom,
! [P: assn > $o,X: assn] :
( ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) )
=> ( P @ ( abs_assn @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_assn_induct
thf(fact_121_Abs__assn__inject,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( ( abs_assn @ X )
= ( abs_assn @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_assn_inject
thf(fact_122_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
= ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).
% mod_h_bot_indep
thf(fact_123_old_Oprod_Oinject,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat] :
( ( ( produc7507926704131184380et_nat @ A2 @ B )
= ( produc7507926704131184380et_nat @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_124_old_Oprod_Oinject,axiom,
! [A2: nat,B: nat,A5: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B )
= ( product_Pair_nat_nat @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_125_old_Oprod_Oinject,axiom,
! [A2: int,B: int,A5: int,B3: int] :
( ( ( product_Pair_int_int @ A2 @ B )
= ( product_Pair_int_int @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_126_old_Oprod_Oinject,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
( ( ( produc5001842942810119800et_nat @ A2 @ B )
= ( produc5001842942810119800et_nat @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_127_old_Oprod_Oinject,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
( ( ( produc2245416461498447860et_nat @ A2 @ B )
= ( produc2245416461498447860et_nat @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_128_prod_Oinject,axiom,
! [X1: heap_e7401611519738050253t_unit,X22: set_nat,Y1: heap_e7401611519738050253t_unit,Y22: set_nat] :
( ( ( produc7507926704131184380et_nat @ X1 @ X22 )
= ( produc7507926704131184380et_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_129_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_130_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y22: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_131_prod_Oinject,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
( ( ( produc5001842942810119800et_nat @ X1 @ X22 )
= ( produc5001842942810119800et_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_132_prod_Oinject,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
( ( ( produc2245416461498447860et_nat @ X1 @ X22 )
= ( produc2245416461498447860et_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_133_sndI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_snd_nat_nat @ X )
= Z ) ) ).
% sndI
thf(fact_134_sndI,axiom,
! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
( ( X
= ( produc5001842942810119800et_nat @ Y @ Z ) )
=> ( ( produc4011572625026189258et_nat @ X )
= Z ) ) ).
% sndI
thf(fact_135_sndI,axiom,
! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
( ( X
= ( produc2245416461498447860et_nat @ Y @ Z ) )
=> ( ( produc5374455773327741254et_nat @ X )
= Z ) ) ).
% sndI
thf(fact_136_sndI,axiom,
! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ Y @ Z ) )
=> ( ( produc8586169260539613262et_nat @ X )
= Z ) ) ).
% sndI
thf(fact_137_sndI,axiom,
! [X: product_prod_int_int,Y: int,Z: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z ) )
=> ( ( product_snd_int_int @ X )
= Z ) ) ).
% sndI
thf(fact_138_snd__eqD,axiom,
! [X: nat,Y: nat,A2: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_139_snd__eqD,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat] :
( ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_140_snd__eqD,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3925858234332021118et_nat] :
( ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_141_snd__eqD,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: set_nat] :
( ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_142_snd__eqD,axiom,
! [X: int,Y: int,A2: int] :
( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_143_snd__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_144_snd__conv,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_145_snd__conv,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_146_snd__conv,axiom,
! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_147_snd__conv,axiom,
! [X1: int,X22: int] :
( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_148_eq__snd__iff,axiom,
! [B: nat,P3: product_prod_nat_nat] :
( ( B
= ( product_snd_nat_nat @ P3 ) )
= ( ? [A6: nat] :
( P3
= ( product_Pair_nat_nat @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_149_eq__snd__iff,axiom,
! [B: produc3658429121746597890et_nat,P3: produc3925858234332021118et_nat] :
( ( B
= ( produc4011572625026189258et_nat @ P3 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o] :
( P3
= ( produc5001842942810119800et_nat @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_150_eq__snd__iff,axiom,
! [B: produc3925858234332021118et_nat,P3: produc2732055786443039994et_nat] :
( ( B
= ( produc5374455773327741254et_nat @ P3 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o] :
( P3
= ( produc2245416461498447860et_nat @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_151_eq__snd__iff,axiom,
! [B: set_nat,P3: produc3658429121746597890et_nat] :
( ( B
= ( produc8586169260539613262et_nat @ P3 ) )
= ( ? [A6: heap_e7401611519738050253t_unit] :
( P3
= ( produc7507926704131184380et_nat @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_152_eq__snd__iff,axiom,
! [B: int,P3: product_prod_int_int] :
( ( B
= ( product_snd_int_int @ P3 ) )
= ( ? [A6: int] :
( P3
= ( product_Pair_int_int @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_153_pure__assn__raw_Oelims_I3_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xa
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( As2 = bot_bot_set_nat )
& X ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_154_pure__assn__raw_Oelims_I2_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( ( pure_a825153325127701367it_nat @ X @ Xa )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xa
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ~ ( ( As2 = bot_bot_set_nat )
& X ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_155_pure__assn__raw_Oelims_I1_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
( ( ( pure_a825153325127701367it_nat @ X @ Xa )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xa
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( Y
= ( ~ ( ( As2 = bot_bot_set_nat )
& X ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_156_pure__assn__raw_Osimps,axiom,
! [B: $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( pure_a825153325127701367it_nat @ B @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( ( As = bot_bot_set_nat )
& B ) ) ).
% pure_assn_raw.simps
thf(fact_157_pure__assn__proper,axiom,
! [B: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B ) ) ).
% pure_assn_proper
thf(fact_158_times__assn__raw_Ocases,axiom,
! [X: produc2732055786443039994et_nat] :
~ ! [P4: produc3658429121746597890et_nat > $o,Q3: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
( X
!= ( produc2245416461498447860et_nat @ P4 @ ( produc5001842942810119800et_nat @ Q3 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ).
% times_assn_raw.cases
thf(fact_159_pure__assn__raw_Ocases,axiom,
! [X: produc6197004810343482825et_nat] :
~ ! [B2: $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
( X
!= ( produc3762353314782720579et_nat @ B2 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ).
% pure_assn_raw.cases
thf(fact_160_old_Oprod_Oexhaust,axiom,
! [Y: produc3658429121746597890et_nat] :
~ ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
( Y
!= ( produc7507926704131184380et_nat @ A4 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_161_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B2: nat] :
( Y
!= ( product_Pair_nat_nat @ A4 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_162_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A4: int,B2: int] :
( Y
!= ( product_Pair_int_int @ A4 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_163_old_Oprod_Oexhaust,axiom,
! [Y: produc3925858234332021118et_nat] :
~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
( Y
!= ( produc5001842942810119800et_nat @ A4 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_164_old_Oprod_Oexhaust,axiom,
! [Y: produc2732055786443039994et_nat] :
~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
( Y
!= ( produc2245416461498447860et_nat @ A4 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_165_surj__pair,axiom,
! [P3: produc3658429121746597890et_nat] :
? [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( P3
= ( produc7507926704131184380et_nat @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_166_surj__pair,axiom,
! [P3: product_prod_nat_nat] :
? [X3: nat,Y2: nat] :
( P3
= ( product_Pair_nat_nat @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_167_surj__pair,axiom,
! [P3: product_prod_int_int] :
? [X3: int,Y2: int] :
( P3
= ( product_Pair_int_int @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_168_surj__pair,axiom,
! [P3: produc3925858234332021118et_nat] :
? [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( P3
= ( produc5001842942810119800et_nat @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_169_surj__pair,axiom,
! [P3: produc2732055786443039994et_nat] :
? [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( P3
= ( produc2245416461498447860et_nat @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_170_prod__cases,axiom,
! [P: produc3658429121746597890et_nat > $o,P3: produc3658429121746597890et_nat] :
( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] : ( P @ ( produc7507926704131184380et_nat @ A4 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_171_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
( ! [A4: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_172_prod__cases,axiom,
! [P: product_prod_int_int > $o,P3: product_prod_int_int] :
( ! [A4: int,B2: int] : ( P @ ( product_Pair_int_int @ A4 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_173_prod__cases,axiom,
! [P: produc3925858234332021118et_nat > $o,P3: produc3925858234332021118et_nat] :
( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_174_prod__cases,axiom,
! [P: produc2732055786443039994et_nat > $o,P3: produc2732055786443039994et_nat] :
( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_175_Pair__inject,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat] :
( ( ( produc7507926704131184380et_nat @ A2 @ B )
= ( produc7507926704131184380et_nat @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_176_Pair__inject,axiom,
! [A2: nat,B: nat,A5: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A2 @ B )
= ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_177_Pair__inject,axiom,
! [A2: int,B: int,A5: int,B3: int] :
( ( ( product_Pair_int_int @ A2 @ B )
= ( product_Pair_int_int @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_178_Pair__inject,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
( ( ( produc5001842942810119800et_nat @ A2 @ B )
= ( produc5001842942810119800et_nat @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_179_Pair__inject,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
( ( ( produc2245416461498447860et_nat @ A2 @ B )
= ( produc2245416461498447860et_nat @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_180_prod__cases3,axiom,
! [Y: produc3925858234332021118et_nat] :
~ ! [A4: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] :
( Y
!= ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_181_prod__cases3,axiom,
! [Y: produc2732055786443039994et_nat] :
~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] :
( Y
!= ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_182_prod__cases4,axiom,
! [Y: produc2732055786443039994et_nat] :
~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] :
( Y
!= ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) ) ).
% prod_cases4
thf(fact_183_prod__induct3,axiom,
! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
( ! [A4: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_184_prod__induct3,axiom,
! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_185_prod__induct4,axiom,
! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_186_one__reorient,axiom,
! [X: assn] :
( ( one_one_assn = X )
= ( X = one_one_assn ) ) ).
% one_reorient
thf(fact_187_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_188_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_189_one__reorient,axiom,
! [X: code_integer] :
( ( one_one_Code_integer = X )
= ( X = one_one_Code_integer ) ) ).
% one_reorient
thf(fact_190_one__reorient,axiom,
! [X: code_natural] :
( ( one_one_Code_natural = X )
= ( X = one_one_Code_natural ) ) ).
% one_reorient
thf(fact_191_type__definition__assn,axiom,
type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).
% type_definition_assn
thf(fact_192_times__assn__proper,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper @ ( times_assn_raw @ P @ Q2 ) ) ) ) ).
% times_assn_proper
thf(fact_193_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn @ one_one_assn ).
% is_pure_assn_basic_simps(2)
thf(fact_194_bijective__def,axiom,
( biject2219996528949639174et_nat
= ( ^ [R: set_Pr3948176798113811640et_nat] :
( ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat,Z2: set_nat] :
( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y3 ) @ R )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R ) )
=> ( Y3 = Z2 ) )
& ! [X2: heap_e7401611519738050253t_unit,Y3: heap_e7401611519738050253t_unit,Z2: set_nat] :
( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y3 @ Z2 ) @ R ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_195_bijective__def,axiom,
( bijective_nat_nat
= ( ^ [R: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat,Z2: nat] :
( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R ) )
=> ( Y3 = Z2 ) )
& ! [X2: nat,Y3: nat,Z2: nat] :
( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_196_bijective__def,axiom,
( bijective_int_int
= ( ^ [R: set_Pr958786334691620121nt_int] :
( ! [X2: int,Y3: int,Z2: int] :
( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R ) )
=> ( Y3 = Z2 ) )
& ! [X2: int,Y3: int,Z2: int] :
( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_197_bijective__def,axiom,
( biject2615096655818420098et_nat
= ( ^ [R: set_Pr3286484037609594932et_nat] :
( ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y3 ) @ R )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R ) )
=> ( Y3 = Z2 ) )
& ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3658429121746597890et_nat] :
( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ Z2 ) @ R ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_198_bijective__def,axiom,
( biject1468766312547416318et_nat
= ( ^ [R: set_Pr8536935166611901872et_nat] :
( ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat,Z2: produc3925858234332021118et_nat] :
( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y3 ) @ R )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R ) )
=> ( Y3 = Z2 ) )
& ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3925858234332021118et_nat] :
( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ Z2 ) @ R ) )
=> ( X2 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_199_ssubst__Pair__rhs,axiom,
! [R2: heap_e7401611519738050253t_unit,S2: set_nat,R3: set_Pr3948176798113811640et_nat,S3: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_200_ssubst__Pair__rhs,axiom,
! [R2: nat,S2: nat,R3: set_Pr1261947904930325089at_nat,S3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_201_ssubst__Pair__rhs,axiom,
! [R2: int,S2: int,R3: set_Pr958786334691620121nt_int,S3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_202_ssubst__Pair__rhs,axiom,
! [R2: produc3658429121746597890et_nat > $o,S2: produc3658429121746597890et_nat,R3: set_Pr3286484037609594932et_nat,S3: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_203_ssubst__Pair__rhs,axiom,
! [R2: produc3658429121746597890et_nat > $o,S2: produc3925858234332021118et_nat,R3: set_Pr8536935166611901872et_nat,S3: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_204_is__singletonI,axiom,
! [X: product_unit] : ( is_sin2160648248035936513t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ).
% is_singletonI
thf(fact_205_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_206_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_207_is__singletonI,axiom,
! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).
% is_singletonI
thf(fact_208_is__singletonI,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( is_sin5180296473474724033_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% is_singletonI
thf(fact_209_curryI,axiom,
! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
( ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) )
=> ( produc1807123492089482958_nat_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_210_curryI,axiom,
! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
( ( F @ ( product_Pair_nat_nat @ A2 @ B ) )
=> ( produc1310100445399344235_nat_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_211_curryI,axiom,
! [F: product_prod_int_int > $o,A2: int,B: int] :
( ( F @ ( product_Pair_int_int @ A2 @ B ) )
=> ( produc175634133007206835_int_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_212_curryI,axiom,
! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) )
=> ( produc6216949301066131538_nat_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_213_curryI,axiom,
! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
( ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) )
=> ( produc5101573711933517782_nat_o @ F @ A2 @ B ) ) ).
% curryI
thf(fact_214_insertCI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) ) ).
% insertCI
thf(fact_215_insertCI,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_216_insertCI,axiom,
! [A2: int,B4: set_int,B: int] :
( ( ~ ( member_int @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_int @ A2 @ ( insert_int @ B @ B4 ) ) ) ).
% insertCI
thf(fact_217_insertCI,axiom,
! [A2: $o,B4: set_o,B: $o] :
( ( ~ ( member_o @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).
% insertCI
thf(fact_218_insertCI,axiom,
! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
( ( ~ ( member_Product_unit @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% insertCI
thf(fact_219_insert__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ A ) )
= ( ( A2 = B )
| ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_220_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_221_insert__iff,axiom,
! [A2: int,B: int,A: set_int] :
( ( member_int @ A2 @ ( insert_int @ B @ A ) )
= ( ( A2 = B )
| ( member_int @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_222_insert__iff,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B @ A ) )
= ( ( A2 = B )
| ( member_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_223_insert__iff,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
= ( ( A2 = B )
| ( member_Product_unit @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_224_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_225_insert__absorb2,axiom,
! [X: int,A: set_int] :
( ( insert_int @ X @ ( insert_int @ X @ A ) )
= ( insert_int @ X @ A ) ) ).
% insert_absorb2
thf(fact_226_insert__absorb2,axiom,
! [X: $o,A: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A ) )
= ( insert_o @ X @ A ) ) ).
% insert_absorb2
thf(fact_227_insert__absorb2,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( insert_Product_unit @ X @ ( insert_Product_unit @ X @ A ) )
= ( insert_Product_unit @ X @ A ) ) ).
% insert_absorb2
thf(fact_228_singletonI,axiom,
! [A2: product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% singletonI
thf(fact_229_singletonI,axiom,
! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_230_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_231_singletonI,axiom,
! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_232_singletonI,axiom,
! [A2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).
% singletonI
thf(fact_233_insertE,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ A ) )
=> ( ( A2 != B )
=> ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_234_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_235_insertE,axiom,
! [A2: int,B: int,A: set_int] :
( ( member_int @ A2 @ ( insert_int @ B @ A ) )
=> ( ( A2 != B )
=> ( member_int @ A2 @ A ) ) ) ).
% insertE
thf(fact_236_insertE,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B @ A ) )
=> ( ( A2 = ~ B )
=> ( member_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_237_insertE,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
=> ( ( A2 != B )
=> ( member_Product_unit @ A2 @ A ) ) ) ).
% insertE
thf(fact_238_insertI1,axiom,
! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) ) ).
% insertI1
thf(fact_239_insertI1,axiom,
! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% insertI1
thf(fact_240_insertI1,axiom,
! [A2: int,B4: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B4 ) ) ).
% insertI1
thf(fact_241_insertI1,axiom,
! [A2: $o,B4: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B4 ) ) ).
% insertI1
thf(fact_242_insertI1,axiom,
! [A2: product_unit,B4: set_Product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ B4 ) ) ).
% insertI1
thf(fact_243_insertI2,axiom,
! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ A2 @ B4 )
=> ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) ) ).
% insertI2
thf(fact_244_insertI2,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( member_nat @ A2 @ B4 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_245_insertI2,axiom,
! [A2: int,B4: set_int,B: int] :
( ( member_int @ A2 @ B4 )
=> ( member_int @ A2 @ ( insert_int @ B @ B4 ) ) ) ).
% insertI2
thf(fact_246_insertI2,axiom,
! [A2: $o,B4: set_o,B: $o] :
( ( member_o @ A2 @ B4 )
=> ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).
% insertI2
thf(fact_247_insertI2,axiom,
! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
( ( member_Product_unit @ A2 @ B4 )
=> ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% insertI2
thf(fact_248_Set_Oset__insert,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ A )
=> ~ ! [B5: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ X @ B5 ) )
=> ( member6576561426505652726_nat_o @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_249_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_250_Set_Oset__insert,axiom,
! [X: int,A: set_int] :
( ( member_int @ X @ A )
=> ~ ! [B5: set_int] :
( ( A
= ( insert_int @ X @ B5 ) )
=> ( member_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_251_Set_Oset__insert,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ~ ! [B5: set_o] :
( ( A
= ( insert_o @ X @ B5 ) )
=> ( member_o @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_252_Set_Oset__insert,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X @ A )
=> ~ ! [B5: set_Product_unit] :
( ( A
= ( insert_Product_unit @ X @ B5 ) )
=> ( member_Product_unit @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_253_insert__ident,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
=> ( ( ( insert5175938949040314269_nat_o @ X @ A )
= ( insert5175938949040314269_nat_o @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_254_insert__ident,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_255_insert__ident,axiom,
! [X: int,A: set_int,B4: set_int] :
( ~ ( member_int @ X @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ A )
= ( insert_int @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_256_insert__ident,axiom,
! [X: $o,A: set_o,B4: set_o] :
( ~ ( member_o @ X @ A )
=> ( ~ ( member_o @ X @ B4 )
=> ( ( ( insert_o @ X @ A )
= ( insert_o @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_257_insert__ident,axiom,
! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A )
=> ( ~ ( member_Product_unit @ X @ B4 )
=> ( ( ( insert_Product_unit @ X @ A )
= ( insert_Product_unit @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_258_insert__absorb,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( insert5175938949040314269_nat_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_259_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_260_insert__absorb,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( ( insert_int @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_261_insert__absorb,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_262_insert__absorb,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( insert_Product_unit @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_263_insert__eq__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ~ ( member6576561426505652726_nat_o @ B @ B4 )
=> ( ( ( insert5175938949040314269_nat_o @ A2 @ A )
= ( insert5175938949040314269_nat_o @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C3: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ B @ C3 ) )
& ~ ( member6576561426505652726_nat_o @ B @ C3 )
& ( B4
= ( insert5175938949040314269_nat_o @ A2 @ C3 ) )
& ~ ( member6576561426505652726_nat_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_264_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B4 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B4
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_265_insert__eq__iff,axiom,
! [A2: int,A: set_int,B: int,B4: set_int] :
( ~ ( member_int @ A2 @ A )
=> ( ~ ( member_int @ B @ B4 )
=> ( ( ( insert_int @ A2 @ A )
= ( insert_int @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C3: set_int] :
( ( A
= ( insert_int @ B @ C3 ) )
& ~ ( member_int @ B @ C3 )
& ( B4
= ( insert_int @ A2 @ C3 ) )
& ~ ( member_int @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_266_insert__eq__iff,axiom,
! [A2: $o,A: set_o,B: $o,B4: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ~ ( member_o @ B @ B4 )
=> ( ( ( insert_o @ A2 @ A )
= ( insert_o @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 = ~ B )
=> ? [C3: set_o] :
( ( A
= ( insert_o @ B @ C3 ) )
& ~ ( member_o @ B @ C3 )
& ( B4
= ( insert_o @ A2 @ C3 ) )
& ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_267_insert__eq__iff,axiom,
! [A2: product_unit,A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ A )
=> ( ~ ( member_Product_unit @ B @ B4 )
=> ( ( ( insert_Product_unit @ A2 @ A )
= ( insert_Product_unit @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C3: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B @ C3 ) )
& ~ ( member_Product_unit @ B @ C3 )
& ( B4
= ( insert_Product_unit @ A2 @ C3 ) )
& ~ ( member_Product_unit @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_268_insert__commute,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_269_insert__commute,axiom,
! [X: int,Y: int,A: set_int] :
( ( insert_int @ X @ ( insert_int @ Y @ A ) )
= ( insert_int @ Y @ ( insert_int @ X @ A ) ) ) ).
% insert_commute
thf(fact_270_insert__commute,axiom,
! [X: $o,Y: $o,A: set_o] :
( ( insert_o @ X @ ( insert_o @ Y @ A ) )
= ( insert_o @ Y @ ( insert_o @ X @ A ) ) ) ).
% insert_commute
thf(fact_271_insert__commute,axiom,
! [X: product_unit,Y: product_unit,A: set_Product_unit] :
( ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ A ) )
= ( insert_Product_unit @ Y @ ( insert_Product_unit @ X @ A ) ) ) ).
% insert_commute
thf(fact_272_mk__disjoint__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ A )
=> ? [B5: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ A2 @ B5 ) )
& ~ ( member6576561426505652726_nat_o @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_273_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_274_mk__disjoint__insert,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ? [B5: set_int] :
( ( A
= ( insert_int @ A2 @ B5 ) )
& ~ ( member_int @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_275_mk__disjoint__insert,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ? [B5: set_o] :
( ( A
= ( insert_o @ A2 @ B5 ) )
& ~ ( member_o @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_276_mk__disjoint__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ? [B5: set_Product_unit] :
( ( A
= ( insert_Product_unit @ A2 @ B5 ) )
& ~ ( member_Product_unit @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_277_singletonD,axiom,
! [B: product_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_278_singletonD,axiom,
! [B: $o,A2: $o] :
( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_279_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_280_singletonD,axiom,
! [B: int,A2: int] :
( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_281_singletonD,axiom,
! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_282_singleton__iff,axiom,
! [B: product_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_283_singleton__iff,axiom,
! [B: $o,A2: $o] :
( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_284_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_285_singleton__iff,axiom,
! [B: int,A2: int] :
( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_286_singleton__iff,axiom,
! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_287_doubleton__eq__iff,axiom,
! [A2: product_unit,B: product_unit,C: product_unit,D2: product_unit] :
( ( ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
= ( insert_Product_unit @ C @ ( insert_Product_unit @ D2 @ bot_bo3957492148770167129t_unit ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_288_doubleton__eq__iff,axiom,
! [A2: $o,B: $o,C: $o,D2: $o] :
( ( ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_289_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_290_doubleton__eq__iff,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_291_doubleton__eq__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat > $o,D2: produc3658429121746597890et_nat > $o] :
( ( ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
= ( insert5175938949040314269_nat_o @ C @ ( insert5175938949040314269_nat_o @ D2 @ bot_bo7824918357723582233_nat_o ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_292_insert__not__empty,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit @ A2 @ A )
!= bot_bo3957492148770167129t_unit ) ).
% insert_not_empty
thf(fact_293_insert__not__empty,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ A )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_294_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_295_insert__not__empty,axiom,
! [A2: int,A: set_int] :
( ( insert_int @ A2 @ A )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_296_insert__not__empty,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( insert5175938949040314269_nat_o @ A2 @ A )
!= bot_bo7824918357723582233_nat_o ) ).
% insert_not_empty
thf(fact_297_singleton__inject,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_298_singleton__inject,axiom,
! [A2: $o,B: $o] :
( ( ( insert_o @ A2 @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_299_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_300_singleton__inject,axiom,
! [A2: int,B: int] :
( ( ( insert_int @ A2 @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_301_singleton__inject,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
= ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_302_curryE,axiom,
! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
( ( produc1807123492089482958_nat_o @ F @ A2 @ B )
=> ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) ) ) ).
% curryE
thf(fact_303_curryE,axiom,
! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A2 @ B )
=> ( F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).
% curryE
thf(fact_304_curryE,axiom,
! [F: product_prod_int_int > $o,A2: int,B: int] :
( ( produc175634133007206835_int_o @ F @ A2 @ B )
=> ( F @ ( product_Pair_int_int @ A2 @ B ) ) ) ).
% curryE
thf(fact_305_curryE,axiom,
! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( produc6216949301066131538_nat_o @ F @ A2 @ B )
=> ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) ) ) ).
% curryE
thf(fact_306_curryE,axiom,
! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
( ( produc5101573711933517782_nat_o @ F @ A2 @ B )
=> ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) ) ) ).
% curryE
thf(fact_307_curryD,axiom,
! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
( ( produc1807123492089482958_nat_o @ F @ A2 @ B )
=> ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) ) ) ).
% curryD
thf(fact_308_curryD,axiom,
! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A2 @ B )
=> ( F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).
% curryD
thf(fact_309_curryD,axiom,
! [F: product_prod_int_int > $o,A2: int,B: int] :
( ( produc175634133007206835_int_o @ F @ A2 @ B )
=> ( F @ ( product_Pair_int_int @ A2 @ B ) ) ) ).
% curryD
thf(fact_310_curryD,axiom,
! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
( ( produc6216949301066131538_nat_o @ F @ A2 @ B )
=> ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) ) ) ).
% curryD
thf(fact_311_curryD,axiom,
! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
( ( produc5101573711933517782_nat_o @ F @ A2 @ B )
=> ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) ) ) ).
% curryD
thf(fact_312_is__singleton__def,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A3: set_Product_unit] :
? [X2: product_unit] :
( A3
= ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% is_singleton_def
thf(fact_313_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
? [X2: $o] :
( A3
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_314_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X2: nat] :
( A3
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_315_is__singleton__def,axiom,
( is_singleton_int
= ( ^ [A3: set_int] :
? [X2: int] :
( A3
= ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% is_singleton_def
thf(fact_316_is__singleton__def,axiom,
( is_sin5180296473474724033_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o] :
? [X2: produc3658429121746597890et_nat > $o] :
( A3
= ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% is_singleton_def
thf(fact_317_is__singletonE,axiom,
! [A: set_Product_unit] :
( ( is_sin2160648248035936513t_unit @ A )
=> ~ ! [X3: product_unit] :
( A
!= ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% is_singletonE
thf(fact_318_is__singletonE,axiom,
! [A: set_o] :
( ( is_singleton_o @ A )
=> ~ ! [X3: $o] :
( A
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_319_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_320_is__singletonE,axiom,
! [A: set_int] :
( ( is_singleton_int @ A )
=> ~ ! [X3: int] :
( A
!= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% is_singletonE
thf(fact_321_is__singletonE,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( is_sin5180296473474724033_nat_o @ A )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
( A
!= ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% is_singletonE
thf(fact_322_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn @ bot_bot_assn ).
% is_pure_assn_basic_simps(1)
thf(fact_323_is__singleton__the__elem,axiom,
( is_sin2160648248035936513t_unit
= ( ^ [A3: set_Product_unit] :
( A3
= ( insert_Product_unit @ ( the_el608902216710161154t_unit @ A3 ) @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% is_singleton_the_elem
thf(fact_324_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
( A3
= ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_325_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_326_is__singleton__the__elem,axiom,
( is_singleton_int
= ( ^ [A3: set_int] :
( A3
= ( insert_int @ ( the_elem_int @ A3 ) @ bot_bot_set_int ) ) ) ) ).
% is_singleton_the_elem
thf(fact_327_is__singleton__the__elem,axiom,
( is_sin5180296473474724033_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o] :
( A3
= ( insert5175938949040314269_nat_o @ ( the_el5959152443890766018_nat_o @ A3 ) @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_328_the__elem__eq,axiom,
! [X: product_unit] :
( ( the_el608902216710161154t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% the_elem_eq
thf(fact_329_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_330_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_331_the__elem__eq,axiom,
! [X: int] :
( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% the_elem_eq
thf(fact_332_the__elem__eq,axiom,
! [X: produc3658429121746597890et_nat > $o] :
( ( the_el5959152443890766018_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X ) ).
% the_elem_eq
thf(fact_333_pure__assn__def,axiom,
( pure_assn
= ( ^ [B6: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B6 ) ) ) ) ).
% pure_assn_def
thf(fact_334_times__assn__def,axiom,
( times_times_assn
= ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% times_assn_def
thf(fact_335_type__definition__def,axiom,
( type_d3909072315231072503_nat_o
= ( ^ [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A3: set_Pr4532377907799695533_nat_o] :
( ! [X2: assn] : ( member6576561426505652726_nat_o @ ( Rep @ X2 ) @ A3 )
& ! [X2: assn] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
& ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ A3 )
=> ( ( Rep @ ( Abs @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_336_type__definition__def,axiom,
( type_d8366093980585677751er_int
= ( ^ [Rep: code_integer > int,Abs: int > code_integer,A3: set_int] :
( ! [X2: code_integer] : ( member_int @ ( Rep @ X2 ) @ A3 )
& ! [X2: code_integer] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
& ! [Y3: int] :
( ( member_int @ Y3 @ A3 )
=> ( ( Rep @ ( Abs @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_337_type__definition__def,axiom,
( type_d6188575255521822967unit_o
= ( ^ [Rep: product_unit > $o,Abs: $o > product_unit,A3: set_o] :
( ! [X2: product_unit] : ( member_o @ ( Rep @ X2 ) @ A3 )
& ! [X2: product_unit] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
& ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ( ( Rep @ ( Abs @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_338_type__definition__def,axiom,
( type_d4410041424927559462al_nat
= ( ^ [Rep: code_natural > nat,Abs: nat > code_natural,A3: set_nat] :
( ! [X2: code_natural] : ( member_nat @ ( Rep @ X2 ) @ A3 )
& ! [X2: code_natural] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( ( Rep @ ( Abs @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_339_type__definition_ORep__inverse,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( Abs2 @ ( Rep2 @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_340_type__definition_ORep__inverse,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( Abs2 @ ( Rep2 @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_341_type__definition_ORep__inverse,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( Abs2 @ ( Rep2 @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_342_type__definition_ORep__inverse,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( Abs2 @ ( Rep2 @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_343_type__definition_OAbs__inverse,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ( ( Rep2 @ ( Abs2 @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_344_type__definition_OAbs__inverse,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( member_int @ Y @ A )
=> ( ( Rep2 @ ( Abs2 @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_345_type__definition_OAbs__inverse,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( member_o @ Y @ A )
=> ( ( Rep2 @ ( Abs2 @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_346_type__definition_OAbs__inverse,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( Rep2 @ ( Abs2 @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_347_type__definition_ORep__inject,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn,Y: assn] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( ( Rep2 @ X )
= ( Rep2 @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_348_type__definition_ORep__inject,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer,Y: code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( ( Rep2 @ X )
= ( Rep2 @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_349_type__definition_ORep__inject,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit,Y: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( ( Rep2 @ X )
= ( Rep2 @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_350_type__definition_ORep__inject,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural,Y: code_natural] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( ( Rep2 @ X )
= ( Rep2 @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_351_type__definition_ORep__induct,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ( ! [X3: assn] : ( P @ ( Rep2 @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_352_type__definition_ORep__induct,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int,P: int > $o] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( member_int @ Y @ A )
=> ( ! [X3: code_integer] : ( P @ ( Rep2 @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_353_type__definition_ORep__induct,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o,P: $o > $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( member_o @ Y @ A )
=> ( ! [X3: product_unit] : ( P @ ( Rep2 @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_354_type__definition_ORep__induct,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat,P: nat > $o] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ! [X3: code_natural] : ( P @ ( Rep2 @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_355_type__definition_OAbs__inject,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ( ( ( Abs2 @ X )
= ( Abs2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_356_type__definition_OAbs__inject,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: int,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ( Abs2 @ X )
= ( Abs2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_357_type__definition_OAbs__inject,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: $o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( member_o @ X @ A )
=> ( ( member_o @ Y @ A )
=> ( ( ( Abs2 @ X )
= ( Abs2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_358_type__definition_OAbs__inject,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: nat,Y: nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( Abs2 @ X )
= ( Abs2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_359_pure__assn__eq__conv,axiom,
! [P: $o,Q2: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q2 ) )
= ( P = Q2 ) ) ).
% pure_assn_eq_conv
thf(fact_360_mult_Oright__neutral,axiom,
! [A2: code_integer] :
( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
= A2 ) ).
% mult.right_neutral
thf(fact_361_mult_Oright__neutral,axiom,
! [A2: code_natural] :
( ( times_2397367101498566445atural @ A2 @ one_one_Code_natural )
= A2 ) ).
% mult.right_neutral
thf(fact_362_mult_Oright__neutral,axiom,
! [A2: assn] :
( ( times_times_assn @ A2 @ one_one_assn )
= A2 ) ).
% mult.right_neutral
thf(fact_363_mult_Oright__neutral,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ one_one_rat )
= A2 ) ).
% mult.right_neutral
thf(fact_364_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_365_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_366_mult__1,axiom,
! [A2: code_integer] :
( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
= A2 ) ).
% mult_1
thf(fact_367_mult__1,axiom,
! [A2: code_natural] :
( ( times_2397367101498566445atural @ one_one_Code_natural @ A2 )
= A2 ) ).
% mult_1
thf(fact_368_mult__1,axiom,
! [A2: assn] :
( ( times_times_assn @ one_one_assn @ A2 )
= A2 ) ).
% mult_1
thf(fact_369_mult__1,axiom,
! [A2: rat] :
( ( times_times_rat @ one_one_rat @ A2 )
= A2 ) ).
% mult_1
thf(fact_370_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_371_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_372_merge__pure__star,axiom,
! [A2: $o,B: $o] :
( ( times_times_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
= ( pure_assn
@ ( A2
& B ) ) ) ).
% merge_pure_star
thf(fact_373_star__false__right,axiom,
! [P: assn] :
( ( times_times_assn @ P @ bot_bot_assn )
= bot_bot_assn ) ).
% star_false_right
thf(fact_374_star__false__left,axiom,
! [P: assn] :
( ( times_times_assn @ bot_bot_assn @ P )
= bot_bot_assn ) ).
% star_false_left
thf(fact_375_pure__false,axiom,
( ( pure_assn @ $false )
= bot_bot_assn ) ).
% pure_false
thf(fact_376_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= bot_bot_assn )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_377_is__pure__assn__starI,axiom,
! [A2: assn,B: assn] :
( ( is_pure_assn @ A2 )
=> ( ( is_pure_assn @ B )
=> ( is_pure_assn @ ( times_times_assn @ A2 @ B ) ) ) ) ).
% is_pure_assn_starI
thf(fact_378_is__pure__assn__pure,axiom,
! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% is_pure_assn_pure
thf(fact_379_pure__true,axiom,
( ( pure_assn @ $true )
= one_one_assn ) ).
% pure_true
thf(fact_380_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= one_one_assn )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_381_assn__times__comm,axiom,
( times_times_assn
= ( ^ [P2: assn,Q: assn] : ( times_times_assn @ Q @ P2 ) ) ) ).
% assn_times_comm
thf(fact_382_assn__times__assoc,axiom,
! [P: assn,Q2: assn,R3: assn] :
( ( times_times_assn @ ( times_times_assn @ P @ Q2 ) @ R3 )
= ( times_times_assn @ P @ ( times_times_assn @ Q2 @ R3 ) ) ) ).
% assn_times_assoc
thf(fact_383_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: assn,A2: assn,C: assn] :
( ( times_times_assn @ B @ ( times_times_assn @ A2 @ C ) )
= ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_384_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: rat,A2: rat,C: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A2 @ C ) )
= ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_385_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_386_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_387_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_assn
= ( ^ [A6: assn,B6: assn] : ( times_times_assn @ B6 @ A6 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_388_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_rat
= ( ^ [A6: rat,B6: rat] : ( times_times_rat @ B6 @ A6 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_389_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_nat
= ( ^ [A6: nat,B6: nat] : ( times_times_nat @ B6 @ A6 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_390_ab__semigroup__mult__class_Omult_Ocommute,axiom,
( times_times_int
= ( ^ [A6: int,B6: int] : ( times_times_int @ B6 @ A6 ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_391_mult_Oassoc,axiom,
! [A2: assn,B: assn,C: assn] :
( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
= ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% mult.assoc
thf(fact_392_mult_Oassoc,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
= ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_393_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_394_mult_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_395_comm__monoid__mult__class_Omult__1,axiom,
! [A2: code_integer] :
( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_396_comm__monoid__mult__class_Omult__1,axiom,
! [A2: code_natural] :
( ( times_2397367101498566445atural @ one_one_Code_natural @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_397_comm__monoid__mult__class_Omult__1,axiom,
! [A2: assn] :
( ( times_times_assn @ one_one_assn @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_398_comm__monoid__mult__class_Omult__1,axiom,
! [A2: rat] :
( ( times_times_rat @ one_one_rat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_399_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_400_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_401_mult_Ocomm__neutral,axiom,
! [A2: code_integer] :
( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
= A2 ) ).
% mult.comm_neutral
thf(fact_402_mult_Ocomm__neutral,axiom,
! [A2: code_natural] :
( ( times_2397367101498566445atural @ A2 @ one_one_Code_natural )
= A2 ) ).
% mult.comm_neutral
thf(fact_403_mult_Ocomm__neutral,axiom,
! [A2: assn] :
( ( times_times_assn @ A2 @ one_one_assn )
= A2 ) ).
% mult.comm_neutral
thf(fact_404_mult_Ocomm__neutral,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ one_one_rat )
= A2 ) ).
% mult.comm_neutral
thf(fact_405_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_406_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_407_mod__starE,axiom,
! [A2: assn,B: assn,H2: produc3658429121746597890et_nat] :
( ( rep_assn @ ( times_times_assn @ A2 @ B ) @ H2 )
=> ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A2 @ X_1 )
=> ! [H_2: produc3658429121746597890et_nat] :
~ ( rep_assn @ B @ H_2 ) ) ) ).
% mod_starE
thf(fact_408_mod__starD,axiom,
! [A: assn,B4: assn,H2: produc3658429121746597890et_nat] :
( ( rep_assn @ ( times_times_assn @ A @ B4 ) @ H2 )
=> ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
( ( rep_assn @ A @ H1 )
& ( rep_assn @ B4 @ H22 ) ) ) ).
% mod_starD
thf(fact_409_assn__one__left,axiom,
! [P: assn] :
( ( times_times_assn @ one_one_assn @ P )
= P ) ).
% assn_one_left
thf(fact_410_is__pure__assnE,axiom,
! [A2: assn] :
( ( is_pure_assn @ A2 )
=> ~ ! [P4: $o] :
( A2
!= ( pure_assn @ P4 ) ) ) ).
% is_pure_assnE
thf(fact_411_is__pure__assn__def,axiom,
( is_pure_assn
= ( ^ [A6: assn] :
? [P2: $o] :
( A6
= ( pure_assn @ P2 ) ) ) ) ).
% is_pure_assn_def
thf(fact_412_type__definition_ORep,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( member6576561426505652726_nat_o @ ( Rep2 @ X ) @ A ) ) ).
% type_definition.Rep
thf(fact_413_type__definition_ORep,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( member_int @ ( Rep2 @ X ) @ A ) ) ).
% type_definition.Rep
thf(fact_414_type__definition_ORep,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( member_o @ ( Rep2 @ X ) @ A ) ) ).
% type_definition.Rep
thf(fact_415_type__definition_ORep,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( member_nat @ ( Rep2 @ X ) @ A ) ) ).
% type_definition.Rep
thf(fact_416_type__definition_Ointro,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn] :
( ! [X3: assn] : ( member6576561426505652726_nat_o @ ( Rep2 @ X3 ) @ A )
=> ( ! [X3: assn] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
=> ( ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( ( Rep2 @ ( Abs2 @ Y2 ) )
= Y2 ) )
=> ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A ) ) ) ) ).
% type_definition.intro
thf(fact_417_type__definition_Ointro,axiom,
! [Rep2: code_integer > int,A: set_int,Abs2: int > code_integer] :
( ! [X3: code_integer] : ( member_int @ ( Rep2 @ X3 ) @ A )
=> ( ! [X3: code_integer] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A )
=> ( ( Rep2 @ ( Abs2 @ Y2 ) )
= Y2 ) )
=> ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A ) ) ) ) ).
% type_definition.intro
thf(fact_418_type__definition_Ointro,axiom,
! [Rep2: product_unit > $o,A: set_o,Abs2: $o > product_unit] :
( ! [X3: product_unit] : ( member_o @ ( Rep2 @ X3 ) @ A )
=> ( ! [X3: product_unit] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ A )
=> ( ( Rep2 @ ( Abs2 @ Y2 ) )
= Y2 ) )
=> ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A ) ) ) ) ).
% type_definition.intro
thf(fact_419_type__definition_Ointro,axiom,
! [Rep2: code_natural > nat,A: set_nat,Abs2: nat > code_natural] :
( ! [X3: code_natural] : ( member_nat @ ( Rep2 @ X3 ) @ A )
=> ( ! [X3: code_natural] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( Rep2 @ ( Abs2 @ Y2 ) )
= Y2 ) )
=> ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A ) ) ) ) ).
% type_definition.intro
thf(fact_420_type__definition_OAbs__cases,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ~ ! [Y2: produc3658429121746597890et_nat > $o] :
( ( X
= ( Abs2 @ Y2 ) )
=> ~ ( member6576561426505652726_nat_o @ Y2 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_421_type__definition_OAbs__cases,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ~ ! [Y2: int] :
( ( X
= ( Abs2 @ Y2 ) )
=> ~ ( member_int @ Y2 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_422_type__definition_OAbs__cases,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ~ ! [Y2: $o] :
( ( X
= ( Abs2 @ Y2 ) )
=> ~ ( member_o @ Y2 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_423_type__definition_OAbs__cases,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ~ ! [Y2: nat] :
( ( X
= ( Abs2 @ Y2 ) )
=> ~ ( member_nat @ Y2 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_424_type__definition_ORep__cases,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ~ ! [X3: assn] :
( Y
!= ( Rep2 @ X3 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_425_type__definition_ORep__cases,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( member_int @ Y @ A )
=> ~ ! [X3: code_integer] :
( Y
!= ( Rep2 @ X3 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_426_type__definition_ORep__cases,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( member_o @ Y @ A )
=> ~ ! [X3: product_unit] :
( Y
= ( ~ ( Rep2 @ X3 ) ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_427_type__definition_ORep__cases,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( member_nat @ Y @ A )
=> ~ ! [X3: code_natural] :
( Y
!= ( Rep2 @ X3 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_428_type__definition_OAbs__induct,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,P: assn > $o,X: assn] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( P @ ( Abs2 @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_429_type__definition_OAbs__induct,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,P: code_integer > $o,X: code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ A )
=> ( P @ ( Abs2 @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_430_type__definition_OAbs__induct,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,P: product_unit > $o,X: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ A )
=> ( P @ ( Abs2 @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_431_type__definition_OAbs__induct,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,P: code_natural > $o,X: code_natural] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( P @ ( Abs2 @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_432_total__on__singleton,axiom,
! [X: product_unit] : ( total_5665700600833547665t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% total_on_singleton
thf(fact_433_total__on__singleton,axiom,
! [X: $o] : ( total_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% total_on_singleton
thf(fact_434_total__on__singleton,axiom,
! [X: nat] : ( total_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% total_on_singleton
thf(fact_435_total__on__singleton,axiom,
! [X: int] : ( total_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% total_on_singleton
thf(fact_436_total__on__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( total_7429123086324524369_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% total_on_singleton
thf(fact_437_refl__on__singleton,axiom,
! [X: product_unit] : ( refl_on_Product_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% refl_on_singleton
thf(fact_438_refl__on__singleton,axiom,
! [X: $o] : ( refl_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% refl_on_singleton
thf(fact_439_refl__on__singleton,axiom,
! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_440_refl__on__singleton,axiom,
! [X: int] : ( refl_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% refl_on_singleton
thf(fact_441_refl__on__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( refl_o1075864025857821388_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% refl_on_singleton
thf(fact_442_Id__on__empty,axiom,
( ( id_on_Product_unit @ bot_bo3957492148770167129t_unit )
= bot_bo1654022762305692595t_unit ) ).
% Id_on_empty
thf(fact_443_Id__on__empty,axiom,
( ( id_on_o @ bot_bot_set_o )
= bot_bo7073875226086086771od_o_o ) ).
% Id_on_empty
thf(fact_444_Id__on__empty,axiom,
( ( id_on_nat @ bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Id_on_empty
thf(fact_445_Id__on__empty,axiom,
( ( id_on_int @ bot_bot_set_int )
= bot_bo1796632182523588997nt_int ) ).
% Id_on_empty
thf(fact_446_Id__on__empty,axiom,
( ( id_on_7438695062414216554_nat_o @ bot_bo7824918357723582233_nat_o )
= bot_bo3657187498359500723_nat_o ) ).
% Id_on_empty
thf(fact_447_linear__order__on__singleton,axiom,
! [X: product_unit] : ( order_888282469287196226t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% linear_order_on_singleton
thf(fact_448_linear__order__on__singleton,axiom,
! [X: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_449_linear__order__on__singleton,axiom,
! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_450_linear__order__on__singleton,axiom,
! [X: int] : ( order_4471489696718655927on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% linear_order_on_singleton
thf(fact_451_linear__order__on__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( order_1145710701291701250_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% linear_order_on_singleton
thf(fact_452_pure__assn__raw_Opelims_I1_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
( ( ( pure_a825153325127701367it_nat @ X @ Xa )
= Y )
=> ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xa
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( Y
= ( ( As2 = bot_bot_set_nat )
& X ) )
=> ~ ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_453_pure__assn__raw_Opelims_I2_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( ( pure_a825153325127701367it_nat @ X @ Xa )
=> ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xa
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
=> ~ ( ( As2 = bot_bot_set_nat )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_454_pure__assn__raw_Opelims_I3_J,axiom,
! [X: $o,Xa: produc3658429121746597890et_nat] :
( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
=> ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xa
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
=> ( ( As2 = bot_bot_set_nat )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_455_Image__singleton__iff,axiom,
! [B: nat,R2: set_Pr1763845938948868674it_nat,A2: product_unit] :
( ( member_nat @ B @ ( image_5657742817519485770it_nat @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( member423003724879500579it_nat @ ( produc4803288329813530246it_nat @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_456_Image__singleton__iff,axiom,
! [B: int,R2: set_Pr8497134830871917726it_int,A2: product_unit] :
( ( member_int @ B @ ( image_5655252347010435494it_int @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( member5468524742225079679it_int @ ( produc4800797859304479970it_int @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_457_Image__singleton__iff,axiom,
! [B: $o,R2: set_Pr5464989305201766866unit_o,A2: product_unit] :
( ( member_o @ B @ ( image_Product_unit_o @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( member3689400097658526619unit_o @ ( produc8487917018914037602unit_o @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_458_Image__singleton__iff,axiom,
! [B: product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( member_Product_unit @ B @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_459_Image__singleton__iff,axiom,
! [B: nat,R2: set_Pr2101469702781467981_o_nat,A2: $o] :
( ( member_nat @ B @ ( image_o_nat @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_460_Image__singleton__iff,axiom,
! [B: int,R2: set_Pr8834758594704517033_o_int,A2: $o] :
( ( member_int @ B @ ( image_o_int @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_461_Image__singleton__iff,axiom,
! [B: $o,R2: set_Product_prod_o_o,A2: $o] :
( ( member_o @ B @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_462_Image__singleton__iff,axiom,
! [B: product_unit,R2: set_Pr6671815714118170108t_unit,A2: $o] :
( ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_463_Image__singleton__iff,axiom,
! [B: int,R2: set_Pr7995236796853374141at_int,A2: nat] :
( ( member_int @ B @ ( image_nat_int @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_464_Image__singleton__iff,axiom,
! [B: $o,R2: set_Pr3149072824959771635_nat_o,A2: nat] :
( ( member_o @ B @ ( image_nat_o @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_465_mult_Oright__assoc,axiom,
! [A2: assn,B: assn,C: assn] :
( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
= ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_466_mult_Oright__assoc,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
= ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_467_mult_Oright__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_468_mult_Oright__assoc,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% mult.right_assoc
thf(fact_469_mult_Oright__commute,axiom,
! [A2: assn,B: assn,C: assn] :
( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
= ( times_times_assn @ ( times_times_assn @ A2 @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_470_mult_Oright__commute,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
= ( times_times_rat @ ( times_times_rat @ A2 @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_471_mult_Oright__commute,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ ( times_times_nat @ A2 @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_472_mult_Oright__commute,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
= ( times_times_int @ ( times_times_int @ A2 @ C ) @ B ) ) ).
% mult.right_commute
thf(fact_473_in__range__empty,axiom,
! [H2: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).
% in_range_empty
thf(fact_474_ImageI,axiom,
! [A2: nat,B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_int @ B @ ( image_nat_int @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_475_ImageI,axiom,
! [A2: nat,B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_476_ImageI,axiom,
! [A2: nat,B: product_unit,R2: set_Pr4334478416066269672t_unit,A: set_nat] :
( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ R2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_Product_unit @ B @ ( image_4288904962331477416t_unit @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_477_ImageI,axiom,
! [A2: int,B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ R2 )
=> ( ( member_int @ A2 @ A )
=> ( member_nat @ B @ ( image_int_nat @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_478_ImageI,axiom,
! [A2: int,B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ R2 )
=> ( ( member_int @ A2 @ A )
=> ( member_o @ B @ ( image_int_o @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_479_ImageI,axiom,
! [A2: int,B: product_unit,R2: set_Pr5889264611543340172t_unit,A: set_int] :
( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ R2 )
=> ( ( member_int @ A2 @ A )
=> ( member_Product_unit @ B @ ( image_1405326909533657676t_unit @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_480_ImageI,axiom,
! [A2: $o,B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
=> ( ( member_o @ A2 @ A )
=> ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_481_ImageI,axiom,
! [A2: $o,B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 )
=> ( ( member_o @ A2 @ A )
=> ( member_int @ B @ ( image_o_int @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_482_ImageI,axiom,
! [A2: $o,B: $o,R2: set_Product_prod_o_o,A: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( ( member_o @ A2 @ A )
=> ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_483_ImageI,axiom,
! [A2: $o,B: product_unit,R2: set_Pr6671815714118170108t_unit,A: set_o] :
( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 )
=> ( ( member_o @ A2 @ A )
=> ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ A ) ) ) ) ).
% ImageI
thf(fact_484_Image__empty2,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ( image_8431784415008483091t_unit @ R3 @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Image_empty2
thf(fact_485_Image__empty2,axiom,
! [R3: set_Pr5464989305201766866unit_o] :
( ( image_Product_unit_o @ R3 @ bot_bo3957492148770167129t_unit )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_486_Image__empty2,axiom,
! [R3: set_Pr1763845938948868674it_nat] :
( ( image_5657742817519485770it_nat @ R3 @ bot_bo3957492148770167129t_unit )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_487_Image__empty2,axiom,
! [R3: set_Pr8497134830871917726it_int] :
( ( image_5655252347010435494it_int @ R3 @ bot_bo3957492148770167129t_unit )
= bot_bot_set_int ) ).
% Image_empty2
thf(fact_488_Image__empty2,axiom,
! [R3: set_Pr6671815714118170108t_unit] :
( ( image_o_Product_unit @ R3 @ bot_bot_set_o )
= bot_bo3957492148770167129t_unit ) ).
% Image_empty2
thf(fact_489_Image__empty2,axiom,
! [R3: set_Product_prod_o_o] :
( ( image_o_o @ R3 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_490_Image__empty2,axiom,
! [R3: set_Pr2101469702781467981_o_nat] :
( ( image_o_nat @ R3 @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% Image_empty2
thf(fact_491_Image__empty2,axiom,
! [R3: set_Pr8834758594704517033_o_int] :
( ( image_o_int @ R3 @ bot_bot_set_o )
= bot_bot_set_int ) ).
% Image_empty2
thf(fact_492_Image__empty2,axiom,
! [R3: set_Pr4334478416066269672t_unit] :
( ( image_4288904962331477416t_unit @ R3 @ bot_bot_set_nat )
= bot_bo3957492148770167129t_unit ) ).
% Image_empty2
thf(fact_493_Image__empty2,axiom,
! [R3: set_Pr3149072824959771635_nat_o] :
( ( image_nat_o @ R3 @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% Image_empty2
thf(fact_494_Id__onI,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A2 ) @ ( id_on_7438695062414216554_nat_o @ A ) ) ) ).
% Id_onI
thf(fact_495_Id__onI,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ ( id_on_o @ A ) ) ) ).
% Id_onI
thf(fact_496_Id__onI,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ ( id_on_Product_unit @ A ) ) ) ).
% Id_onI
thf(fact_497_Id__onI,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( id_on_nat @ A ) ) ) ).
% Id_onI
thf(fact_498_Id__onI,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ ( id_on_int @ A ) ) ) ).
% Id_onI
thf(fact_499_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_500_refl__on__domain,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( refl_o1075864025857821388_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
& ( member6576561426505652726_nat_o @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_501_refl__on__domain,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( refl_on_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( ( member_o @ A2 @ A )
& ( member_o @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_502_refl__on__domain,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( refl_on_Product_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
=> ( ( member_Product_unit @ A2 @ A )
& ( member_Product_unit @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_503_refl__on__domain,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( refl_on_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member_nat @ A2 @ A )
& ( member_nat @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_504_refl__on__domain,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( refl_on_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member_int @ A2 @ A )
& ( member_int @ B @ A ) ) ) ) ).
% refl_on_domain
thf(fact_505_rev__ImageI,axiom,
! [A2: nat,A: set_nat,B: int,R2: set_Pr7995236796853374141at_int] :
( ( member_nat @ A2 @ A )
=> ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 )
=> ( member_int @ B @ ( image_nat_int @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_506_rev__ImageI,axiom,
! [A2: nat,A: set_nat,B: $o,R2: set_Pr3149072824959771635_nat_o] :
( ( member_nat @ A2 @ A )
=> ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
=> ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_507_rev__ImageI,axiom,
! [A2: nat,A: set_nat,B: product_unit,R2: set_Pr4334478416066269672t_unit] :
( ( member_nat @ A2 @ A )
=> ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ R2 )
=> ( member_Product_unit @ B @ ( image_4288904962331477416t_unit @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_508_rev__ImageI,axiom,
! [A2: int,A: set_int,B: nat,R2: set_Pr3448869479623346877nt_nat] :
( ( member_int @ A2 @ A )
=> ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ R2 )
=> ( member_nat @ B @ ( image_int_nat @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_509_rev__ImageI,axiom,
! [A2: int,A: set_int,B: $o,R2: set_Pr903927857289325719_int_o] :
( ( member_int @ A2 @ A )
=> ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ R2 )
=> ( member_o @ B @ ( image_int_o @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_510_rev__ImageI,axiom,
! [A2: int,A: set_int,B: product_unit,R2: set_Pr5889264611543340172t_unit] :
( ( member_int @ A2 @ A )
=> ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ R2 )
=> ( member_Product_unit @ B @ ( image_1405326909533657676t_unit @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_511_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: nat,R2: set_Pr2101469702781467981_o_nat] :
( ( member_o @ A2 @ A )
=> ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
=> ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_512_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: int,R2: set_Pr8834758594704517033_o_int] :
( ( member_o @ A2 @ A )
=> ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 )
=> ( member_int @ B @ ( image_o_int @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_513_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: $o,R2: set_Product_prod_o_o] :
( ( member_o @ A2 @ A )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_514_rev__ImageI,axiom,
! [A2: $o,A: set_o,B: product_unit,R2: set_Pr6671815714118170108t_unit] :
( ( member_o @ A2 @ A )
=> ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 )
=> ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ A ) ) ) ) ).
% rev_ImageI
thf(fact_515_Image__iff,axiom,
! [B: set_nat,R2: set_Pr3948176798113811640et_nat,A: set_he8644174446132254637t_unit] :
( ( member_set_nat @ B @ ( image_3942343239045813184et_nat @ R2 @ A ) )
= ( ? [X2: heap_e7401611519738050253t_unit] :
( ( member242389114213480694t_unit @ X2 @ A )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_516_Image__iff,axiom,
! [B: nat,R2: set_Pr1261947904930325089at_nat,A: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R2 @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_517_Image__iff,axiom,
! [B: int,R2: set_Pr958786334691620121nt_int,A: set_int] :
( ( member_int @ B @ ( image_int_int @ R2 @ A ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_518_Image__iff,axiom,
! [B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,A: set_Pr4532377907799695533_nat_o] :
( ( member6260224972018164377et_nat @ B @ ( image_1814916760386648892et_nat @ R2 @ A ) )
= ( ? [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_519_Image__iff,axiom,
! [B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,A: set_Pr4532377907799695533_nat_o] :
( ( member1996754912294343701et_nat @ B @ ( image_7240710759097644216et_nat @ R2 @ A ) )
= ( ? [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_520_ImageE,axiom,
! [B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
( ( member_nat @ B @ ( image_int_nat @ R2 @ A ) )
=> ~ ! [X3: int] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X3 @ B ) @ R2 )
=> ~ ( member_int @ X3 @ A ) ) ) ).
% ImageE
thf(fact_521_ImageE,axiom,
! [B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
( ( member_nat @ B @ ( image_o_nat @ R2 @ A ) )
=> ~ ! [X3: $o] :
( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X3 @ B ) @ R2 )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% ImageE
thf(fact_522_ImageE,axiom,
! [B: nat,R2: set_Pr1763845938948868674it_nat,A: set_Product_unit] :
( ( member_nat @ B @ ( image_5657742817519485770it_nat @ R2 @ A ) )
=> ~ ! [X3: product_unit] :
( ( member423003724879500579it_nat @ ( produc4803288329813530246it_nat @ X3 @ B ) @ R2 )
=> ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% ImageE
thf(fact_523_ImageE,axiom,
! [B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
( ( member_int @ B @ ( image_nat_int @ R2 @ A ) )
=> ~ ! [X3: nat] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X3 @ B ) @ R2 )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_524_ImageE,axiom,
! [B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
( ( member_int @ B @ ( image_o_int @ R2 @ A ) )
=> ~ ! [X3: $o] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X3 @ B ) @ R2 )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% ImageE
thf(fact_525_ImageE,axiom,
! [B: int,R2: set_Pr8497134830871917726it_int,A: set_Product_unit] :
( ( member_int @ B @ ( image_5655252347010435494it_int @ R2 @ A ) )
=> ~ ! [X3: product_unit] :
( ( member5468524742225079679it_int @ ( produc4800797859304479970it_int @ X3 @ B ) @ R2 )
=> ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% ImageE
thf(fact_526_ImageE,axiom,
! [B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
( ( member_o @ B @ ( image_nat_o @ R2 @ A ) )
=> ~ ! [X3: nat] :
( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ X3 @ B ) @ R2 )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% ImageE
thf(fact_527_ImageE,axiom,
! [B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
( ( member_o @ B @ ( image_int_o @ R2 @ A ) )
=> ~ ! [X3: int] :
( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ X3 @ B ) @ R2 )
=> ~ ( member_int @ X3 @ A ) ) ) ).
% ImageE
thf(fact_528_ImageE,axiom,
! [B: $o,R2: set_Product_prod_o_o,A: set_o] :
( ( member_o @ B @ ( image_o_o @ R2 @ A ) )
=> ~ ! [X3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R2 )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% ImageE
thf(fact_529_ImageE,axiom,
! [B: $o,R2: set_Pr5464989305201766866unit_o,A: set_Product_unit] :
( ( member_o @ B @ ( image_Product_unit_o @ R2 @ A ) )
=> ~ ! [X3: product_unit] :
( ( member3689400097658526619unit_o @ ( produc8487917018914037602unit_o @ X3 @ B ) @ R2 )
=> ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% ImageE
thf(fact_530_refl__onD2,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( refl_o1075864025857821388_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
=> ( member6576561426505652726_nat_o @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_531_refl__onD2,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
( ( refl_on_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
=> ( member_o @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_532_refl__onD2,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
( ( refl_on_Product_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
=> ( member_Product_unit @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_533_refl__onD2,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( refl_on_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( member_nat @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_534_refl__onD2,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( refl_on_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( member_int @ Y @ A ) ) ) ).
% refl_onD2
thf(fact_535_refl__onD1,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( refl_o1075864025857821388_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
=> ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% refl_onD1
thf(fact_536_refl__onD1,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
( ( refl_on_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
=> ( member_o @ X @ A ) ) ) ).
% refl_onD1
thf(fact_537_refl__onD1,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
( ( refl_on_Product_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
=> ( member_Product_unit @ X @ A ) ) ) ).
% refl_onD1
thf(fact_538_refl__onD1,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( refl_on_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( member_nat @ X @ A ) ) ) ).
% refl_onD1
thf(fact_539_refl__onD1,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( refl_on_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( member_int @ X @ A ) ) ) ).
% refl_onD1
thf(fact_540_refl__onD,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( refl_o1075864025857821388_nat_o @ A @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_541_refl__onD,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
( ( refl_on_o @ A @ R2 )
=> ( ( member_o @ A2 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_542_refl__onD,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ A @ R2 )
=> ( ( member_Product_unit @ A2 @ A )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_543_refl__onD,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
( ( refl_on_nat @ A @ R2 )
=> ( ( member_nat @ A2 @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_544_refl__onD,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
( ( refl_on_int @ A @ R2 )
=> ( ( member_int @ A2 @ A )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_545_total__on__def,axiom,
( total_on_nat
= ( ^ [A3: set_nat,R4: set_Pr1261947904930325089at_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( ( X2 != Y3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_546_total__on__def,axiom,
( total_on_int
= ( ^ [A3: set_int,R4: set_Pr958786334691620121nt_int] :
! [X2: int] :
( ( member_int @ X2 @ A3 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ A3 )
=> ( ( X2 != Y3 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_547_total__onI,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
| ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_7429123086324524369_nat_o @ A @ R2 ) ) ).
% total_onI
thf(fact_548_total__onI,axiom,
! [A: set_o,R2: set_Product_prod_o_o] :
( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y2 ) @ R2 )
| ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_o @ A @ R2 ) ) ).
% total_onI
thf(fact_549_total__onI,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
( ! [X3: product_unit,Y2: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( member_Product_unit @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ Y2 ) @ R2 )
| ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_5665700600833547665t_unit @ A @ R2 ) ) ).
% total_onI
thf(fact_550_total__onI,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_nat @ A @ R2 ) ) ).
% total_onI
thf(fact_551_total__onI,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y2: int] :
( ( member_int @ X3 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_int @ A @ R2 ) ) ).
% total_onI
thf(fact_552_total__on__empty,axiom,
! [R2: set_Pr5094982260447487303t_unit] : ( total_5665700600833547665t_unit @ bot_bo3957492148770167129t_unit @ R2 ) ).
% total_on_empty
thf(fact_553_total__on__empty,axiom,
! [R2: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R2 ) ).
% total_on_empty
thf(fact_554_total__on__empty,axiom,
! [R2: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R2 ) ).
% total_on_empty
thf(fact_555_total__on__empty,axiom,
! [R2: set_Pr958786334691620121nt_int] : ( total_on_int @ bot_bot_set_int @ R2 ) ).
% total_on_empty
thf(fact_556_total__on__empty,axiom,
! [R2: set_Pr2161125870931222855_nat_o] : ( total_7429123086324524369_nat_o @ bot_bo7824918357723582233_nat_o @ R2 ) ).
% total_on_empty
thf(fact_557_models__in__range,axiom,
! [P: assn,H2: produc3658429121746597890et_nat] :
( ( rep_assn @ P @ H2 )
=> ( in_range @ H2 ) ) ).
% models_in_range
thf(fact_558_Id__on__iff,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( id_on_7438695062414216554_nat_o @ A ) )
= ( ( X = Y )
& ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_559_Id__on__iff,axiom,
! [X: $o,Y: $o,A: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( id_on_o @ A ) )
= ( ( X = Y )
& ( member_o @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_560_Id__on__iff,axiom,
! [X: product_unit,Y: product_unit,A: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( id_on_Product_unit @ A ) )
= ( ( X = Y )
& ( member_Product_unit @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_561_Id__on__iff,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A ) )
= ( ( X = Y )
& ( member_nat @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_562_Id__on__iff,axiom,
! [X: int,Y: int,A: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( id_on_int @ A ) )
= ( ( X = Y )
& ( member_int @ X @ A ) ) ) ).
% Id_on_iff
thf(fact_563_Id__on__eqI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( A2 = B )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ ( id_on_7438695062414216554_nat_o @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_564_Id__on__eqI,axiom,
! [A2: $o,B: $o,A: set_o] :
( ( A2 = B )
=> ( ( member_o @ A2 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( id_on_o @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_565_Id__on__eqI,axiom,
! [A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( A2 = B )
=> ( ( member_Product_unit @ A2 @ A )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ ( id_on_Product_unit @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_566_Id__on__eqI,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( A2 = B )
=> ( ( member_nat @ A2 @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( id_on_nat @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_567_Id__on__eqI,axiom,
! [A2: int,B: int,A: set_int] :
( ( A2 = B )
=> ( ( member_int @ A2 @ A )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( id_on_int @ A ) ) ) ) ).
% Id_on_eqI
thf(fact_568_Id__onE,axiom,
! [C: produc4928098042776334183_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ C @ ( id_on_7438695062414216554_nat_o @ A ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( C
!= ( produc7368190662567826135_nat_o @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_569_Id__onE,axiom,
! [C: product_prod_o_o,A: set_o] :
( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( C
!= ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_570_Id__onE,axiom,
! [C: produc8459935480633519975t_unit,A: set_Product_unit] :
( ( member7821138191923116944t_unit @ C @ ( id_on_Product_unit @ A ) )
=> ~ ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( C
!= ( produc5692694829384537815t_unit @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_571_Id__onE,axiom,
! [C: product_prod_nat_nat,A: set_nat] :
( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( C
!= ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_572_Id__onE,axiom,
! [C: product_prod_int_int,A: set_int] :
( ( member5262025264175285858nt_int @ C @ ( id_on_int @ A ) )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( C
!= ( product_Pair_int_int @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_573_lnear__order__on__empty,axiom,
order_888282469287196226t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% lnear_order_on_empty
thf(fact_574_lnear__order__on__empty,axiom,
order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% lnear_order_on_empty
thf(fact_575_lnear__order__on__empty,axiom,
order_4473980167227706203on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% lnear_order_on_empty
thf(fact_576_lnear__order__on__empty,axiom,
order_4471489696718655927on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% lnear_order_on_empty
thf(fact_577_lnear__order__on__empty,axiom,
order_1145710701291701250_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% lnear_order_on_empty
thf(fact_578_refl__on__empty,axiom,
refl_on_Product_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% refl_on_empty
thf(fact_579_refl__on__empty,axiom,
refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% refl_on_empty
thf(fact_580_refl__on__empty,axiom,
refl_on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% refl_on_empty
thf(fact_581_refl__on__empty,axiom,
refl_on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% refl_on_empty
thf(fact_582_refl__on__empty,axiom,
refl_o1075864025857821388_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% refl_on_empty
thf(fact_583_properD1,axiom,
! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( proper @ P )
=> ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% properD1
thf(fact_584_proper__iff,axiom,
! [P: produc3658429121746597890et_nat > $o,As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
( ( proper @ P )
=> ( ( relH @ As @ H2 @ H3 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).
% proper_iff
thf(fact_585_proper__def,axiom,
( proper
= ( ^ [P2: produc3658429121746597890et_nat > $o] :
! [H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As3: set_nat] :
( ( ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) ) )
& ( ( ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
& ( relH @ As3 @ H4 @ H5 )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) )
=> ( P2 @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ) ) ) ).
% proper_def
thf(fact_586_properD2,axiom,
! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat,H3: heap_e7401611519738050253t_unit] :
( ( proper @ P )
=> ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
=> ( ( relH @ As @ H2 @ H3 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
=> ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).
% properD2
thf(fact_587_properI,axiom,
! [P: produc3658429121746597890et_nat > $o] :
( ! [As2: set_nat,H: heap_e7401611519738050253t_unit] :
( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
=> ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit] :
( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( relH @ As2 @ H @ H6 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
=> ( P @ ( produc7507926704131184380et_nat @ H6 @ As2 ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_588_pair__vimage__is__Image,axiom,
! [U: heap_e7401611519738050253t_unit,E: set_Pr3948176798113811640et_nat] :
( ( vimage5412618565253405765et_nat @ ( produc7507926704131184380et_nat @ U ) @ E )
= ( image_3942343239045813184et_nat @ E @ ( insert8563697753282147485t_unit @ U @ bot_bo3686245532977126937t_unit ) ) ) ).
% pair_vimage_is_Image
thf(fact_589_pair__vimage__is__Image,axiom,
! [U: nat,E: set_Pr1261947904930325089at_nat] :
( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ U ) @ E )
= ( image_nat_nat @ E @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ).
% pair_vimage_is_Image
thf(fact_590_pair__vimage__is__Image,axiom,
! [U: int,E: set_Pr958786334691620121nt_int] :
( ( vimage7872673496510460816nt_int @ ( product_Pair_int_int @ U ) @ E )
= ( image_int_int @ E @ ( insert_int @ U @ bot_bot_set_int ) ) ) ).
% pair_vimage_is_Image
thf(fact_591_pair__vimage__is__Image,axiom,
! [U: produc3658429121746597890et_nat > $o,E: set_Pr3286484037609594932et_nat] :
( ( vimage1990136491886319165et_nat @ ( produc5001842942810119800et_nat @ U ) @ E )
= ( image_1814916760386648892et_nat @ E @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ).
% pair_vimage_is_Image
thf(fact_592_pair__vimage__is__Image,axiom,
! [U: produc3658429121746597890et_nat > $o,E: set_Pr8536935166611901872et_nat] :
( ( vimage7160910655466636085et_nat @ ( produc2245416461498447860et_nat @ U ) @ E )
= ( image_7240710759097644216et_nat @ E @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ).
% pair_vimage_is_Image
thf(fact_593_Range__insert,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( range_9132311542094983838et_nat @ ( insert1271470778893645362et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 ) )
= ( insert_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) ) ) ).
% Range_insert
thf(fact_594_Range__insert,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
= ( insert_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).
% Range_insert
thf(fact_595_Range__insert,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( range_int_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
= ( insert_int @ B @ ( range_int_int @ R2 ) ) ) ).
% Range_insert
thf(fact_596_Range__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( range_3398330936903821850et_nat @ ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 ) )
= ( insert1271470778893645362et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) ) ) ).
% Range_insert
thf(fact_597_Range__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( range_7260058354244137878et_nat @ ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 ) )
= ( insert3111075074545270702et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) ) ) ).
% Range_insert
thf(fact_598_Domain__insert,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( domain2007548326387822151et_nat @ ( insert1271470778893645362et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 ) )
= ( insert8563697753282147485t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) ) ) ).
% Domain_insert
thf(fact_599_Domain__insert,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
= ( insert_nat @ A2 @ ( domain_nat_nat @ R2 ) ) ) ).
% Domain_insert
thf(fact_600_Domain__insert,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( domain_int_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
= ( insert_int @ A2 @ ( domain_int_int @ R2 ) ) ) ).
% Domain_insert
thf(fact_601_Domain__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( domain2358961613337345475et_nat @ ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 ) )
= ( insert5175938949040314269_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) ) ) ).
% Domain_insert
thf(fact_602_Domain__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( domain3618434998481624895et_nat @ ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 ) )
= ( insert5175938949040314269_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) ) ) ).
% Domain_insert
thf(fact_603_preorder__on__empty,axiom,
order_7534304569044981540t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% preorder_on_empty
thf(fact_604_preorder__on__empty,axiom,
order_preorder_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% preorder_on_empty
thf(fact_605_preorder__on__empty,axiom,
order_4861654808422542329on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% preorder_on_empty
thf(fact_606_preorder__on__empty,axiom,
order_4859164337913492053on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% preorder_on_empty
thf(fact_607_preorder__on__empty,axiom,
order_174704566861754596_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% preorder_on_empty
thf(fact_608_trancl__single,axiom,
! [A2: nat,B: nat] :
( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ bot_bo2099793752762293965at_nat ) )
= ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ bot_bo2099793752762293965at_nat ) ) ).
% trancl_single
thf(fact_609_trancl__single,axiom,
! [A2: int,B: int] :
( ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ bot_bo1796632182523588997nt_int ) )
= ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ bot_bo1796632182523588997nt_int ) ) ).
% trancl_single
thf(fact_610_trans__singleton,axiom,
! [A2: nat] : ( trans_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% trans_singleton
thf(fact_611_trans__singleton,axiom,
! [A2: int] : ( trans_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ bot_bo1796632182523588997nt_int ) ) ).
% trans_singleton
thf(fact_612_mult_Osafe__commute,axiom,
! [X: assn,Y: assn,A2: assn,B: assn] :
( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A2 )
=> ( ( times_times_assn @ A2 @ B )
= ( times_times_assn @ B @ A2 ) ) ) ).
% mult.safe_commute
thf(fact_613_mult_Osafe__commute,axiom,
! [X: rat,Y: rat,A2: rat,B: rat] :
( ( syntax3730441303064801268at_rat @ ( times_times_rat @ X @ Y ) @ A2 )
=> ( ( times_times_rat @ A2 @ B )
= ( times_times_rat @ B @ A2 ) ) ) ).
% mult.safe_commute
thf(fact_614_mult_Osafe__commute,axiom,
! [X: nat,Y: nat,A2: nat,B: nat] :
( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A2 )
=> ( ( times_times_nat @ A2 @ B )
= ( times_times_nat @ B @ A2 ) ) ) ).
% mult.safe_commute
thf(fact_615_mult_Osafe__commute,axiom,
! [X: int,Y: int,A2: int,B: int] :
( ( syntax5678989248478167196nt_int @ ( times_times_int @ X @ Y ) @ A2 )
=> ( ( times_times_int @ A2 @ B )
= ( times_times_int @ B @ A2 ) ) ) ).
% mult.safe_commute
thf(fact_616_vimage__eq,axiom,
! [A2: nat,F: nat > nat,B4: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) )
= ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_617_vimage__eq,axiom,
! [A2: nat,F: nat > int,B4: set_int] :
( ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) )
= ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_618_vimage__eq,axiom,
! [A2: nat,F: nat > $o,B4: set_o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) )
= ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_619_vimage__eq,axiom,
! [A2: nat,F: nat > product_unit,B4: set_Product_unit] :
( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
= ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_620_vimage__eq,axiom,
! [A2: int,F: int > nat,B4: set_nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) )
= ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_621_vimage__eq,axiom,
! [A2: int,F: int > int,B4: set_int] :
( ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) )
= ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_622_vimage__eq,axiom,
! [A2: int,F: int > $o,B4: set_o] :
( ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) )
= ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_623_vimage__eq,axiom,
! [A2: int,F: int > product_unit,B4: set_Product_unit] :
( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ B4 ) )
= ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_624_vimage__eq,axiom,
! [A2: $o,F: $o > nat,B4: set_nat] :
( ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) )
= ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_625_vimage__eq,axiom,
! [A2: $o,F: $o > int,B4: set_int] :
( ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) )
= ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_626_vimageI,axiom,
! [F: nat > nat,A2: nat,B: nat,B4: set_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_nat @ B @ B4 )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_627_vimageI,axiom,
! [F: int > nat,A2: int,B: nat,B4: set_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_nat @ B @ B4 )
=> ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_628_vimageI,axiom,
! [F: $o > nat,A2: $o,B: nat,B4: set_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_nat @ B @ B4 )
=> ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_629_vimageI,axiom,
! [F: product_unit > nat,A2: product_unit,B: nat,B4: set_nat] :
( ( ( F @ A2 )
= B )
=> ( ( member_nat @ B @ B4 )
=> ( member_Product_unit @ A2 @ ( vimage6253328473476588386it_nat @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_630_vimageI,axiom,
! [F: nat > int,A2: nat,B: int,B4: set_int] :
( ( ( F @ A2 )
= B )
=> ( ( member_int @ B @ B4 )
=> ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_631_vimageI,axiom,
! [F: int > int,A2: int,B: int,B4: set_int] :
( ( ( F @ A2 )
= B )
=> ( ( member_int @ B @ B4 )
=> ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_632_vimageI,axiom,
! [F: $o > int,A2: $o,B: int,B4: set_int] :
( ( ( F @ A2 )
= B )
=> ( ( member_int @ B @ B4 )
=> ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_633_vimageI,axiom,
! [F: product_unit > int,A2: product_unit,B: int,B4: set_int] :
( ( ( F @ A2 )
= B )
=> ( ( member_int @ B @ B4 )
=> ( member_Product_unit @ A2 @ ( vimage6250838002967538110it_int @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_634_vimageI,axiom,
! [F: nat > $o,A2: nat,B: $o,B4: set_o] :
( ( ( F @ A2 )
= B )
=> ( ( member_o @ B @ B4 )
=> ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_635_vimageI,axiom,
! [F: int > $o,A2: int,B: $o,B4: set_o] :
( ( ( F @ A2 )
= B )
=> ( ( member_o @ B @ B4 )
=> ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_636_vimage__empty,axiom,
! [F: product_unit > product_unit] :
( ( vimage7995052115951654139t_unit @ F @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% vimage_empty
thf(fact_637_vimage__empty,axiom,
! [F: $o > product_unit] :
( ( vimage4222377557387472176t_unit @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_638_vimage__empty,axiom,
! [F: nat > product_unit] :
( ( vimage4884490618288580032t_unit @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_639_vimage__empty,axiom,
! [F: int > product_unit] :
( ( vimage2000912565490760292t_unit @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_int ) ).
% vimage_empty
thf(fact_640_vimage__empty,axiom,
! [F: product_unit > $o] :
( ( vimage8808348054653608838unit_o @ F @ bot_bot_set_o )
= bot_bo3957492148770167129t_unit ) ).
% vimage_empty
thf(fact_641_vimage__empty,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_642_vimage__empty,axiom,
! [F: nat > $o] :
( ( vimage_nat_o @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_643_vimage__empty,axiom,
! [F: int > $o] :
( ( vimage_int_o @ F @ bot_bot_set_o )
= bot_bot_set_int ) ).
% vimage_empty
thf(fact_644_vimage__empty,axiom,
! [F: product_unit > nat] :
( ( vimage6253328473476588386it_nat @ F @ bot_bot_set_nat )
= bot_bo3957492148770167129t_unit ) ).
% vimage_empty
thf(fact_645_vimage__empty,axiom,
! [F: $o > nat] :
( ( vimage_o_nat @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_646_relH__sym,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H2 @ H3 )
=> ( relH @ As @ H3 @ H2 ) ) ).
% relH_sym
thf(fact_647_relH__trans,axiom,
! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H12 @ H23 )
=> ( ( relH @ As @ H23 @ H32 )
=> ( relH @ As @ H12 @ H32 ) ) ) ).
% relH_trans
thf(fact_648_vimage__Collect,axiom,
! [P: nat > $o,F: nat > nat,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_649_vimage__Collect,axiom,
! [P: nat > $o,F: int > nat,Q2: int > $o] :
( ! [X3: int] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_int_nat @ F @ ( collect_nat @ P ) )
= ( collect_int @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_650_vimage__Collect,axiom,
! [P: nat > $o,F: $o > nat,Q2: $o > $o] :
( ! [X3: $o] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_o_nat @ F @ ( collect_nat @ P ) )
= ( collect_o @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_651_vimage__Collect,axiom,
! [P: nat > $o,F: product_unit > nat,Q2: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage6253328473476588386it_nat @ F @ ( collect_nat @ P ) )
= ( collect_Product_unit @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_652_vimage__Collect,axiom,
! [P: int > $o,F: nat > int,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_nat_int @ F @ ( collect_int @ P ) )
= ( collect_nat @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_653_vimage__Collect,axiom,
! [P: int > $o,F: int > int,Q2: int > $o] :
( ! [X3: int] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_int_int @ F @ ( collect_int @ P ) )
= ( collect_int @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_654_vimage__Collect,axiom,
! [P: int > $o,F: $o > int,Q2: $o > $o] :
( ! [X3: $o] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_o_int @ F @ ( collect_int @ P ) )
= ( collect_o @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_655_vimage__Collect,axiom,
! [P: int > $o,F: product_unit > int,Q2: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage6250838002967538110it_int @ F @ ( collect_int @ P ) )
= ( collect_Product_unit @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_656_vimage__Collect,axiom,
! [P: $o > $o,F: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_nat_o @ F @ ( collect_o @ P ) )
= ( collect_nat @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_657_vimage__Collect,axiom,
! [P: $o > $o,F: int > $o,Q2: int > $o] :
( ! [X3: int] :
( ( P @ ( F @ X3 ) )
= ( Q2 @ X3 ) )
=> ( ( vimage_int_o @ F @ ( collect_o @ P ) )
= ( collect_int @ Q2 ) ) ) ).
% vimage_Collect
thf(fact_658_vimageI2,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_659_vimageI2,axiom,
! [F: int > nat,A2: int,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_int @ A2 @ ( vimage_int_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_660_vimageI2,axiom,
! [F: $o > nat,A2: $o,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_661_vimageI2,axiom,
! [F: product_unit > nat,A2: product_unit,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_Product_unit @ A2 @ ( vimage6253328473476588386it_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_662_vimageI2,axiom,
! [F: nat > int,A2: nat,A: set_int] :
( ( member_int @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_int @ F @ A ) ) ) ).
% vimageI2
thf(fact_663_vimageI2,axiom,
! [F: int > int,A2: int,A: set_int] :
( ( member_int @ ( F @ A2 ) @ A )
=> ( member_int @ A2 @ ( vimage_int_int @ F @ A ) ) ) ).
% vimageI2
thf(fact_664_vimageI2,axiom,
! [F: $o > int,A2: $o,A: set_int] :
( ( member_int @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_int @ F @ A ) ) ) ).
% vimageI2
thf(fact_665_vimageI2,axiom,
! [F: product_unit > int,A2: product_unit,A: set_int] :
( ( member_int @ ( F @ A2 ) @ A )
=> ( member_Product_unit @ A2 @ ( vimage6250838002967538110it_int @ F @ A ) ) ) ).
% vimageI2
thf(fact_666_vimageI2,axiom,
! [F: nat > $o,A2: nat,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_667_vimageI2,axiom,
! [F: int > $o,A2: int,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member_int @ A2 @ ( vimage_int_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_668_vimageE,axiom,
! [A2: nat,F: nat > nat,B4: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) )
=> ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_669_vimageE,axiom,
! [A2: nat,F: nat > int,B4: set_int] :
( ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) )
=> ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_670_vimageE,axiom,
! [A2: nat,F: nat > $o,B4: set_o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) )
=> ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_671_vimageE,axiom,
! [A2: nat,F: nat > product_unit,B4: set_Product_unit] :
( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
=> ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_672_vimageE,axiom,
! [A2: int,F: int > nat,B4: set_nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) )
=> ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_673_vimageE,axiom,
! [A2: int,F: int > int,B4: set_int] :
( ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) )
=> ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_674_vimageE,axiom,
! [A2: int,F: int > $o,B4: set_o] :
( ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) )
=> ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_675_vimageE,axiom,
! [A2: int,F: int > product_unit,B4: set_Product_unit] :
( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ B4 ) )
=> ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_676_vimageE,axiom,
! [A2: $o,F: $o > nat,B4: set_nat] :
( ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) )
=> ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_677_vimageE,axiom,
! [A2: $o,F: $o > int,B4: set_int] :
( ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) )
=> ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_678_vimageD,axiom,
! [A2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_679_vimageD,axiom,
! [A2: nat,F: nat > int,A: set_int] :
( ( member_nat @ A2 @ ( vimage_nat_int @ F @ A ) )
=> ( member_int @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_680_vimageD,axiom,
! [A2: nat,F: nat > $o,A: set_o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_681_vimageD,axiom,
! [A2: nat,F: nat > product_unit,A: set_Product_unit] :
( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ A ) )
=> ( member_Product_unit @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_682_vimageD,axiom,
! [A2: int,F: int > nat,A: set_nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_683_vimageD,axiom,
! [A2: int,F: int > int,A: set_int] :
( ( member_int @ A2 @ ( vimage_int_int @ F @ A ) )
=> ( member_int @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_684_vimageD,axiom,
! [A2: int,F: int > $o,A: set_o] :
( ( member_int @ A2 @ ( vimage_int_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_685_vimageD,axiom,
! [A2: int,F: int > product_unit,A: set_Product_unit] :
( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ A ) )
=> ( member_Product_unit @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_686_vimageD,axiom,
! [A2: $o,F: $o > nat,A: set_nat] :
( ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_687_vimageD,axiom,
! [A2: $o,F: $o > int,A: set_int] :
( ( member_o @ A2 @ ( vimage_o_int @ F @ A ) )
=> ( member_int @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_688_transD,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
( ( trans_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_689_transD,axiom,
! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
( ( trans_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_690_transE,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
( ( trans_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_691_transE,axiom,
! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
( ( trans_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_692_transI,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_nat @ R2 ) ) ).
% transI
thf(fact_693_transI,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans_int @ R2 ) ) ).
% transI
thf(fact_694_trans__def,axiom,
( trans_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [X2: nat,Y3: nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R4 ) ) ) ) ) ).
% trans_def
thf(fact_695_trans__def,axiom,
( trans_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [X2: int,Y3: int,Z2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R4 ) ) ) ) ) ).
% trans_def
thf(fact_696_Domain_Ocases,axiom,
! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
=> ~ ! [B2: set_nat] :
~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 ) ) ).
% Domain.cases
thf(fact_697_Domain_Ocases,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
=> ~ ! [B2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ).
% Domain.cases
thf(fact_698_Domain_Ocases,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
=> ~ ! [B2: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) ) ).
% Domain.cases
thf(fact_699_Domain_Ocases,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat] :
~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 ) ) ).
% Domain.cases
thf(fact_700_Domain_Ocases,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
=> ~ ! [B2: produc3925858234332021118et_nat] :
~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 ) ) ).
% Domain.cases
thf(fact_701_Domain_Osimps,axiom,
! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
= ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
( ( A2 = A6 )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_702_Domain_Osimps,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
= ( ? [A6: nat,B6: nat] :
( ( A2 = A6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_703_Domain_Osimps,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
= ( ? [A6: int,B6: int] :
( ( A2 = A6 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_704_Domain_Osimps,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( A2 = A6 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_705_Domain_Osimps,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
( ( A2 = A6 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_706_Domain_ODomainI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
=> ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_707_Domain_ODomainI,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_708_Domain_ODomainI,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( member_int @ A2 @ ( domain_int_int @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_709_Domain_ODomainI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
=> ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_710_Domain_ODomainI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
=> ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_711_DomainE,axiom,
! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
=> ~ ! [B2: set_nat] :
~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 ) ) ).
% DomainE
thf(fact_712_DomainE,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
=> ~ ! [B2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ).
% DomainE
thf(fact_713_DomainE,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
=> ~ ! [B2: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) ) ).
% DomainE
thf(fact_714_DomainE,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat] :
~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 ) ) ).
% DomainE
thf(fact_715_DomainE,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
=> ~ ! [B2: produc3925858234332021118et_nat] :
~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 ) ) ).
% DomainE
thf(fact_716_Domain__iff,axiom,
! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
= ( ? [Y3: set_nat] : ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_717_Domain__iff,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
= ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_718_Domain__iff,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
= ( ? [Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_719_Domain__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
= ( ? [Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_720_Domain__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
= ( ? [Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_721_Range_Ocases,axiom,
! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
=> ~ ! [A4: heap_e7401611519738050253t_unit] :
~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ A2 ) @ R2 ) ) ).
% Range.cases
thf(fact_722_Range_Ocases,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
=> ~ ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A2 ) @ R2 ) ) ).
% Range.cases
thf(fact_723_Range_Ocases,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( range_int_int @ R2 ) )
=> ~ ! [A4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ A2 ) @ R2 ) ) ).
% Range.cases
thf(fact_724_Range_Ocases,axiom,
! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
=> ~ ! [A4: produc3658429121746597890et_nat > $o] :
~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ A2 ) @ R2 ) ) ).
% Range.cases
thf(fact_725_Range_Ocases,axiom,
! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
=> ~ ! [A4: produc3658429121746597890et_nat > $o] :
~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ A2 ) @ R2 ) ) ).
% Range.cases
thf(fact_726_Range_Osimps,axiom,
! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
= ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
( ( A2 = B6 )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_727_Range_Osimps,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
= ( ? [A6: nat,B6: nat] :
( ( A2 = B6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_728_Range_Osimps,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( range_int_int @ R2 ) )
= ( ? [A6: int,B6: int] :
( ( A2 = B6 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_729_Range_Osimps,axiom,
! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( A2 = B6 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_730_Range_Osimps,axiom,
! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
( ( A2 = B6 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_731_Range_Ointros,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
=> ( member_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) ) ) ).
% Range.intros
thf(fact_732_Range_Ointros,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( member_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).
% Range.intros
thf(fact_733_Range_Ointros,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( member_int @ B @ ( range_int_int @ R2 ) ) ) ).
% Range.intros
thf(fact_734_Range_Ointros,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
=> ( member6260224972018164377et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) ) ) ).
% Range.intros
thf(fact_735_Range_Ointros,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
=> ( member1996754912294343701et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) ) ) ).
% Range.intros
thf(fact_736_RangeE,axiom,
! [B: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) )
=> ~ ! [A4: heap_e7401611519738050253t_unit] :
~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ B ) @ R2 ) ) ).
% RangeE
thf(fact_737_RangeE,axiom,
! [B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ B @ ( range_nat_nat @ R2 ) )
=> ~ ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ R2 ) ) ).
% RangeE
thf(fact_738_RangeE,axiom,
! [B: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ B @ ( range_int_int @ R2 ) )
=> ~ ! [A4: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B ) @ R2 ) ) ).
% RangeE
thf(fact_739_RangeE,axiom,
! [B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member6260224972018164377et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) )
=> ~ ! [A4: produc3658429121746597890et_nat > $o] :
~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B ) @ R2 ) ) ).
% RangeE
thf(fact_740_RangeE,axiom,
! [B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member1996754912294343701et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) )
=> ~ ! [A4: produc3658429121746597890et_nat > $o] :
~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B ) @ R2 ) ) ).
% RangeE
thf(fact_741_Range__iff,axiom,
! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
= ( ? [Y3: heap_e7401611519738050253t_unit] : ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_742_Range__iff,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
= ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_743_Range__iff,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ A2 @ ( range_int_int @ R2 ) )
= ( ? [Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ A2 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_744_Range__iff,axiom,
! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
= ( ? [Y3: produc3658429121746597890et_nat > $o] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_745_Range__iff,axiom,
! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
= ( ? [Y3: produc3658429121746597890et_nat > $o] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_746_vimage__singleton__eq,axiom,
! [A2: nat,F: nat > product_unit,B: product_unit] :
( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_747_vimage__singleton__eq,axiom,
! [A2: int,F: int > product_unit,B: product_unit] :
( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_748_vimage__singleton__eq,axiom,
! [A2: $o,F: $o > product_unit,B: product_unit] :
( ( member_o @ A2 @ ( vimage4222377557387472176t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_749_vimage__singleton__eq,axiom,
! [A2: product_unit,F: product_unit > product_unit,B: product_unit] :
( ( member_Product_unit @ A2 @ ( vimage7995052115951654139t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_750_vimage__singleton__eq,axiom,
! [A2: nat,F: nat > $o,B: $o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_751_vimage__singleton__eq,axiom,
! [A2: int,F: int > $o,B: $o] :
( ( member_int @ A2 @ ( vimage_int_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_752_vimage__singleton__eq,axiom,
! [A2: $o,F: $o > $o,B: $o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_753_vimage__singleton__eq,axiom,
! [A2: product_unit,F: product_unit > $o,B: $o] :
( ( member_Product_unit @ A2 @ ( vimage8808348054653608838unit_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_754_vimage__singleton__eq,axiom,
! [A2: nat,F: nat > nat,B: nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_755_vimage__singleton__eq,axiom,
! [A2: int,F: int > nat,B: nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A2 )
= B ) ) ).
% vimage_singleton_eq
thf(fact_756_relH__in__rangeI_I2_J,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H2 @ H3 )
=> ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ).
% relH_in_rangeI(2)
thf(fact_757_relH__in__rangeI_I1_J,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
( ( relH @ As @ H2 @ H3 )
=> ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ).
% relH_in_rangeI(1)
thf(fact_758_relH__refl,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
=> ( relH @ As @ H2 @ H2 ) ) ).
% relH_refl
thf(fact_759_Image__empty__trancl__Image__empty,axiom,
! [R3: set_Pr5094982260447487303t_unit,V: product_unit] :
( ( ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ R3 ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_760_Image__empty__trancl__Image__empty,axiom,
! [R3: set_Product_prod_o_o,V: $o] :
( ( ( image_o_o @ R3 @ ( insert_o @ V @ bot_bot_set_o ) )
= bot_bot_set_o )
=> ( ( image_o_o @ ( transitive_trancl_o @ R3 ) @ ( insert_o @ V @ bot_bot_set_o ) )
= bot_bot_set_o ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_761_Image__empty__trancl__Image__empty,axiom,
! [R3: set_Pr1261947904930325089at_nat,V: nat] :
( ( ( image_nat_nat @ R3 @ ( insert_nat @ V @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_762_Image__empty__trancl__Image__empty,axiom,
! [R3: set_Pr958786334691620121nt_int,V: int] :
( ( ( image_int_int @ R3 @ ( insert_int @ V @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( image_int_int @ ( transi6261509568448316235cl_int @ R3 ) @ ( insert_int @ V @ bot_bot_set_int ) )
= bot_bot_set_int ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_763_Image__empty__trancl__Image__empty,axiom,
! [R3: set_Pr2161125870931222855_nat_o,V: produc3658429121746597890et_nat > $o] :
( ( ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_764_mod__relH,axiom,
! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,P: assn] :
( ( relH @ As @ H2 @ H3 )
=> ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).
% mod_relH
thf(fact_765_for__in__RI,axiom,
! [X: heap_e7401611519738050253t_unit,R3: set_Pr3948176798113811640et_nat] :
( ( member242389114213480694t_unit @ X @ ( domain2007548326387822151et_nat @ R3 ) )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ ( fun_of5878246503291251064et_nat @ R3 @ X ) ) @ R3 ) ) ).
% for_in_RI
thf(fact_766_for__in__RI,axiom,
! [X: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X @ ( domain_nat_nat @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ ( fun_of_rel_nat_nat @ R3 @ X ) ) @ R3 ) ) ).
% for_in_RI
thf(fact_767_for__in__RI,axiom,
! [X: int,R3: set_Pr958786334691620121nt_int] :
( ( member_int @ X @ ( domain_int_int @ R3 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ ( fun_of_rel_int_int @ R3 @ X ) ) @ R3 ) ) ).
% for_in_RI
thf(fact_768_for__in__RI,axiom,
! [X: produc3658429121746597890et_nat > $o,R3: set_Pr3286484037609594932et_nat] :
( ( member6576561426505652726_nat_o @ X @ ( domain2358961613337345475et_nat @ R3 ) )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ ( fun_of3673525222065131764et_nat @ R3 @ X ) ) @ R3 ) ) ).
% for_in_RI
thf(fact_769_for__in__RI,axiom,
! [X: produc3658429121746597890et_nat > $o,R3: set_Pr8536935166611901872et_nat] :
( ( member6576561426505652726_nat_o @ X @ ( domain3618434998481624895et_nat @ R3 ) )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ ( fun_of384177504262425200et_nat @ R3 @ X ) ) @ R3 ) ) ).
% for_in_RI
thf(fact_770_trancl_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 )
=> ~ ! [B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_771_trancl_Ocases,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ R2 )
=> ~ ! [B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_772_trancl_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
= ( ? [A6: nat,B6: nat] :
( ( A1 = A6 )
& ( A22 = B6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) )
| ? [A6: nat,B6: nat,C4: nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_773_trancl_Osimps,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi6261509568448316235cl_int @ R2 ) )
= ( ? [A6: int,B6: int] :
( ( A1 = A6 )
& ( A22 = B6 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) )
| ? [A6: int,B6: int,C4: int] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ ( transi6261509568448316235cl_int @ R2 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_774_trancl_Or__into__trancl,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_775_trancl_Or__into__trancl,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_776_tranclE,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ~ ! [C2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C2 @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_777_tranclE,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ~ ! [C2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ C2 @ B ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_778_trancl__trans,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_779_trancl__trans,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_780_trancl__induct,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_781_trancl__induct,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% trancl_induct
thf(fact_782_r__r__into__trancl,axiom,
! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_783_r__r__into__trancl,axiom,
! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_784_converse__tranclE,axiom,
! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
=> ~ ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_785_converse__tranclE,axiom,
! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 )
=> ~ ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_786_trancl__induct2,axiom,
! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi4137310414928362273et_nat @ R2 ) )
=> ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ R2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ ( transi4137310414928362273et_nat @ R2 ) )
=> ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_787_trancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R2 ) )
=> ( ! [A4: nat,B2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ R2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ ( transi243908449541399842at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_788_trancl__induct2,axiom,
! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
=> ( ! [A4: int,B2: int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ R2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: int,B2: int,Aa: int,Ba: int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_789_trancl__induct2,axiom,
! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ R2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
=> ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_790_trancl__induct2,axiom,
! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ R2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
=> ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_791_converse__trancl__induct,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_792_converse__trancl__induct,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ B ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_793_trancl__trans__induct,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
=> ( P @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( P @ X3 @ Y2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( P @ Y2 @ Z3 )
=> ( P @ X3 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_794_trancl__trans__induct,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,P: int > int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ! [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
=> ( P @ X3 @ Y2 ) )
=> ( ! [X3: int,Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( P @ X3 @ Y2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( P @ Y2 @ Z3 )
=> ( P @ X3 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_795_trancl__into__trancl2,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_796_trancl__into__trancl2,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_797_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_798_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_799_irrefl__trancl__rD,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_800_irrefl__trancl__rD,axiom,
! [R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_801_trancl__Image__advance__ss,axiom,
! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ E )
=> ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ E ) @ ( insert_Product_unit @ U @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_802_trancl__Image__advance__ss,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
=> ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_803_trancl__Image__advance__ss,axiom,
! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ E )
=> ( ord_less_eq_set_int @ ( image_int_int @ ( transi6261509568448316235cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) @ ( image_int_int @ ( transi6261509568448316235cl_int @ E ) @ ( insert_int @ U @ bot_bot_set_int ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_804_trancl__Image__advance__ss,axiom,
! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ E )
=> ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_805_trancl__Image__advance__ss,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_806_wand__assnI,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,R3: assn] :
( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
=> ( ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( ( inf_inf_set_nat @ As @ As4 )
= bot_bot_set_nat )
=> ( ( relH @ As @ H2 @ H6 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
=> ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
=> ( rep_assn @ R3 @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q2 @ R3 ) @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% wand_assnI
thf(fact_807_trancl__Image__in__Range,axiom,
! [R3: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ V2 ) @ ( range_nat_nat @ R3 ) ) ).
% trancl_Image_in_Range
thf(fact_808_Pair__vimage__Sigma,axiom,
! [X: nat,A: set_nat,F: nat > set_Product_unit] :
( ( ( member_nat @ X @ A )
=> ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X ) @ ( produc5506539147108004339t_unit @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X ) @ ( produc5506539147108004339t_unit @ A @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_809_Pair__vimage__Sigma,axiom,
! [X: int,A: set_int,F: int > set_Product_unit] :
( ( ( member_int @ X @ A )
=> ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X ) @ ( produc2622961094310184599t_unit @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X ) @ ( produc2622961094310184599t_unit @ A @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_810_Pair__vimage__Sigma,axiom,
! [X: $o,A: set_o,F: $o > set_Product_unit] :
( ( ( member_o @ X @ A )
=> ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X ) @ ( produc411009820408249917t_unit @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X ) @ ( produc411009820408249917t_unit @ A @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_811_Pair__vimage__Sigma,axiom,
! [X: product_unit,A: set_Product_unit,F: product_unit > set_Product_unit] :
( ( ( member_Product_unit @ X @ A )
=> ( ( vimage8114940889160087125t_unit @ ( produc5692694829384537815t_unit @ X ) @ ( produc6121607364029222536t_unit @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( vimage8114940889160087125t_unit @ ( produc5692694829384537815t_unit @ X ) @ ( produc6121607364029222536t_unit @ A @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_812_Pair__vimage__Sigma,axiom,
! [X: nat,A: set_nat,F: nat > set_o] :
( ( ( member_nat @ X @ A )
=> ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_813_Pair__vimage__Sigma,axiom,
! [X: int,A: set_int,F: int > set_o] :
( ( ( member_int @ X @ A )
=> ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X ) @ ( product_Sigma_int_o @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X ) @ ( product_Sigma_int_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_814_Pair__vimage__Sigma,axiom,
! [X: $o,A: set_o,F: $o > set_o] :
( ( ( member_o @ X @ A )
=> ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X ) @ ( product_Sigma_o_o @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X ) @ ( product_Sigma_o_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_815_Pair__vimage__Sigma,axiom,
! [X: product_unit,A: set_Product_unit,F: product_unit > set_o] :
( ( ( member_Product_unit @ X @ A )
=> ( ( vimage9156843197228287893unit_o @ ( produc8487917018914037602unit_o @ X ) @ ( produc4996980317674386579unit_o @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( vimage9156843197228287893unit_o @ ( produc8487917018914037602unit_o @ X ) @ ( produc4996980317674386579unit_o @ A @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_816_Pair__vimage__Sigma,axiom,
! [X: int,A: set_int,F: int > set_nat] :
( ( ( member_int @ X @ A )
=> ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X ) @ ( produc1456381018704787142nt_nat @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X ) @ ( produc1456381018704787142nt_nat @ A @ F ) )
= bot_bot_set_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_817_Pair__vimage__Sigma,axiom,
! [X: $o,A: set_o,F: $o > set_nat] :
( ( ( member_o @ X @ A )
=> ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X ) @ ( product_Sigma_o_nat @ A @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X ) @ ( product_Sigma_o_nat @ A @ F ) )
= bot_bot_set_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_818_relH__def,axiom,
( relH
= ( ^ [As3: set_nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) )
& ! [T: typerep,X2: nat] :
( ( member_nat @ X2 @ As3 )
=> ( ( ( refs_Product_unit @ H4 @ T @ X2 )
= ( refs_Product_unit @ H5 @ T @ X2 ) )
& ( ( arrays_Product_unit @ H4 @ T @ X2 )
= ( arrays_Product_unit @ H5 @ T @ X2 ) ) ) ) ) ) ) ).
% relH_def
thf(fact_819_vimage__insert,axiom,
! [F: nat > product_unit,A2: product_unit,B4: set_Product_unit] :
( ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ A2 @ B4 ) )
= ( sup_sup_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( vimage4884490618288580032t_unit @ F @ B4 ) ) ) ).
% vimage_insert
thf(fact_820_vimage__insert,axiom,
! [F: nat > $o,A2: $o,B4: set_o] :
( ( vimage_nat_o @ F @ ( insert_o @ A2 @ B4 ) )
= ( sup_sup_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( vimage_nat_o @ F @ B4 ) ) ) ).
% vimage_insert
thf(fact_821_vimage__insert,axiom,
! [F: nat > nat,A2: nat,B4: set_nat] :
( ( vimage_nat_nat @ F @ ( insert_nat @ A2 @ B4 ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% vimage_insert
thf(fact_822_vimage__insert,axiom,
! [F: nat > int,A2: int,B4: set_int] :
( ( vimage_nat_int @ F @ ( insert_int @ A2 @ B4 ) )
= ( sup_sup_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( vimage_nat_int @ F @ B4 ) ) ) ).
% vimage_insert
thf(fact_823_vimage__insert,axiom,
! [F: nat > produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( vimage8201027140005291904_nat_o @ F @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( sup_sup_set_nat @ ( vimage8201027140005291904_nat_o @ F @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( vimage8201027140005291904_nat_o @ F @ B4 ) ) ) ).
% vimage_insert
thf(fact_824_trancl__over__edgeE,axiom,
! [U: nat,W: nat,V1: nat,V22: nat,E: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ W ) @ ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ E ) ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ W ) @ ( transi6264000038957366511cl_nat @ E ) )
=> ~ ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V1 ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V22 @ W ) @ ( transi2905341329935302413cl_nat @ E ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_825_trancl__over__edgeE,axiom,
! [U: int,W: int,V1: int,V22: int,E: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ W ) @ ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ V1 @ V22 ) @ E ) ) )
=> ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ W ) @ ( transi6261509568448316235cl_int @ E ) )
=> ~ ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V1 ) @ ( transi2902850859426252137cl_int @ E ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V22 @ W ) @ ( transi2902850859426252137cl_int @ E ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_826_partial__order__on__empty,axiom,
order_2505591618835473410t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% partial_order_on_empty
thf(fact_827_partial__order__on__empty,axiom,
order_8538205470752950541r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% partial_order_on_empty
thf(fact_828_partial__order__on__empty,axiom,
order_5251275573222108571on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% partial_order_on_empty
thf(fact_829_partial__order__on__empty,axiom,
order_5248785102713058295on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% partial_order_on_empty
thf(fact_830_partial__order__on__empty,axiom,
order_3690095022144090562_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% partial_order_on_empty
thf(fact_831_prod__set__simps_I2_J,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( basic_2698842327037663054et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
= ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) ).
% prod_set_simps(2)
thf(fact_832_prod__set__simps_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( basic_8479756702517612234et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( insert1271470778893645362et_nat @ Y @ bot_bo3771712512152092492et_nat ) ) ).
% prod_set_simps(2)
thf(fact_833_prod__set__simps_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( basic_2768200282441914438et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( insert3111075074545270702et_nat @ Y @ bot_bo1481135142794719944et_nat ) ) ).
% prod_set_simps(2)
thf(fact_834_prod__set__simps_I2_J,axiom,
! [X: nat,Y: nat] :
( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= ( insert_nat @ Y @ bot_bot_set_nat ) ) ).
% prod_set_simps(2)
thf(fact_835_prod__set__simps_I2_J,axiom,
! [X: int,Y: int] :
( ( basic_snds_int_int @ ( product_Pair_int_int @ X @ Y ) )
= ( insert_int @ Y @ bot_bot_set_int ) ) ).
% prod_set_simps(2)
thf(fact_836_dual__order_Orefl,axiom,
! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_837_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_838_dual__order_Orefl,axiom,
! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_839_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_840_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_841_order__refl,axiom,
! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).
% order_refl
thf(fact_842_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_843_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_844_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_845_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_846_subset__antisym,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_847_subsetI,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( member6576561426505652726_nat_o @ X3 @ B4 ) )
=> ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ).
% subsetI
thf(fact_848_subsetI,axiom,
! [A: set_int,B4: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_int @ X3 @ B4 ) )
=> ( ord_less_eq_set_int @ A @ B4 ) ) ).
% subsetI
thf(fact_849_subsetI,axiom,
! [A: set_o,B4: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B4 ) )
=> ( ord_less_eq_set_o @ A @ B4 ) ) ).
% subsetI
thf(fact_850_subsetI,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( member_Product_unit @ X3 @ B4 ) )
=> ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ).
% subsetI
thf(fact_851_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_852_Int__iff,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
= ( ( member6576561426505652726_nat_o @ C @ A )
& ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_853_Int__iff,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
= ( ( member_int @ C @ A )
& ( member_int @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_854_Int__iff,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
= ( ( member_o @ C @ A )
& ( member_o @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_855_Int__iff,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
= ( ( member_Product_unit @ C @ A )
& ( member_Product_unit @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_856_Int__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_857_IntI,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ A )
=> ( ( member6576561426505652726_nat_o @ C @ B4 )
=> ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_858_IntI,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ A )
=> ( ( member_int @ C @ B4 )
=> ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_859_IntI,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ A )
=> ( ( member_o @ C @ B4 )
=> ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_860_IntI,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ A )
=> ( ( member_Product_unit @ C @ B4 )
=> ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_861_IntI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_862_Un__iff,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( ( member6576561426505652726_nat_o @ C @ A )
| ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_863_Un__iff,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
= ( ( member_int @ C @ A )
| ( member_int @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_864_Un__iff,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) )
= ( ( member_o @ C @ A )
| ( member_o @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_865_Un__iff,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( ( member_Product_unit @ C @ A )
| ( member_Product_unit @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_866_Un__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_867_UnCI,axiom,
! [C: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( ~ ( member6576561426505652726_nat_o @ C @ B4 )
=> ( member6576561426505652726_nat_o @ C @ A ) )
=> ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% UnCI
thf(fact_868_UnCI,axiom,
! [C: int,B4: set_int,A: set_int] :
( ( ~ ( member_int @ C @ B4 )
=> ( member_int @ C @ A ) )
=> ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% UnCI
thf(fact_869_UnCI,axiom,
! [C: $o,B4: set_o,A: set_o] :
( ( ~ ( member_o @ C @ B4 )
=> ( member_o @ C @ A ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% UnCI
thf(fact_870_UnCI,axiom,
! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
( ( ~ ( member_Product_unit @ C @ B4 )
=> ( member_Product_unit @ C @ A ) )
=> ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% UnCI
thf(fact_871_UnCI,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnCI
thf(fact_872_empty__subsetI,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% empty_subsetI
thf(fact_873_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_874_empty__subsetI,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% empty_subsetI
thf(fact_875_empty__subsetI,axiom,
! [A: set_Pr4532377907799695533_nat_o] : ( ord_le2965882846123202637_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ).
% empty_subsetI
thf(fact_876_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_877_subset__empty,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% subset_empty
thf(fact_878_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_879_subset__empty,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_880_subset__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o )
= ( A = bot_bo7824918357723582233_nat_o ) ) ).
% subset_empty
thf(fact_881_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_882_insert__subset,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
= ( ( member6576561426505652726_nat_o @ X @ B4 )
& ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_883_insert__subset,axiom,
! [X: int,A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( ( member_int @ X @ B4 )
& ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_884_insert__subset,axiom,
! [X: $o,A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B4 )
= ( ( member_o @ X @ B4 )
& ( ord_less_eq_set_o @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_885_insert__subset,axiom,
! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
= ( ( member_Product_unit @ X @ B4 )
& ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_886_insert__subset,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( ( member_nat @ X @ B4 )
& ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_887_SigmaI,axiom,
! [A2: nat,A: set_nat,B: int,B4: nat > set_int] :
( ( member_nat @ A2 @ A )
=> ( ( member_int @ B @ ( B4 @ A2 ) )
=> ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_888_SigmaI,axiom,
! [A2: nat,A: set_nat,B: $o,B4: nat > set_o] :
( ( member_nat @ A2 @ A )
=> ( ( member_o @ B @ ( B4 @ A2 ) )
=> ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_889_SigmaI,axiom,
! [A2: nat,A: set_nat,B: product_unit,B4: nat > set_Product_unit] :
( ( member_nat @ A2 @ A )
=> ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
=> ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_890_SigmaI,axiom,
! [A2: int,A: set_int,B: nat,B4: int > set_nat] :
( ( member_int @ A2 @ A )
=> ( ( member_nat @ B @ ( B4 @ A2 ) )
=> ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_891_SigmaI,axiom,
! [A2: int,A: set_int,B: $o,B4: int > set_o] :
( ( member_int @ A2 @ A )
=> ( ( member_o @ B @ ( B4 @ A2 ) )
=> ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_892_SigmaI,axiom,
! [A2: int,A: set_int,B: product_unit,B4: int > set_Product_unit] :
( ( member_int @ A2 @ A )
=> ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
=> ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_893_SigmaI,axiom,
! [A2: $o,A: set_o,B: nat,B4: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_nat @ B @ ( B4 @ A2 ) )
=> ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_894_SigmaI,axiom,
! [A2: $o,A: set_o,B: int,B4: $o > set_int] :
( ( member_o @ A2 @ A )
=> ( ( member_int @ B @ ( B4 @ A2 ) )
=> ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_895_SigmaI,axiom,
! [A2: $o,A: set_o,B: $o,B4: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B @ ( B4 @ A2 ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_896_SigmaI,axiom,
! [A2: $o,A: set_o,B: product_unit,B4: $o > set_Product_unit] :
( ( member_o @ A2 @ A )
=> ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
=> ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) ) ) ) ).
% SigmaI
thf(fact_897_mem__Sigma__iff,axiom,
! [A2: nat,B: int,A: set_nat,B4: nat > set_int] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) )
= ( ( member_nat @ A2 @ A )
& ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_898_mem__Sigma__iff,axiom,
! [A2: nat,B: $o,A: set_nat,B4: nat > set_o] :
( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) )
= ( ( member_nat @ A2 @ A )
& ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_899_mem__Sigma__iff,axiom,
! [A2: nat,B: product_unit,A: set_nat,B4: nat > set_Product_unit] :
( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) )
= ( ( member_nat @ A2 @ A )
& ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_900_mem__Sigma__iff,axiom,
! [A2: int,B: nat,A: set_int,B4: int > set_nat] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
= ( ( member_int @ A2 @ A )
& ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_901_mem__Sigma__iff,axiom,
! [A2: int,B: $o,A: set_int,B4: int > set_o] :
( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) )
= ( ( member_int @ A2 @ A )
& ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_902_mem__Sigma__iff,axiom,
! [A2: int,B: product_unit,A: set_int,B4: int > set_Product_unit] :
( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) )
= ( ( member_int @ A2 @ A )
& ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_903_mem__Sigma__iff,axiom,
! [A2: $o,B: nat,A: set_o,B4: $o > set_nat] :
( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) )
= ( ( member_o @ A2 @ A )
& ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_904_mem__Sigma__iff,axiom,
! [A2: $o,B: int,A: set_o,B4: $o > set_int] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) )
= ( ( member_o @ A2 @ A )
& ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_905_mem__Sigma__iff,axiom,
! [A2: $o,B: $o,A: set_o,B4: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) )
= ( ( member_o @ A2 @ A )
& ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_906_mem__Sigma__iff,axiom,
! [A2: $o,B: product_unit,A: set_o,B4: $o > set_Product_unit] :
( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) )
= ( ( member_o @ A2 @ A )
& ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_907_Un__empty,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ( A = bot_bo3957492148770167129t_unit )
& ( B4 = bot_bo3957492148770167129t_unit ) ) ) ).
% Un_empty
thf(fact_908_Un__empty,axiom,
! [A: set_o,B4: set_o] :
( ( ( sup_sup_set_o @ A @ B4 )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B4 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_909_Un__empty,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_910_Un__empty,axiom,
! [A: set_int,B4: set_int] :
( ( ( sup_sup_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ( A = bot_bot_set_int )
& ( B4 = bot_bot_set_int ) ) ) ).
% Un_empty
thf(fact_911_Un__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( sup_su5209123915105501825_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
= ( ( A = bot_bo7824918357723582233_nat_o )
& ( B4 = bot_bo7824918357723582233_nat_o ) ) ) ).
% Un_empty
thf(fact_912_Int__subset__iff,axiom,
! [C5: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A @ B4 ) )
= ( ( ord_less_eq_set_nat @ C5 @ A )
& ( ord_less_eq_set_nat @ C5 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_913_Int__insert__right__if1,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_914_Int__insert__right__if1,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_915_Int__insert__right__if1,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_916_Int__insert__right__if1,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_917_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_918_Int__insert__right__if0,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_919_Int__insert__right__if0,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ~ ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( inf_inf_set_int @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_920_Int__insert__right__if0,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( inf_inf_set_o @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_921_Int__insert__right__if0,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_922_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_923_insert__inter__insert,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_924_insert__inter__insert,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B4 ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_925_insert__inter__insert,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ ( insert_Product_unit @ A2 @ B4 ) )
= ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_926_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_927_Int__insert__left__if1,axiom,
! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ C5 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
= ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_928_Int__insert__left__if1,axiom,
! [A2: int,C5: set_int,B4: set_int] :
( ( member_int @ A2 @ C5 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
= ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_929_Int__insert__left__if1,axiom,
! [A2: $o,C5: set_o,B4: set_o] :
( ( member_o @ A2 @ C5 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_930_Int__insert__left__if1,axiom,
! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ A2 @ C5 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
= ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_931_Int__insert__left__if1,axiom,
! [A2: nat,C5: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_932_Int__insert__left__if0,axiom,
! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ A2 @ C5 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
= ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_933_Int__insert__left__if0,axiom,
! [A2: int,C5: set_int,B4: set_int] :
( ~ ( member_int @ A2 @ C5 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_int @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_934_Int__insert__left__if0,axiom,
! [A2: $o,C5: set_o,B4: set_o] :
( ~ ( member_o @ A2 @ C5 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_o @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_935_Int__insert__left__if0,axiom,
! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ A2 @ C5 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
= ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_936_Int__insert__left__if0,axiom,
! [A2: nat,C5: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_937_Un__subset__iff,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
= ( ( ord_less_eq_set_nat @ A @ C5 )
& ( ord_less_eq_set_nat @ B4 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_938_Un__insert__right,axiom,
! [A: set_int,A2: int,B4: set_int] :
( ( sup_sup_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ A2 @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% Un_insert_right
thf(fact_939_Un__insert__right,axiom,
! [A: set_o,A2: $o,B4: set_o] :
( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% Un_insert_right
thf(fact_940_Un__insert__right,axiom,
! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( insert_Product_unit @ A2 @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% Un_insert_right
thf(fact_941_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% Un_insert_right
thf(fact_942_Un__insert__left,axiom,
! [A2: int,B4: set_int,C5: set_int] :
( ( sup_sup_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
= ( insert_int @ A2 @ ( sup_sup_set_int @ B4 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_943_Un__insert__left,axiom,
! [A2: $o,B4: set_o,C5: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
= ( insert_o @ A2 @ ( sup_sup_set_o @ B4 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_944_Un__insert__left,axiom,
! [A2: product_unit,B4: set_Product_unit,C5: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
= ( insert_Product_unit @ A2 @ ( sup_su793286257634532545t_unit @ B4 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_945_Un__insert__left,axiom,
! [A2: nat,B4: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% Un_insert_left
thf(fact_946_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_947_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_948_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_949_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_950_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_951_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_952_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_953_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_954_vimage__Int,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% vimage_Int
thf(fact_955_vimage__Un,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% vimage_Un
thf(fact_956_relH__dist__union,axiom,
! [As: set_nat,As5: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
( ( relH @ ( sup_sup_set_nat @ As @ As5 ) @ H2 @ H3 )
= ( ( relH @ As @ H2 @ H3 )
& ( relH @ As5 @ H2 @ H3 ) ) ) ).
% relH_dist_union
thf(fact_957_singleton__insert__inj__eq,axiom,
! [B: product_unit,A2: product_unit,A: set_Product_unit] :
( ( ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_958_singleton__insert__inj__eq,axiom,
! [B: $o,A2: $o,A: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_959_singleton__insert__inj__eq,axiom,
! [B: int,A2: int,A: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_960_singleton__insert__inj__eq,axiom,
! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o )
= ( insert5175938949040314269_nat_o @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_961_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_962_singleton__insert__inj__eq_H,axiom,
! [A2: product_unit,A: set_Product_unit,B: product_unit] :
( ( ( insert_Product_unit @ A2 @ A )
= ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
= ( ( A2 = B )
& ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_963_singleton__insert__inj__eq_H,axiom,
! [A2: $o,A: set_o,B: $o] :
( ( ( insert_o @ A2 @ A )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A2 = B )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_964_singleton__insert__inj__eq_H,axiom,
! [A2: int,A: set_int,B: int] :
( ( ( insert_int @ A2 @ A )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A2 = B )
& ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_965_singleton__insert__inj__eq_H,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( ( insert5175938949040314269_nat_o @ A2 @ A )
= ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
= ( ( A2 = B )
& ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_966_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_967_disjoint__insert_I2_J,axiom,
! [A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) )
= ( ~ ( member_Product_unit @ B @ A )
& ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_968_disjoint__insert_I2_J,axiom,
! [A: set_o,B: $o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A @ ( insert_o @ B @ B4 ) ) )
= ( ~ ( member_o @ B @ A )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_969_disjoint__insert_I2_J,axiom,
! [A: set_nat,B: nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B4 ) ) )
= ( ~ ( member_nat @ B @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_970_disjoint__insert_I2_J,axiom,
! [A: set_int,B: int,B4: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ A @ ( insert_int @ B @ B4 ) ) )
= ( ~ ( member_int @ B @ A )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_971_disjoint__insert_I2_J,axiom,
! [A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) )
= ( ~ ( member6576561426505652726_nat_o @ B @ A )
& ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_972_disjoint__insert_I1_J,axiom,
! [B4: set_Product_unit,A2: product_unit,A: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ B4 @ ( insert_Product_unit @ A2 @ A ) )
= bot_bo3957492148770167129t_unit )
= ( ~ ( member_Product_unit @ A2 @ B4 )
& ( ( inf_in4660618365625256667t_unit @ B4 @ A )
= bot_bo3957492148770167129t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_973_disjoint__insert_I1_J,axiom,
! [B4: set_o,A2: $o,A: set_o] :
( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A2 @ A ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B4 )
& ( ( inf_inf_set_o @ B4 @ A )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_974_disjoint__insert_I1_J,axiom,
! [B4: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B4 )
& ( ( inf_inf_set_nat @ B4 @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_975_disjoint__insert_I1_J,axiom,
! [B4: set_int,A2: int,A: set_int] :
( ( ( inf_inf_set_int @ B4 @ ( insert_int @ A2 @ A ) )
= bot_bot_set_int )
= ( ~ ( member_int @ A2 @ B4 )
& ( ( inf_inf_set_int @ B4 @ A )
= bot_bot_set_int ) ) ) ).
% disjoint_insert(1)
thf(fact_976_disjoint__insert_I1_J,axiom,
! [B4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ B4 @ ( insert5175938949040314269_nat_o @ A2 @ A ) )
= bot_bo7824918357723582233_nat_o )
= ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
& ( ( inf_in1906310914598751387_nat_o @ B4 @ A )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% disjoint_insert(1)
thf(fact_977_insert__disjoint_I2_J,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B4 ) )
= ( ~ ( member_Product_unit @ A2 @ B4 )
& ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_978_insert__disjoint_I2_J,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B4 ) )
= ( ~ ( member_o @ A2 @ B4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_979_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 ) )
= ( ~ ( member_nat @ A2 @ B4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_980_insert__disjoint_I2_J,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 ) )
= ( ~ ( member_int @ A2 @ B4 )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_981_insert__disjoint_I2_J,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B4 ) )
= ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
& ( bot_bo7824918357723582233_nat_o
= ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_982_insert__disjoint_I1_J,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ~ ( member_Product_unit @ A2 @ B4 )
& ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_983_insert__disjoint_I1_J,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B4 )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B4 )
& ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_984_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B4 )
& ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_985_insert__disjoint_I1_J,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 )
= bot_bot_set_int )
= ( ~ ( member_int @ A2 @ B4 )
& ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int ) ) ) ).
% insert_disjoint(1)
thf(fact_986_insert__disjoint_I1_J,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B4 )
= bot_bo7824918357723582233_nat_o )
= ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
& ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% insert_disjoint(1)
thf(fact_987_Image__Id__on,axiom,
! [A: set_nat,B4: set_nat] :
( ( image_nat_nat @ ( id_on_nat @ A ) @ B4 )
= ( inf_inf_set_nat @ A @ B4 ) ) ).
% Image_Id_on
thf(fact_988_in__range__dist__union,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat,As5: set_nat] :
( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As5 ) ) )
= ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
& ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As5 ) ) ) ) ).
% in_range_dist_union
thf(fact_989_reachable__mono,axiom,
! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,X4: set_nat,X5: set_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R5 )
=> ( ( ord_less_eq_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ X4 ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R5 ) @ X5 ) ) ) ) ).
% reachable_mono
thf(fact_990_order__antisym__conv,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( ( ord_le2510731241096832064er_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_991_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_992_order__antisym__conv,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_993_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_994_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_995_linorder__le__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_eq_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_996_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_997_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_998_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_999_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1000_ord__le__eq__subst,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1001_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1002_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1003_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1004_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > rat,C: rat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1005_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1006_ord__le__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1007_ord__le__eq__subst,axiom,
! [A2: filter_nat,B: filter_nat,F: filter_nat > rat,C: rat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1008_ord__eq__le__subst,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1009_ord__eq__le__subst,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1010_ord__eq__le__subst,axiom,
! [A2: int,F: rat > int,B: rat,C: rat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1011_ord__eq__le__subst,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1012_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1013_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1014_ord__eq__le__subst,axiom,
! [A2: rat,F: int > rat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1015_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1016_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1017_ord__eq__le__subst,axiom,
! [A2: rat,F: filter_nat > rat,B: filter_nat,C: filter_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ! [X3: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1018_linorder__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_linear
thf(fact_1019_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_1020_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_1021_subset__Collect__conv,axiom,
! [S: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ord_le2965882846123202637_nat_o @ S @ ( collec939566748876313656_nat_o @ P ) )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_1022_subset__Collect__conv,axiom,
! [S: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ S @ ( collect_int @ P ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_1023_subset__Collect__conv,axiom,
! [S: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ S @ ( collect_o @ P ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_1024_subset__Collect__conv,axiom,
! [S: set_Product_unit,P: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ S @ ( collect_Product_unit @ P ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_1025_subset__Collect__conv,axiom,
! [S: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_1026_order__eq__refl,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( X = Y )
=> ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1027_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1028_order__eq__refl,axiom,
! [X: rat,Y: rat] :
( ( X = Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1029_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1030_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_1031_ord__eq__le__eq__trans,axiom,
! [A2: filter_nat,B: filter_nat,C: filter_nat,D2: filter_nat] :
( ( A2 = B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ( C = D2 )
=> ( ord_le2510731241096832064er_nat @ A2 @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1032_ord__eq__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( C = D2 )
=> ( ord_less_eq_set_nat @ A2 @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1033_ord__eq__le__eq__trans,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( A2 = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ( C = D2 )
=> ( ord_less_eq_rat @ A2 @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1034_ord__eq__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( C = D2 )
=> ( ord_less_eq_nat @ A2 @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1035_ord__eq__le__eq__trans,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ( C = D2 )
=> ( ord_less_eq_int @ A2 @ D2 ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_1036_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1037_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1038_order__subst2,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1039_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1040_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1041_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1042_order__subst2,axiom,
! [A2: int,B: int,F: int > rat,C: rat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1043_order__subst2,axiom,
! [A2: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1044_order__subst2,axiom,
! [A2: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1045_order__subst2,axiom,
! [A2: filter_nat,B: filter_nat,F: filter_nat > rat,C: rat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: filter_nat,Y2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_1046_order__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1047_order__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1048_order__subst1,axiom,
! [A2: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1049_order__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1050_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1051_order__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1052_order__subst1,axiom,
! [A2: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1053_order__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1054_order__subst1,axiom,
! [A2: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1055_order__subst1,axiom,
! [A2: filter_nat,F: rat > filter_nat,B: rat,C: rat] :
( ( ord_le2510731241096832064er_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_le2510731241096832064er_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2510731241096832064er_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_1056_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
= ( ^ [A6: filter_nat,B6: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
& ( ord_le2510731241096832064er_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1057_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1058_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [A6: rat,B6: rat] :
( ( ord_less_eq_rat @ A6 @ B6 )
& ( ord_less_eq_rat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1059_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1060_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1061_inter__eq__subsetI,axiom,
! [S: set_nat,S4: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ S @ S4 )
=> ( ( ( inf_inf_set_nat @ A @ S4 )
= ( inf_inf_set_nat @ B4 @ S4 ) )
=> ( ( inf_inf_set_nat @ A @ S )
= ( inf_inf_set_nat @ B4 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_1062_Int__left__commute,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
= ( inf_inf_set_nat @ B4 @ ( inf_inf_set_nat @ A @ C5 ) ) ) ).
% Int_left_commute
thf(fact_1063_Int__Collect__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) ) @ ( inf_in1906310914598751387_nat_o @ B4 @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1064_Int__Collect__mono,axiom,
! [A: set_int,B4: set_int,P: int > $o,Q2: int > $o] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B4 @ ( collect_int @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1065_Int__Collect__mono,axiom,
! [A: set_o,B4: set_o,P: $o > $o,Q2: $o > $o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B4 @ ( collect_o @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1066_Int__Collect__mono,axiom,
! [A: set_Product_unit,B4: set_Product_unit,P: product_unit > $o,Q2: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le3507040750410214029t_unit @ ( inf_in4660618365625256667t_unit @ A @ ( collect_Product_unit @ P ) ) @ ( inf_in4660618365625256667t_unit @ B4 @ ( collect_Product_unit @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1067_Int__Collect__mono,axiom,
! [A: set_nat,B4: set_nat,P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1068_Collect__mono__iff,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1069_Collect__mono__iff,axiom,
! [P: int > $o,Q2: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1070_Collect__mono__iff,axiom,
! [P: $o > $o,Q2: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q2 ) )
= ( ! [X2: $o] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1071_Collect__mono__iff,axiom,
! [P: product_unit > $o,Q2: product_unit > $o] :
( ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q2 ) )
= ( ! [X2: product_unit] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1072_Collect__mono__iff,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1073_Un__left__commute,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
= ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A @ C5 ) ) ) ).
% Un_left_commute
thf(fact_1074_Un__Int__distrib2,axiom,
! [B4: set_nat,C5: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ C5 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ A ) @ ( sup_sup_set_nat @ C5 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1075_Un__Int__assoc__eq,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) )
= ( ord_less_eq_set_nat @ C5 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1076_Int__left__absorb,axiom,
! [A: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ).
% Int_left_absorb
thf(fact_1077_Int__Un__distrib2,axiom,
! [B4: set_nat,C5: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ C5 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ A ) @ ( inf_inf_set_nat @ C5 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1078_Un__left__absorb,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_left_absorb
thf(fact_1079_Un__Int__distrib,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ A @ C5 ) ) ) ).
% Un_Int_distrib
thf(fact_1080_Int__Un__distrib,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ A @ C5 ) ) ) ).
% Int_Un_distrib
thf(fact_1081_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [A3: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_1082_antisym,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_1083_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_1084_antisym,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_1085_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_1086_antisym,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_1087_subset__trans,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C5 )
=> ( ord_less_eq_set_nat @ A @ C5 ) ) ) ).
% subset_trans
thf(fact_1088_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
( ( sup_sup_set_nat @ A3 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_1089_Un__Int__crazy,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ B4 @ C5 ) ) @ ( inf_inf_set_nat @ C5 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ B4 @ C5 ) ) @ ( sup_sup_set_nat @ C5 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1090_Int__greatest,axiom,
! [C5: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( ( ord_less_eq_set_nat @ C5 @ B4 )
=> ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_1091_Collect__mono,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ! [X3: produc3658429121746597890et_nat > $o] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% Collect_mono
thf(fact_1092_Collect__mono,axiom,
! [P: int > $o,Q2: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) ) ) ).
% Collect_mono
thf(fact_1093_Collect__mono,axiom,
! [P: $o > $o,Q2: $o > $o] :
( ! [X3: $o] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q2 ) ) ) ).
% Collect_mono
thf(fact_1094_Collect__mono,axiom,
! [P: product_unit > $o,Q2: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q2 ) ) ) ).
% Collect_mono
thf(fact_1095_Collect__mono,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_1096_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_1097_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B7: set_nat] : ( inf_inf_set_nat @ B7 @ A3 ) ) ) ).
% Int_commute
thf(fact_1098_Int__absorb2,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( inf_inf_set_nat @ A @ B4 )
= A ) ) ).
% Int_absorb2
thf(fact_1099_Int__absorb1,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( inf_inf_set_nat @ A @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_1100_subset__iff,axiom,
( ord_le2965882846123202637_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
! [T: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ T @ A3 )
=> ( member6576561426505652726_nat_o @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1101_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B7: set_int] :
! [T: int] :
( ( member_int @ T @ A3 )
=> ( member_int @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1102_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B7: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1103_subset__iff,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
! [T: product_unit] :
( ( member_Product_unit @ T @ A3 )
=> ( member_Product_unit @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1104_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1105_subset__UnE,axiom,
! [C5: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ ( sup_sup_set_nat @ A @ B4 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A )
=> ! [B8: set_nat] :
( ( ord_less_eq_set_nat @ B8 @ B4 )
=> ( C5
!= ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1106_equalityD2,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% equalityD2
thf(fact_1107_equalityD1,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% equalityD1
thf(fact_1108_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A3 ) ) ) ).
% Un_commute
thf(fact_1109_Un__absorb2,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( sup_sup_set_nat @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_1110_Un__absorb1,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( sup_sup_set_nat @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_1111_Int__lower2,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_1112_Int__lower1,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ A ) ).
% Int_lower1
thf(fact_1113_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_1114_subset__eq,axiom,
( ord_le2965882846123202637_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A3 )
=> ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1115_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B7: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A3 )
=> ( member_int @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1116_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B7: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1117_subset__eq,axiom,
( ord_le3507040750410214029t_unit
= ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A3 )
=> ( member_Product_unit @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1118_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1119_equalityE,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% equalityE
thf(fact_1120_Un__upper2,axiom,
! [B4: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_upper2
thf(fact_1121_Un__upper1,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_upper1
thf(fact_1122_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_1123_Int__assoc,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% Int_assoc
thf(fact_1124_Un__least,axiom,
! [A: set_nat,C5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ C5 )
=> ( ( ord_less_eq_set_nat @ B4 @ C5 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 ) ) ) ).
% Un_least
thf(fact_1125_Un__assoc,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% Un_assoc
thf(fact_1126_Int__mono,axiom,
! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C5 )
=> ( ( ord_less_eq_set_nat @ B4 @ D3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ C5 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1127_subsetD,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ( member6576561426505652726_nat_o @ C @ A )
=> ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1128_subsetD,axiom,
! [A: set_int,B4: set_int,C: int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1129_subsetD,axiom,
! [A: set_o,B4: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1130_subsetD,axiom,
! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( member_Product_unit @ C @ A )
=> ( member_Product_unit @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1131_subsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1132_in__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1133_in__mono,axiom,
! [A: set_int,B4: set_int,X: int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_int @ X @ A )
=> ( member_int @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1134_in__mono,axiom,
! [A: set_o,B4: set_o,X: $o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( member_o @ X @ A )
=> ( member_o @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1135_in__mono,axiom,
! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( member_Product_unit @ X @ A )
=> ( member_Product_unit @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1136_in__mono,axiom,
! [A: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1137_ball__Un,axiom,
! [A: set_nat,B4: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1138_Un__mono,axiom,
! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C5 )
=> ( ( ord_less_eq_set_nat @ B4 @ D3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ C5 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_1139_bex__Un,axiom,
! [A: set_nat,B4: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B4 ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B4 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1140_IntD2,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
=> ( member6576561426505652726_nat_o @ C @ B4 ) ) ).
% IntD2
thf(fact_1141_IntD2,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
=> ( member_int @ C @ B4 ) ) ).
% IntD2
thf(fact_1142_IntD2,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
=> ( member_o @ C @ B4 ) ) ).
% IntD2
thf(fact_1143_IntD2,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
=> ( member_Product_unit @ C @ B4 ) ) ).
% IntD2
thf(fact_1144_IntD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_1145_IntD1,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
=> ( member6576561426505652726_nat_o @ C @ A ) ) ).
% IntD1
thf(fact_1146_IntD1,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
=> ( member_int @ C @ A ) ) ).
% IntD1
thf(fact_1147_IntD1,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
=> ( member_o @ C @ A ) ) ).
% IntD1
thf(fact_1148_IntD1,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
=> ( member_Product_unit @ C @ A ) ) ).
% IntD1
thf(fact_1149_IntD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_1150_UnI2,axiom,
! [C: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ B4 )
=> ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% UnI2
thf(fact_1151_UnI2,axiom,
! [C: int,B4: set_int,A: set_int] :
( ( member_int @ C @ B4 )
=> ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% UnI2
thf(fact_1152_UnI2,axiom,
! [C: $o,B4: set_o,A: set_o] :
( ( member_o @ C @ B4 )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% UnI2
thf(fact_1153_UnI2,axiom,
! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
( ( member_Product_unit @ C @ B4 )
=> ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% UnI2
thf(fact_1154_UnI2,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI2
thf(fact_1155_UnI1,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ A )
=> ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% UnI1
thf(fact_1156_UnI1,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ A )
=> ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% UnI1
thf(fact_1157_UnI1,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ A )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% UnI1
thf(fact_1158_UnI1,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ A )
=> ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% UnI1
thf(fact_1159_UnI1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI1
thf(fact_1160_IntE,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
=> ~ ( ( member6576561426505652726_nat_o @ C @ A )
=> ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% IntE
thf(fact_1161_IntE,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
=> ~ ( ( member_int @ C @ A )
=> ~ ( member_int @ C @ B4 ) ) ) ).
% IntE
thf(fact_1162_IntE,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
=> ~ ( ( member_o @ C @ A )
=> ~ ( member_o @ C @ B4 ) ) ) ).
% IntE
thf(fact_1163_IntE,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
=> ~ ( ( member_Product_unit @ C @ A )
=> ~ ( member_Product_unit @ C @ B4 ) ) ) ).
% IntE
thf(fact_1164_IntE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_1165_UnE,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
=> ( ~ ( member6576561426505652726_nat_o @ C @ A )
=> ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% UnE
thf(fact_1166_UnE,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
=> ( ~ ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% UnE
thf(fact_1167_UnE,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) )
=> ( ~ ( member_o @ C @ A )
=> ( member_o @ C @ B4 ) ) ) ).
% UnE
thf(fact_1168_UnE,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
=> ( ~ ( member_Product_unit @ C @ A )
=> ( member_Product_unit @ C @ B4 ) ) ) ).
% UnE
thf(fact_1169_UnE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_1170_dual__order_Otrans,axiom,
! [B: filter_nat,A2: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ C @ B )
=> ( ord_le2510731241096832064er_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1171_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1172_dual__order_Otrans,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1173_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1174_dual__order_Otrans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_1175_dual__order_Oantisym,axiom,
! [B: filter_nat,A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_1176_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_1177_dual__order_Oantisym,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_1178_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_1179_dual__order_Oantisym,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_1180_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
= ( ^ [A6: filter_nat,B6: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
& ( ord_le2510731241096832064er_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1181_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A6 )
& ( ord_less_eq_set_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1182_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [A6: rat,B6: rat] :
( ( ord_less_eq_rat @ B6 @ A6 )
& ( ord_less_eq_rat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1183_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1184_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1185_linorder__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B: rat] :
( ! [A4: rat,B2: rat] :
( ( ord_less_eq_rat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: rat,B2: rat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_1186_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat,B2: nat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_1187_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: int,B2: int] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_1188_order__trans,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ Y @ Z )
=> ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1189_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1190_order__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_eq_rat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1191_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_1192_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_1193_order_Otrans,axiom,
! [A2: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_1194_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_1195_order_Otrans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_1196_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_1197_order_Otrans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_1198_order__antisym,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1199_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1200_order__antisym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1201_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1202_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1203_ord__le__eq__trans,axiom,
! [A2: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1204_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1205_ord__le__eq__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1206_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1207_ord__le__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1208_ord__eq__le__trans,axiom,
! [A2: filter_nat,B: filter_nat,C: filter_nat] :
( ( A2 = B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1209_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1210_ord__eq__le__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( A2 = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1211_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1212_ord__eq__le__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1213_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
& ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1214_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1215_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
& ( ord_less_eq_rat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1216_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1217_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1218_le__cases3,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ( ord_less_eq_rat @ X @ Y )
=> ~ ( ord_less_eq_rat @ Y @ Z ) )
=> ( ( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_eq_rat @ X @ Z ) )
=> ( ( ( ord_less_eq_rat @ X @ Z )
=> ~ ( ord_less_eq_rat @ Z @ Y ) )
=> ( ( ( ord_less_eq_rat @ Z @ Y )
=> ~ ( ord_less_eq_rat @ Y @ X ) )
=> ( ( ( ord_less_eq_rat @ Y @ Z )
=> ~ ( ord_less_eq_rat @ Z @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z @ X )
=> ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1219_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1220_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1221_nle__le,axiom,
! [A2: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A2 @ B ) )
= ( ( ord_less_eq_rat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_1222_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_1223_nle__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B ) )
= ( ( ord_less_eq_int @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_1224_Sigma__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,C5: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_nat,D3: ( produc3658429121746597890et_nat > $o ) > set_nat] :
( ( ord_le2965882846123202637_nat_o @ A @ C5 )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
=> ( ord_le4460016228128491106_o_nat @ ( produc3142209312014666069_o_nat @ A @ B4 ) @ ( produc3142209312014666069_o_nat @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_1225_Sigma__mono,axiom,
! [A: set_int,C5: set_int,B4: int > set_nat,D3: int > set_nat] :
( ( ord_less_eq_set_int @ A @ C5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
=> ( ord_le5333435103577920093nt_nat @ ( produc1456381018704787142nt_nat @ A @ B4 ) @ ( produc1456381018704787142nt_nat @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_1226_Sigma__mono,axiom,
! [A: set_o,C5: set_o,B4: $o > set_nat,D3: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ C5 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
=> ( ord_le6472378536984071597_o_nat @ ( product_Sigma_o_nat @ A @ B4 ) @ ( product_Sigma_o_nat @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_1227_Sigma__mono,axiom,
! [A: set_Product_unit,C5: set_Product_unit,B4: product_unit > set_nat,D3: product_unit > set_nat] :
( ( ord_le3507040750410214029t_unit @ A @ C5 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
=> ( ord_le8121013740822022818it_nat @ ( produc6875377002296012693it_nat @ A @ B4 ) @ ( produc6875377002296012693it_nat @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_1228_Sigma__mono,axiom,
! [A: set_nat,C5: set_nat,B4: nat > set_nat,D3: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ C5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A @ B4 ) @ ( produc457027306803732586at_nat @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_1229_rtrancl__Un__separator__converseE,axiom,
! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P @ Q2 ) ) )
=> ( ! [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ ( transi2905341329935302413cl_nat @ P ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ Q2 )
=> ( Y2 = X3 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_1230_rtrancl__Un__separator__converseE,axiom,
! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ P @ Q2 ) ) )
=> ( ! [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ B ) @ ( transi2902850859426252137cl_int @ P ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ Q2 )
=> ( Y2 = X3 ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_1231_rtrancl__Un__separatorE,axiom,
! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P @ Q2 ) ) )
=> ( ! [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X3 ) @ ( transi2905341329935302413cl_nat @ P ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Q2 )
=> ( X3 = Y2 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_1232_rtrancl__Un__separatorE,axiom,
! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ P @ Q2 ) ) )
=> ( ! [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X3 ) @ ( transi2902850859426252137cl_int @ P ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ Q2 )
=> ( X3 = Y2 ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_1233_disjoint__mono,axiom,
! [A2: set_Product_unit,A5: set_Product_unit,B: set_Product_unit,B3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ A5 )
=> ( ( ord_le3507040750410214029t_unit @ B @ B3 )
=> ( ( ( inf_in4660618365625256667t_unit @ A5 @ B3 )
= bot_bo3957492148770167129t_unit )
=> ( ( inf_in4660618365625256667t_unit @ A2 @ B )
= bot_bo3957492148770167129t_unit ) ) ) ) ).
% disjoint_mono
thf(fact_1234_disjoint__mono,axiom,
! [A2: set_o,A5: set_o,B: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A2 @ A5 )
=> ( ( ord_less_eq_set_o @ B @ B3 )
=> ( ( ( inf_inf_set_o @ A5 @ B3 )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o ) ) ) ) ).
% disjoint_mono
thf(fact_1235_disjoint__mono,axiom,
! [A2: set_int,A5: set_int,B: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A2 @ A5 )
=> ( ( ord_less_eq_set_int @ B @ B3 )
=> ( ( ( inf_inf_set_int @ A5 @ B3 )
= bot_bot_set_int )
=> ( ( inf_inf_set_int @ A2 @ B )
= bot_bot_set_int ) ) ) ) ).
% disjoint_mono
thf(fact_1236_disjoint__mono,axiom,
! [A2: set_Pr4532377907799695533_nat_o,A5: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A2 @ A5 )
=> ( ( ord_le2965882846123202637_nat_o @ B @ B3 )
=> ( ( ( inf_in1906310914598751387_nat_o @ A5 @ B3 )
= bot_bo7824918357723582233_nat_o )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ B )
= bot_bo7824918357723582233_nat_o ) ) ) ) ).
% disjoint_mono
thf(fact_1237_disjoint__mono,axiom,
! [A2: set_nat,A5: set_nat,B: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ A5 )
=> ( ( ord_less_eq_set_nat @ B @ B3 )
=> ( ( ( inf_inf_set_nat @ A5 @ B3 )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ) ).
% disjoint_mono
thf(fact_1238_rtrancl__image__unfold__right,axiom,
! [E: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ V2 ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ V2 ) ) ).
% rtrancl_image_unfold_right
thf(fact_1239_rtrancl__reachable__induct,axiom,
! [I: set_nat,INV: set_nat,E: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_set_nat @ I @ INV )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ INV ) @ INV )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ I ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_1240_Image__Int__subset,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ R3 @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ).
% Image_Int_subset
thf(fact_1241_Image__mono,axiom,
! [R6: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,A8: set_nat,A: set_nat] :
( ( ord_le3146513528884898305at_nat @ R6 @ R2 )
=> ( ( ord_less_eq_set_nat @ A8 @ A )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R6 @ A8 ) @ ( image_nat_nat @ R2 @ A ) ) ) ) ).
% Image_mono
thf(fact_1242_refl__on__Un,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
( ( refl_on_nat @ A @ R2 )
=> ( ( refl_on_nat @ B4 @ S2 )
=> ( refl_on_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_1243_refl__on__Int,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
( ( refl_on_nat @ A @ R2 )
=> ( ( refl_on_nat @ B4 @ S2 )
=> ( refl_on_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_in2572325071724192079at_nat @ R2 @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_1244_trancl__union__outside,axiom,
! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,U2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) )
=> ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ E ) )
=> ? [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ X3 ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ U2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ W ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_1245_trancl__union__outside,axiom,
! [V: int,W: int,E: set_Pr958786334691620121nt_int,U2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) )
=> ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ E ) )
=> ? [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ X3 ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ U2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ W ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_1246_SigmaE,axiom,
! [C: product_prod_nat_int,A: set_nat,B4: nat > set_int] :
( ( member4262671552274231302at_int @ C @ ( produc454536836294682310at_int @ A @ B4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y2: int] :
( ( member_int @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_nat_int @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1247_SigmaE,axiom,
! [C: product_prod_nat_o,A: set_nat,B4: nat > set_o] :
( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A @ B4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_nat_o @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1248_SigmaE,axiom,
! [C: produc526117512892780850t_unit,A: set_nat,B4: nat > set_Product_unit] :
( ( member348563419383025609t_unit @ C @ ( produc5506539147108004339t_unit @ A @ B4 ) )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( produc3434450474625521892t_unit @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1249_SigmaE,axiom,
! [C: product_prod_int_nat,A: set_int,B4: int > set_nat] :
( ( member216504246829706758nt_nat @ C @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_int_nat @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1250_SigmaE,axiom,
! [C: product_prod_int_o,A: set_int,B4: int > set_o] :
( ( member4489920277610959864_int_o @ C @ ( product_Sigma_int_o @ A @ B4 ) )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ A )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_int_o @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1251_SigmaE,axiom,
! [C: produc6865911496949736918t_unit,A: set_int,B4: int > set_Product_unit] :
( ( member6688357403439981677t_unit @ C @ ( produc2622961094310184599t_unit @ A @ B4 ) )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ A )
=> ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( produc550872421827702152t_unit @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1252_SigmaE,axiom,
! [C: product_prod_o_nat,A: set_o,B4: $o > set_nat] :
( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A @ B4 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_o_nat @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1253_SigmaE,axiom,
! [C: product_prod_o_int,A: set_o,B4: $o > set_int] :
( ( member7847949116333733898_o_int @ C @ ( product_Sigma_o_int @ A @ B4 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y2: int] :
( ( member_int @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_o_int @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1254_SigmaE,axiom,
! [C: product_prod_o_o,A: set_o,B4: $o > set_o] :
( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A @ B4 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( product_Pair_o_o @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1255_SigmaE,axiom,
! [C: produc3788275437394581276t_unit,A: set_o,B4: $o > set_Product_unit] :
( ( member9215336023991391429t_unit @ C @ ( produc411009820408249917t_unit @ A @ B4 ) )
=> ~ ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
=> ( C
!= ( produc3901946521647900940t_unit @ X3 @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_1256_SigmaD1,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: heap_e7401611519738050253t_unit > set_set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc5260988393495149579et_nat @ A @ B4 ) )
=> ( member242389114213480694t_unit @ A2 @ A ) ) ).
% SigmaD1
thf(fact_1257_SigmaD1,axiom,
! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
=> ( member_nat @ A2 @ A ) ) ).
% SigmaD1
thf(fact_1258_SigmaD1,axiom,
! [A2: int,B: int,A: set_int,B4: int > set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( produc1453890548195736866nt_int @ A @ B4 ) )
=> ( member_int @ A2 @ A ) ) ).
% SigmaD1
thf(fact_1259_SigmaD1,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc8052021730408606599et_nat @ A @ B4 ) )
=> ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% SigmaD1
thf(fact_1260_SigmaD1,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc7016705354093686019et_nat @ A @ B4 ) )
=> ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% SigmaD1
thf(fact_1261_SigmaD2,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: heap_e7401611519738050253t_unit > set_set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc5260988393495149579et_nat @ A @ B4 ) )
=> ( member_set_nat @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_1262_SigmaD2,axiom,
! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
=> ( member_nat @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_1263_SigmaD2,axiom,
! [A2: int,B: int,A: set_int,B4: int > set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( produc1453890548195736866nt_int @ A @ B4 ) )
=> ( member_int @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_1264_SigmaD2,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc8052021730408606599et_nat @ A @ B4 ) )
=> ( member6260224972018164377et_nat @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_1265_SigmaD2,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc7016705354093686019et_nat @ A @ B4 ) )
=> ( member1996754912294343701et_nat @ B @ ( B4 @ A2 ) ) ) ).
% SigmaD2
thf(fact_1266_SigmaE2,axiom,
! [A2: nat,B: int,A: set_nat,B4: nat > set_int] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1267_SigmaE2,axiom,
! [A2: nat,B: $o,A: set_nat,B4: nat > set_o] :
( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1268_SigmaE2,axiom,
! [A2: nat,B: product_unit,A: set_nat,B4: nat > set_Product_unit] :
( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) )
=> ~ ( ( member_nat @ A2 @ A )
=> ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1269_SigmaE2,axiom,
! [A2: int,B: nat,A: set_int,B4: int > set_nat] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
=> ~ ( ( member_int @ A2 @ A )
=> ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1270_SigmaE2,axiom,
! [A2: int,B: $o,A: set_int,B4: int > set_o] :
( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) )
=> ~ ( ( member_int @ A2 @ A )
=> ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1271_SigmaE2,axiom,
! [A2: int,B: product_unit,A: set_int,B4: int > set_Product_unit] :
( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) )
=> ~ ( ( member_int @ A2 @ A )
=> ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1272_SigmaE2,axiom,
! [A2: $o,B: nat,A: set_o,B4: $o > set_nat] :
( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1273_SigmaE2,axiom,
! [A2: $o,B: int,A: set_o,B4: $o > set_int] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1274_SigmaE2,axiom,
! [A2: $o,B: $o,A: set_o,B4: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1275_SigmaE2,axiom,
! [A2: $o,B: product_unit,A: set_o,B4: $o > set_Product_unit] :
( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) )
=> ~ ( ( member_o @ A2 @ A )
=> ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_1276_trancl__image__by__rtrancl,axiom,
! [E: set_Pr1261947904930325089at_nat,Vi: set_nat] :
( ( sup_sup_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ Vi ) @ Vi )
= ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_1277_converse__rtranclE_H,axiom,
! [U: nat,V: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( U != V )
=> ~ ! [Vh: nat] :
( ( U != Vh )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ Vh ) @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Vh @ V ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_1278_converse__rtranclE_H,axiom,
! [U: int,V: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ R3 ) )
=> ( ( U != V )
=> ~ ! [Vh: int] :
( ( U != Vh )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ Vh ) @ R3 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Vh @ V ) @ ( transi2902850859426252137cl_int @ R3 ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_1279_rtrancl_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_1280_rtrancl_Ocases,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_1281_rtrancl_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
= ( ? [A6: nat] :
( ( A1 = A6 )
& ( A22 = A6 ) )
| ? [A6: nat,B6: nat,C4: nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_1282_rtrancl_Osimps,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi2902850859426252137cl_int @ R2 ) )
= ( ? [A6: int] :
( ( A1 = A6 )
& ( A22 = A6 ) )
| ? [A6: int,B6: int,C4: int] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ ( transi2902850859426252137cl_int @ R2 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_1283_rtrancl_Ortrancl__refl,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_1284_rtrancl_Ortrancl__refl,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ ( transi2902850859426252137cl_int @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_1285_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_1286_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_1287_rtranclE,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( A2 != B )
=> ~ ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_1288_rtranclE,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( A2 != B )
=> ~ ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_1289_rtrancl__trans,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_1290_rtrancl__trans,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_1291_rtrancl__induct,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P @ A2 )
=> ( ! [Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_1292_rtrancl__induct,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( P @ A2 )
=> ( ! [Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_1293_converse__rtranclE,axiom,
! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( X != Z )
=> ~ ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_1294_converse__rtranclE,axiom,
! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( X != Z )
=> ~ ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_1295_converse__rtrancl__induct,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P @ B )
=> ( ! [Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_1296_converse__rtrancl__induct,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( P @ B )
=> ( ! [Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_1297_converse__rtrancl__into__rtrancl,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_1298_converse__rtrancl__into__rtrancl,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_1299_rtrancl__induct2,axiom,
! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
=> ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1300_rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1301_rtrancl__induct2,axiom,
! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: int,B2: int,Aa: int,Ba: int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1302_rtrancl__induct2,axiom,
! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
=> ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1303_rtrancl__induct2,axiom,
! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ ( transi344104063936286263et_nat @ R2 ) )
=> ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P @ A4 @ B2 )
=> ( P @ Aa @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1304_converse__rtranclE2,axiom,
! [Xa: heap_e7401611519738050253t_unit,Xb: set_nat,Za: heap_e7401611519738050253t_unit,Zb: set_nat,R2: set_Pr719794911490849221et_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Xa @ Xb ) @ ( produc7507926704131184380et_nat @ Za @ Zb ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
=> ( ( ( produc7507926704131184380et_nat @ Xa @ Xb )
!= ( produc7507926704131184380et_nat @ Za @ Zb ) )
=> ~ ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Xa @ Xb ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ R2 )
=> ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Za @ Zb ) ) @ ( transi6919426408697524543et_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1305_converse__rtranclE2,axiom,
! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( ( product_Pair_nat_nat @ Xa @ Xb )
!= ( product_Pair_nat_nat @ Za @ Zb ) )
=> ~ ! [A4: nat,B2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ R2 )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1306_converse__rtranclE2,axiom,
! [Xa: int,Xb: int,Za: int,Zb: int,R2: set_Pr2560585780119916871nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Xa @ Xb ) @ ( product_Pair_int_int @ Za @ Zb ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
=> ( ( ( product_Pair_int_int @ Xa @ Xb )
!= ( product_Pair_int_int @ Za @ Zb ) )
=> ~ ! [A4: int,B2: int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Xa @ Xb ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ R2 )
=> ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Za @ Zb ) ) @ ( transi5430920176653612092nt_int @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1307_converse__rtranclE2,axiom,
! [Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Za: produc3658429121746597890et_nat > $o,Zb: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Xa @ Xb ) @ ( produc5001842942810119800et_nat @ Za @ Zb ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
=> ( ( ( produc5001842942810119800et_nat @ Xa @ Xb )
!= ( produc5001842942810119800et_nat @ Za @ Zb ) )
=> ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Xa @ Xb ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ R2 )
=> ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Za @ Zb ) ) @ ( transi2414916928880341691et_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1308_converse__rtranclE2,axiom,
! [Xa: produc3658429121746597890et_nat > $o,Xb: produc3925858234332021118et_nat,Za: produc3658429121746597890et_nat > $o,Zb: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Xa @ Xb ) @ ( produc2245416461498447860et_nat @ Za @ Zb ) ) @ ( transi344104063936286263et_nat @ R2 ) )
=> ( ( ( produc2245416461498447860et_nat @ Xa @ Xb )
!= ( produc2245416461498447860et_nat @ Za @ Zb ) )
=> ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Xa @ Xb ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ R2 )
=> ~ ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Za @ Zb ) ) @ ( transi344104063936286263et_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1309_converse__rtrancl__induct2,axiom,
! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Aa @ Ba ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A4 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1310_converse__rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A4 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1311_converse__rtrancl__induct2,axiom,
! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: int,B2: int,Aa: int,Ba: int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Aa @ Ba ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A4 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1312_converse__rtrancl__induct2,axiom,
! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Aa @ Ba ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A4 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1313_converse__rtrancl__induct2,axiom,
! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Aa @ Ba ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
=> ( ( P @ Aa @ Ba )
=> ( P @ A4 @ B2 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1314_Un__empty__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ bot_bo3957492148770167129t_unit )
= A ) ).
% Un_empty_right
thf(fact_1315_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_1316_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_1317_Un__empty__right,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ A @ bot_bot_set_int )
= A ) ).
% Un_empty_right
thf(fact_1318_Un__empty__right,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ A @ bot_bo7824918357723582233_nat_o )
= A ) ).
% Un_empty_right
thf(fact_1319_Un__empty__left,axiom,
! [B4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1320_Un__empty__left,axiom,
! [B4: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1321_Un__empty__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1322_Un__empty__left,axiom,
! [B4: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1323_Un__empty__left,axiom,
! [B4: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1324_disjoint__iff__not__equal,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ! [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1325_disjoint__iff__not__equal,axiom,
! [A: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ B4 )
=> ( X2 = ~ Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1326_disjoint__iff__not__equal,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1327_disjoint__iff__not__equal,axiom,
! [A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ! [Y3: int] :
( ( member_int @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1328_disjoint__iff__not__equal,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
=> ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1329_Int__empty__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Int_empty_right
thf(fact_1330_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_1331_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1332_Int__empty__right,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ A @ bot_bot_set_int )
= bot_bot_set_int ) ).
% Int_empty_right
thf(fact_1333_Int__empty__right,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ A @ bot_bo7824918357723582233_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Int_empty_right
thf(fact_1334_Int__empty__left,axiom,
! [B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ B4 )
= bot_bo3957492148770167129t_unit ) ).
% Int_empty_left
thf(fact_1335_Int__empty__left,axiom,
! [B4: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B4 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_1336_Int__empty__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1337_Int__empty__left,axiom,
! [B4: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ B4 )
= bot_bot_set_int ) ).
% Int_empty_left
thf(fact_1338_Int__empty__left,axiom,
! [B4: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ B4 )
= bot_bo7824918357723582233_nat_o ) ).
% Int_empty_left
thf(fact_1339_disjoint__iff,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ~ ( member_Product_unit @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1340_disjoint__iff,axiom,
! [A: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1341_disjoint__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1342_disjoint__iff,axiom,
! [A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ~ ( member_int @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1343_disjoint__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
=> ~ ( member6576561426505652726_nat_o @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1344_Int__emptyI,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ~ ( member_Product_unit @ X3 @ B4 ) )
=> ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit ) ) ).
% Int_emptyI
thf(fact_1345_Int__emptyI,axiom,
! [A: set_o,B4: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B4 ) )
=> ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1346_Int__emptyI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B4 ) )
=> ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1347_Int__emptyI,axiom,
! [A: set_int,B4: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ~ ( member_int @ X3 @ B4 ) )
=> ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int ) ) ).
% Int_emptyI
thf(fact_1348_Int__emptyI,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ~ ( member6576561426505652726_nat_o @ X3 @ B4 ) )
=> ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o ) ) ).
% Int_emptyI
thf(fact_1349_disjointI,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ~ ( member_Product_unit @ X3 @ B ) )
=> ( ( inf_in4660618365625256667t_unit @ A2 @ B )
= bot_bo3957492148770167129t_unit ) ) ).
% disjointI
thf(fact_1350_disjointI,axiom,
! [A2: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ~ ( member_o @ X3 @ B ) )
=> ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o ) ) ).
% disjointI
thf(fact_1351_disjointI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% disjointI
thf(fact_1352_disjointI,axiom,
! [A2: set_int,B: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ~ ( member_int @ X3 @ B ) )
=> ( ( inf_inf_set_int @ A2 @ B )
= bot_bot_set_int ) ) ).
% disjointI
thf(fact_1353_disjointI,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A2 )
=> ~ ( member6576561426505652726_nat_o @ X3 @ B ) )
=> ( ( inf_in1906310914598751387_nat_o @ A2 @ B )
= bot_bo7824918357723582233_nat_o ) ) ).
% disjointI
thf(fact_1354_Int__insert__right,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) )
& ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1355_Int__insert__right,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) )
& ( ~ ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1356_Int__insert__right,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1357_Int__insert__right,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ A )
=> ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1358_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1359_Int__insert__left,axiom,
! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( member6576561426505652726_nat_o @ A2 @ C5 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
= ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) )
& ( ~ ( member6576561426505652726_nat_o @ A2 @ C5 )
=> ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
= ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1360_Int__insert__left,axiom,
! [A2: int,C5: set_int,B4: set_int] :
( ( ( member_int @ A2 @ C5 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
= ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C5 ) ) ) )
& ( ~ ( member_int @ A2 @ C5 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_int @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1361_Int__insert__left,axiom,
! [A2: $o,C5: set_o,B4: set_o] :
( ( ( member_o @ A2 @ C5 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B4 @ C5 ) ) ) )
& ( ~ ( member_o @ A2 @ C5 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_o @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1362_Int__insert__left,axiom,
! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
( ( ( member_Product_unit @ A2 @ C5 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
= ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ C5 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
= ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1363_Int__insert__left,axiom,
! [A2: nat,C5: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) )
& ( ~ ( member_nat @ A2 @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1364_subrelI,axiom,
! [R2: set_Pr3948176798113811640et_nat,S2: set_Pr3948176798113811640et_nat] :
( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le8090042543978450200et_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_1365_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_1366_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% subrelI
thf(fact_1367_subrelI,axiom,
! [R2: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le5966269811547037844et_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_1368_subrelI,axiom,
! [R2: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ S2 ) )
=> ( ord_le4763372923235995152et_nat @ R2 @ S2 ) ) ).
% subrelI
thf(fact_1369_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
=> ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_1370_bot_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
=> ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1371_bot_Oextremum__uniqueI,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
=> ( A2 = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_1372_bot_Oextremum__uniqueI,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
=> ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1373_bot_Oextremum__uniqueI,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
=> ( A2 = bot_bot_filter_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1374_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1375_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1376_bot_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
= ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_unique
thf(fact_1377_bot_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_1378_bot_Oextremum__unique,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_1379_bot_Oextremum__unique,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
= ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% bot.extremum_unique
thf(fact_1380_bot_Oextremum__unique,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
= ( A2 = bot_bot_filter_nat ) ) ).
% bot.extremum_unique
thf(fact_1381_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1382_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1383_bot_Oextremum,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ).
% bot.extremum
thf(fact_1384_bot_Oextremum,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% bot.extremum
thf(fact_1385_bot_Oextremum,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% bot.extremum
thf(fact_1386_bot_Oextremum,axiom,
! [A2: set_Pr4532377907799695533_nat_o] : ( ord_le2965882846123202637_nat_o @ bot_bo7824918357723582233_nat_o @ A2 ) ).
% bot.extremum
thf(fact_1387_bot_Oextremum,axiom,
! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A2 ) ).
% bot.extremum
thf(fact_1388_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_1389_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_1390_subset__emptyI,axiom,
! [A: set_Product_unit] :
( ! [X3: product_unit] :
~ ( member_Product_unit @ X3 @ A )
=> ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% subset_emptyI
thf(fact_1391_subset__emptyI,axiom,
! [A: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1392_subset__emptyI,axiom,
! [A: set_int] :
( ! [X3: int] :
~ ( member_int @ X3 @ A )
=> ( ord_less_eq_set_int @ A @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_1393_subset__emptyI,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% subset_emptyI
thf(fact_1394_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1395_insert__subsetI,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,X4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( ord_le2965882846123202637_nat_o @ X4 @ A )
=> ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1396_insert__subsetI,axiom,
! [X: int,A: set_int,X4: set_int] :
( ( member_int @ X @ A )
=> ( ( ord_less_eq_set_int @ X4 @ A )
=> ( ord_less_eq_set_int @ ( insert_int @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1397_insert__subsetI,axiom,
! [X: $o,A: set_o,X4: set_o] :
( ( member_o @ X @ A )
=> ( ( ord_less_eq_set_o @ X4 @ A )
=> ( ord_less_eq_set_o @ ( insert_o @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1398_insert__subsetI,axiom,
! [X: product_unit,A: set_Product_unit,X4: set_Product_unit] :
( ( member_Product_unit @ X @ A )
=> ( ( ord_le3507040750410214029t_unit @ X4 @ A )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1399_insert__subsetI,axiom,
! [X: nat,A: set_nat,X4: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1400_subset__insertI2,axiom,
! [A: set_int,B4: set_int,B: int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ord_less_eq_set_int @ A @ ( insert_int @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1401_subset__insertI2,axiom,
! [A: set_o,B4: set_o,B: $o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1402_subset__insertI2,axiom,
! [A: set_Product_unit,B4: set_Product_unit,B: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1403_subset__insertI2,axiom,
! [A: set_nat,B4: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1404_subset__insertI,axiom,
! [B4: set_int,A2: int] : ( ord_less_eq_set_int @ B4 @ ( insert_int @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_1405_subset__insertI,axiom,
! [B4: set_o,A2: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_1406_subset__insertI,axiom,
! [B4: set_Product_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ B4 @ ( insert_Product_unit @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_1407_subset__insertI,axiom,
! [B4: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_1408_subset__insert,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
= ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1409_subset__insert,axiom,
! [X: int,A: set_int,B4: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1410_subset__insert,axiom,
! [X: $o,A: set_o,B4: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) )
= ( ord_less_eq_set_o @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1411_subset__insert,axiom,
! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
= ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1412_subset__insert,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1413_insert__mono,axiom,
! [C5: set_int,D3: set_int,A2: int] :
( ( ord_less_eq_set_int @ C5 @ D3 )
=> ( ord_less_eq_set_int @ ( insert_int @ A2 @ C5 ) @ ( insert_int @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_1414_insert__mono,axiom,
! [C5: set_o,D3: set_o,A2: $o] :
( ( ord_less_eq_set_o @ C5 @ D3 )
=> ( ord_less_eq_set_o @ ( insert_o @ A2 @ C5 ) @ ( insert_o @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_1415_insert__mono,axiom,
! [C5: set_Product_unit,D3: set_Product_unit,A2: product_unit] :
( ( ord_le3507040750410214029t_unit @ C5 @ D3 )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ C5 ) @ ( insert_Product_unit @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_1416_insert__mono,axiom,
! [C5: set_nat,D3: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C5 @ D3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C5 ) @ ( insert_nat @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_1417_Image__Un,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
( ( image_nat_nat @ R3 @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ).
% Image_Un
thf(fact_1418_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_less_eq_set_nat @ A @ ( vimage8201027140005291904_nat_o @ F @ B4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member6576561426505652726_nat_o @ ( F @ X2 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1419_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1420_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > int,B4: set_int] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_int @ F @ B4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1421_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > $o,B4: set_o] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_o @ F @ B4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1422_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > product_unit,B4: set_Product_unit] :
( ( ord_less_eq_set_nat @ A @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_Product_unit @ ( F @ X2 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_1423_vimage__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% vimage_mono
thf(fact_1424_relH__subset,axiom,
! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As: set_nat] :
( ( relH @ Bs @ H2 @ H3 )
=> ( ( ord_less_eq_set_nat @ As @ Bs )
=> ( relH @ As @ H2 @ H3 ) ) ) ).
% relH_subset
thf(fact_1425_rtrancl__apply__insert,axiom,
! [R3: set_Pr5094982260447487303t_unit,X: product_unit,S: set_Product_unit] :
( ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( insert_Product_unit @ X @ S ) )
= ( insert_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( sup_su793286257634532545t_unit @ S @ ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1426_rtrancl__apply__insert,axiom,
! [R3: set_Product_prod_o_o,X: $o,S: set_o] :
( ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( insert_o @ X @ S ) )
= ( insert_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( sup_sup_set_o @ S @ ( image_o_o @ R3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1427_rtrancl__apply__insert,axiom,
! [R3: set_Pr1261947904930325089at_nat,X: nat,S: set_nat] :
( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( insert_nat @ X @ S ) )
= ( insert_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( sup_sup_set_nat @ S @ ( image_nat_nat @ R3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1428_rtrancl__apply__insert,axiom,
! [R3: set_Pr958786334691620121nt_int,X: int,S: set_int] :
( ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( insert_int @ X @ S ) )
= ( insert_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( sup_sup_set_int @ S @ ( image_int_int @ R3 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1429_rtrancl__apply__insert,axiom,
! [R3: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
( ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ X @ S ) )
= ( insert5175938949040314269_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( sup_su5209123915105501825_nat_o @ S @ ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_1430_antisymD,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( antisym_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 )
=> ( A2 = B ) ) ) ) ).
% antisymD
thf(fact_1431_antisymD,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( antisym_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 )
=> ( A2 = B ) ) ) ) ).
% antisymD
thf(fact_1432_antisymI,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 )
=> ( X3 = Y2 ) ) )
=> ( antisym_nat @ R2 ) ) ).
% antisymI
thf(fact_1433_antisymI,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 )
=> ( X3 = Y2 ) ) )
=> ( antisym_int @ R2 ) ) ).
% antisymI
thf(fact_1434_antisym__def,axiom,
( antisym_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [X2: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_1435_antisym__def,axiom,
( antisym_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [X2: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
=> ( X2 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_1436_Image__absorb__rtrancl,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_nat,C5: set_nat] :
( ( trans_nat @ A )
=> ( ( refl_on_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ C5 @ B4 )
=> ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ A ) @ C5 )
= ( image_nat_nat @ A @ C5 ) ) ) ) ) ).
% Image_absorb_rtrancl
thf(fact_1437_times__assn__raw_Osimps,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( times_assn_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( ? [As1: set_nat,As22: set_nat] :
( ( As
= ( sup_sup_set_nat @ As1 @ As22 ) )
& ( ( inf_inf_set_nat @ As1 @ As22 )
= bot_bot_set_nat )
& ( P @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
& ( Q2 @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_1438_times__assn__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( Y
= ( ~ ? [As1: set_nat,As22: set_nat] :
( ( As2
= ( sup_sup_set_nat @ As1 @ As22 ) )
& ( ( inf_inf_set_nat @ As1 @ As22 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_1439_times__assn__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ~ ? [As12: set_nat,As23: set_nat] :
( ( As2
= ( sup_sup_set_nat @ As12 @ As23 ) )
& ( ( inf_inf_set_nat @ As12 @ As23 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_1440_times__assn__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ? [As13: set_nat,As24: set_nat] :
( ( As2
= ( sup_sup_set_nat @ As13 @ As24 ) )
& ( ( inf_inf_set_nat @ As13 @ As24 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_1441_rtrancl__Image__advance__ss,axiom,
! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ E )
=> ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ U @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1442_rtrancl__Image__advance__ss,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
=> ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1443_rtrancl__Image__advance__ss,axiom,
! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ E )
=> ( ord_less_eq_set_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ U @ bot_bot_set_int ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1444_rtrancl__Image__advance__ss,axiom,
! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ E )
=> ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1445_rtrancl__Image__advance__ss,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_1446_rtrancl__image__advance__rtrancl,axiom,
! [Q4: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Q0: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Q4 @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Q4 @ X ) @ ( transi712980445759334096_nat_o @ R3 ) )
=> ( member6576561426505652726_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1447_rtrancl__image__advance__rtrancl,axiom,
! [Q4: $o,R3: set_Product_prod_o_o,Q0: set_o,X: $o] :
( ( member_o @ Q4 @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q4 @ X ) @ ( transitive_rtrancl_o @ R3 ) )
=> ( member_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1448_rtrancl__image__advance__rtrancl,axiom,
! [Q4: product_unit,R3: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
( ( member_Product_unit @ Q4 @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Q4 @ X ) @ ( transi6481887417741755664t_unit @ R3 ) )
=> ( member_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1449_rtrancl__image__advance__rtrancl,axiom,
! [Q4: nat,R3: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
( ( member_nat @ Q4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q4 @ X ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( member_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1450_rtrancl__image__advance__rtrancl,axiom,
! [Q4: int,R3: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
( ( member_int @ Q4 @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Q4 @ X ) @ ( transi2902850859426252137cl_int @ R3 ) )
=> ( member_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_1451_rtrancl__image__advance,axiom,
! [Q4: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Q0: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Q4 @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Q4 @ X ) @ R3 )
=> ( member6576561426505652726_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1452_rtrancl__image__advance,axiom,
! [Q4: $o,R3: set_Product_prod_o_o,Q0: set_o,X: $o] :
( ( member_o @ Q4 @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q4 @ X ) @ R3 )
=> ( member_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1453_rtrancl__image__advance,axiom,
! [Q4: product_unit,R3: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
( ( member_Product_unit @ Q4 @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Q4 @ X ) @ R3 )
=> ( member_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1454_rtrancl__image__advance,axiom,
! [Q4: nat,R3: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
( ( member_nat @ Q4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q4 @ X ) @ R3 )
=> ( member_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1455_rtrancl__image__advance,axiom,
! [Q4: int,R3: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
( ( member_int @ Q4 @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Q4 @ X ) @ R3 )
=> ( member_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_1456_trancl__rtrancl__trancl,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_1457_trancl__rtrancl__trancl,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_1458_rtrancl__trancl__trancl,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_1459_rtrancl__trancl__trancl,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_1460_rtrancl__into__trancl2,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_1461_rtrancl__into__trancl2,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_1462_rtrancl__into__trancl1,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_1463_rtrancl__into__trancl1,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_1464_rtrancl__eq__or__trancl,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_1465_rtrancl__eq__or__trancl,axiom,
! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_1466_trancl__into__rtrancl,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_1467_trancl__into__rtrancl,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_1468_tranclD2,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ? [Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_1469_tranclD2,axiom,
! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) )
=> ? [Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ ( transi2902850859426252137cl_int @ R3 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ Y ) @ R3 ) ) ) ).
% tranclD2
thf(fact_1470_rtranclD,axiom,
! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( A2 = B )
| ( ( A2 != B )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_1471_rtranclD,axiom,
! [A2: int,B: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R3 ) )
=> ( ( A2 = B )
| ( ( A2 != B )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_1472_tranclD,axiom,
! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
=> ? [Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% tranclD
thf(fact_1473_tranclD,axiom,
! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) )
=> ? [Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ R3 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) ) ) ) ).
% tranclD
thf(fact_1474_insert__is__Un,axiom,
( insert_Product_unit
= ( ^ [A6: product_unit] : ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A6 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% insert_is_Un
thf(fact_1475_insert__is__Un,axiom,
( insert_o
= ( ^ [A6: $o] : ( sup_sup_set_o @ ( insert_o @ A6 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_1476_insert__is__Un,axiom,
( insert_nat
= ( ^ [A6: nat] : ( sup_sup_set_nat @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1477_insert__is__Un,axiom,
( insert_int
= ( ^ [A6: int] : ( sup_sup_set_int @ ( insert_int @ A6 @ bot_bot_set_int ) ) ) ) ).
% insert_is_Un
thf(fact_1478_insert__is__Un,axiom,
( insert5175938949040314269_nat_o
= ( ^ [A6: produc3658429121746597890et_nat > $o] : ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ A6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% insert_is_Un
thf(fact_1479_Un__singleton__iff,axiom,
! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
( ( ( sup_su793286257634532545t_unit @ A @ B4 )
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= ( ( ( A = bot_bo3957492148770167129t_unit )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4 = bot_bo3957492148770167129t_unit ) )
| ( ( A
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1480_Un__singleton__iff,axiom,
! [A: set_o,B4: set_o,X: $o] :
( ( ( sup_sup_set_o @ A @ B4 )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A = bot_bot_set_o )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4 = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1481_Un__singleton__iff,axiom,
! [A: set_nat,B4: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B4 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1482_Un__singleton__iff,axiom,
! [A: set_int,B4: set_int,X: int] :
( ( ( sup_sup_set_int @ A @ B4 )
= ( insert_int @ X @ bot_bot_set_int ) )
= ( ( ( A = bot_bot_set_int )
& ( B4
= ( insert_int @ X @ bot_bot_set_int ) ) )
| ( ( A
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B4 = bot_bot_set_int ) )
| ( ( A
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B4
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1483_Un__singleton__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( ( sup_su5209123915105501825_nat_o @ A @ B4 )
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= ( ( ( A = bot_bo7824918357723582233_nat_o )
& ( B4
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
| ( ( A
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
& ( B4 = bot_bo7824918357723582233_nat_o ) )
| ( ( A
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
& ( B4
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1484_singleton__Un__iff,axiom,
! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit )
= ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( ( ( A = bot_bo3957492148770167129t_unit )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4 = bot_bo3957492148770167129t_unit ) )
| ( ( A
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1485_singleton__Un__iff,axiom,
! [X: $o,A: set_o,B4: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A @ B4 ) )
= ( ( ( A = bot_bot_set_o )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4 = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1486_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B4 ) )
= ( ( ( A = bot_bot_set_nat )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4 = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1487_singleton__Un__iff,axiom,
! [X: int,A: set_int,B4: set_int] :
( ( ( insert_int @ X @ bot_bot_set_int )
= ( sup_sup_set_int @ A @ B4 ) )
= ( ( ( A = bot_bot_set_int )
& ( B4
= ( insert_int @ X @ bot_bot_set_int ) ) )
| ( ( A
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B4 = bot_bot_set_int ) )
| ( ( A
= ( insert_int @ X @ bot_bot_set_int ) )
& ( B4
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1488_singleton__Un__iff,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o )
= ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( ( ( A = bot_bo7824918357723582233_nat_o )
& ( B4
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
| ( ( A
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
& ( B4 = bot_bo7824918357723582233_nat_o ) )
| ( ( A
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
& ( B4
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1489_Not__Domain__rtrancl,axiom,
! [X: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Y: produc3658429121746597890et_nat > $o] :
( ~ ( member6576561426505652726_nat_o @ X @ ( domain4881241903163524940_nat_o @ R3 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi712980445759334096_nat_o @ R3 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1490_Not__Domain__rtrancl,axiom,
! [X: $o,R3: set_Product_prod_o_o,Y: $o] :
( ~ ( member_o @ X @ ( domain_o_o @ R3 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_rtrancl_o @ R3 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1491_Not__Domain__rtrancl,axiom,
! [X: product_unit,R3: set_Pr5094982260447487303t_unit,Y: product_unit] :
( ~ ( member_Product_unit @ X @ ( domain4292024440186670924t_unit @ R3 ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi6481887417741755664t_unit @ R3 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1492_Not__Domain__rtrancl,axiom,
! [X: nat,R3: set_Pr1261947904930325089at_nat,Y: nat] :
( ~ ( member_nat @ X @ ( domain_nat_nat @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1493_Not__Domain__rtrancl,axiom,
! [X: int,R3: set_Pr958786334691620121nt_int,Y: int] :
( ~ ( member_int @ X @ ( domain_int_int @ R3 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1494_subset__singletonD,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
=> ( ( A = bot_bo3957492148770167129t_unit )
| ( A
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_1495_subset__singletonD,axiom,
! [A: set_o,X: $o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A = bot_bot_set_o )
| ( A
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1496_subset__singletonD,axiom,
! [A: set_int,X: int] :
( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( A = bot_bot_set_int )
| ( A
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_1497_subset__singletonD,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
=> ( ( A = bot_bo7824918357723582233_nat_o )
| ( A
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% subset_singletonD
thf(fact_1498_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1499_subset__singleton__iff,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( ord_le3507040750410214029t_unit @ X4 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( ( X4 = bot_bo3957492148770167129t_unit )
| ( X4
= ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_1500_subset__singleton__iff,axiom,
! [X4: set_o,A2: $o] :
( ( ord_less_eq_set_o @ X4 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( ( X4 = bot_bot_set_o )
| ( X4
= ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1501_subset__singleton__iff,axiom,
! [X4: set_int,A2: int] :
( ( ord_less_eq_set_int @ X4 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( ( X4 = bot_bot_set_int )
| ( X4
= ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_1502_subset__singleton__iff,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( ord_le2965882846123202637_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
= ( ( X4 = bot_bo7824918357723582233_nat_o )
| ( X4
= ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1503_subset__singleton__iff,axiom,
! [X4: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1504_mod__star__conv,axiom,
! [A: assn,B4: assn,H2: produc3658429121746597890et_nat] :
( ( rep_assn @ ( times_times_assn @ A @ B4 ) @ H2 )
= ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
( ( H2
= ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
& ( ( inf_inf_set_nat @ As1 @ As22 )
= bot_bot_set_nat )
& ( rep_assn @ A @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
& ( rep_assn @ B4 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_1505_star__assnI,axiom,
! [P: assn,H2: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,As5: set_nat] :
( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
=> ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As5 ) )
=> ( ( ( inf_inf_set_nat @ As @ As5 )
= bot_bot_set_nat )
=> ( rep_assn @ ( times_times_assn @ P @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ).
% star_assnI
thf(fact_1506_in__range__subset,axiom,
! [As: set_nat,As5: set_nat,H2: heap_e7401611519738050253t_unit] :
( ( ord_less_eq_set_nat @ As @ As5 )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As5 ) )
=> ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% in_range_subset
thf(fact_1507_wand__raw_Osimps,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( wand_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
& ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( ( inf_inf_set_nat @ As @ As6 )
= bot_bot_set_nat )
& ( relH @ As @ H2 @ H5 )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
& ( P @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
=> ( Q2 @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_1508_wand__raw_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( Y
= ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
& ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( ( inf_inf_set_nat @ As2 @ As6 )
= bot_bot_set_nat )
& ( relH @ As2 @ H @ H5 )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
& ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_1509_wand__raw_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
& ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
( ( ( ( inf_inf_set_nat @ As2 @ As7 )
= bot_bot_set_nat )
& ( relH @ As2 @ H @ H7 )
& ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
& ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_1510_wand__raw_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
& ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( ( ( inf_inf_set_nat @ As2 @ As4 )
= bot_bot_set_nat )
& ( relH @ As2 @ H @ H6 )
& ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
& ( X @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_1511_Image__empty__rtrancl__Image__id,axiom,
! [R3: set_Pr5094982260447487303t_unit,V: product_unit] :
( ( ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
= ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1512_Image__empty__rtrancl__Image__id,axiom,
! [R3: set_Product_prod_o_o,V: $o] :
( ( ( image_o_o @ R3 @ ( insert_o @ V @ bot_bot_set_o ) )
= bot_bot_set_o )
=> ( ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( insert_o @ V @ bot_bot_set_o ) )
= ( insert_o @ V @ bot_bot_set_o ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1513_Image__empty__rtrancl__Image__id,axiom,
! [R3: set_Pr1261947904930325089at_nat,V: nat] :
( ( ( image_nat_nat @ R3 @ ( insert_nat @ V @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1514_Image__empty__rtrancl__Image__id,axiom,
! [R3: set_Pr958786334691620121nt_int,V: int] :
( ( ( image_int_int @ R3 @ ( insert_int @ V @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( insert_int @ V @ bot_bot_set_int ) )
= ( insert_int @ V @ bot_bot_set_int ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1515_Image__empty__rtrancl__Image__id,axiom,
! [R3: set_Pr2161125870931222855_nat_o,V: produc3658429121746597890et_nat > $o] :
( ( ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
= ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_1516_sup__bot_Oright__neutral,axiom,
! [A2: assn] :
( ( sup_sup_assn @ A2 @ bot_bot_assn )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1517_sup__bot_Oright__neutral,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ A2 @ bot_bot_Product_unit )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1518_sup__bot_Oright__neutral,axiom,
! [A2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A2 @ bot_bo3957492148770167129t_unit )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1519_sup__bot_Oright__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1520_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1521_sup__bot_Oright__neutral,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1522_sup__bot_Oright__neutral,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_1523_sup__bot_Oneutr__eq__iff,axiom,
! [A2: assn,B: assn] :
( ( bot_bot_assn
= ( sup_sup_assn @ A2 @ B ) )
= ( ( A2 = bot_bot_assn )
& ( B = bot_bot_assn ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1524_sup__bot_Oneutr__eq__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A2 @ B ) )
= ( ( A2 = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1525_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( sup_su793286257634532545t_unit @ A2 @ B ) )
= ( ( A2 = bot_bo3957492148770167129t_unit )
& ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1526_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_o,B: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A2 @ B ) )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1527_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1528_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_int,B: set_int] :
( ( bot_bot_set_int
= ( sup_sup_set_int @ A2 @ B ) )
= ( ( A2 = bot_bot_set_int )
& ( B = bot_bot_set_int ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1529_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( sup_su5209123915105501825_nat_o @ A2 @ B ) )
= ( ( A2 = bot_bo7824918357723582233_nat_o )
& ( B = bot_bo7824918357723582233_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1530_sup__bot_Oleft__neutral,axiom,
! [A2: assn] :
( ( sup_sup_assn @ bot_bot_assn @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1531_sup__bot_Oleft__neutral,axiom,
! [A2: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1532_sup__bot_Oleft__neutral,axiom,
! [A2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1533_sup__bot_Oleft__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1534_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1535_sup__bot_Oleft__neutral,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1536_sup__bot_Oleft__neutral,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_1537_sup__bot_Oeq__neutr__iff,axiom,
! [A2: assn,B: assn] :
( ( ( sup_sup_assn @ A2 @ B )
= bot_bot_assn )
= ( ( A2 = bot_bot_assn )
& ( B = bot_bot_assn ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1538_sup__bot_Oeq__neutr__iff,axiom,
! [A2: product_unit,B: product_unit] :
( ( ( sup_sup_Product_unit @ A2 @ B )
= bot_bot_Product_unit )
= ( ( A2 = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1539_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_unit,B: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A2 @ B )
= bot_bo3957492148770167129t_unit )
= ( ( A2 = bot_bo3957492148770167129t_unit )
& ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1540_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1541_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1542_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_int,B: set_int] :
( ( ( sup_sup_set_int @ A2 @ B )
= bot_bot_set_int )
= ( ( A2 = bot_bot_set_int )
& ( B = bot_bot_set_int ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1543_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
( ( ( sup_su5209123915105501825_nat_o @ A2 @ B )
= bot_bo7824918357723582233_nat_o )
= ( ( A2 = bot_bo7824918357723582233_nat_o )
& ( B = bot_bo7824918357723582233_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1544_sup__eq__bot__iff,axiom,
! [X: assn,Y: assn] :
( ( ( sup_sup_assn @ X @ Y )
= bot_bot_assn )
= ( ( X = bot_bot_assn )
& ( Y = bot_bot_assn ) ) ) ).
% sup_eq_bot_iff
thf(fact_1545_sup__eq__bot__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( sup_sup_Product_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_1546_sup__eq__bot__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ X @ Y )
= bot_bo3957492148770167129t_unit )
= ( ( X = bot_bo3957492148770167129t_unit )
& ( Y = bot_bo3957492148770167129t_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_1547_sup__eq__bot__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( sup_sup_set_o @ X @ Y )
= bot_bot_set_o )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_1548_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1549_sup__eq__bot__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ( sup_sup_set_int @ X @ Y )
= bot_bot_set_int )
= ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% sup_eq_bot_iff
thf(fact_1550_sup__eq__bot__iff,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( ( sup_su5209123915105501825_nat_o @ X @ Y )
= bot_bo7824918357723582233_nat_o )
= ( ( X = bot_bo7824918357723582233_nat_o )
& ( Y = bot_bo7824918357723582233_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_1551_bot__eq__sup__iff,axiom,
! [X: assn,Y: assn] :
( ( bot_bot_assn
= ( sup_sup_assn @ X @ Y ) )
= ( ( X = bot_bot_assn )
& ( Y = bot_bot_assn ) ) ) ).
% bot_eq_sup_iff
thf(fact_1552_bot__eq__sup__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X @ Y ) )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_1553_bot__eq__sup__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( sup_su793286257634532545t_unit @ X @ Y ) )
= ( ( X = bot_bo3957492148770167129t_unit )
& ( Y = bot_bo3957492148770167129t_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_1554_bot__eq__sup__iff,axiom,
! [X: set_o,Y: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X @ Y ) )
= ( ( X = bot_bot_set_o )
& ( Y = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_1555_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1556_bot__eq__sup__iff,axiom,
! [X: set_int,Y: set_int] :
( ( bot_bot_set_int
= ( sup_sup_set_int @ X @ Y ) )
= ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% bot_eq_sup_iff
thf(fact_1557_bot__eq__sup__iff,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( sup_su5209123915105501825_nat_o @ X @ Y ) )
= ( ( X = bot_bo7824918357723582233_nat_o )
& ( Y = bot_bo7824918357723582233_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_1558_sup__bot__right,axiom,
! [X: assn] :
( ( sup_sup_assn @ X @ bot_bot_assn )
= X ) ).
% sup_bot_right
thf(fact_1559_sup__bot__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% sup_bot_right
thf(fact_1560_sup__bot__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ bot_bo3957492148770167129t_unit )
= X ) ).
% sup_bot_right
thf(fact_1561_sup__bot__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% sup_bot_right
thf(fact_1562_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_1563_sup__bot__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ bot_bot_set_int )
= X ) ).
% sup_bot_right
thf(fact_1564_sup__bot__right,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ X @ bot_bo7824918357723582233_nat_o )
= X ) ).
% sup_bot_right
thf(fact_1565_sup__bot__left,axiom,
! [X: assn] :
( ( sup_sup_assn @ bot_bot_assn @ X )
= X ) ).
% sup_bot_left
thf(fact_1566_sup__bot__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X )
= X ) ).
% sup_bot_left
thf(fact_1567_sup__bot__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ X )
= X ) ).
% sup_bot_left
thf(fact_1568_sup__bot__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X )
= X ) ).
% sup_bot_left
thf(fact_1569_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_1570_sup__bot__left,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ X )
= X ) ).
% sup_bot_left
thf(fact_1571_sup__bot__left,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ X )
= X ) ).
% sup_bot_left
thf(fact_1572_boolean__algebra_Oconj__zero__right,axiom,
! [X: assn] :
( ( inf_inf_assn @ X @ bot_bot_assn )
= bot_bot_assn ) ).
% boolean_algebra.conj_zero_right
thf(fact_1573_boolean__algebra_Oconj__zero__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_1574_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_1575_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1576_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1577_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ bot_bot_set_int )
= bot_bot_set_int ) ).
% boolean_algebra.conj_zero_right
thf(fact_1578_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ X @ bot_bo7824918357723582233_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1579_boolean__algebra_Oconj__zero__left,axiom,
! [X: assn] :
( ( inf_inf_assn @ bot_bot_assn @ X )
= bot_bot_assn ) ).
% boolean_algebra.conj_zero_left
thf(fact_1580_boolean__algebra_Oconj__zero__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_1581_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_1582_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1583_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1584_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ X )
= bot_bot_set_int ) ).
% boolean_algebra.conj_zero_left
thf(fact_1585_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ X )
= bot_bo7824918357723582233_nat_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1586_inf__bot__right,axiom,
! [X: assn] :
( ( inf_inf_assn @ X @ bot_bot_assn )
= bot_bot_assn ) ).
% inf_bot_right
thf(fact_1587_inf__bot__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_1588_inf__bot__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% inf_bot_right
thf(fact_1589_inf__bot__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_1590_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1591_inf__bot__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ bot_bot_set_int )
= bot_bot_set_int ) ).
% inf_bot_right
thf(fact_1592_inf__bot__right,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ X @ bot_bo7824918357723582233_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% inf_bot_right
thf(fact_1593_merge__pure__or,axiom,
! [A2: $o,B: $o] :
( ( sup_sup_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
= ( pure_assn
@ ( A2
| B ) ) ) ).
% merge_pure_or
thf(fact_1594_merge__pure__and,axiom,
! [A2: $o,B: $o] :
( ( inf_inf_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
= ( pure_assn
@ ( A2
& B ) ) ) ).
% merge_pure_and
thf(fact_1595_inf__bot__left,axiom,
! [X: assn] :
( ( inf_inf_assn @ bot_bot_assn @ X )
= bot_bot_assn ) ).
% inf_bot_left
thf(fact_1596_inf__bot__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_1597_inf__bot__left,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X )
= bot_bo3957492148770167129t_unit ) ).
% inf_bot_left
thf(fact_1598_inf__bot__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_1599_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1600_inf__bot__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ bot_bot_set_int @ X )
= bot_bot_set_int ) ).
% inf_bot_left
thf(fact_1601_inf__bot__left,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ X )
= bot_bo7824918357723582233_nat_o ) ).
% inf_bot_left
thf(fact_1602_boolean__algebra_Odisj__zero__right,axiom,
! [X: assn] :
( ( sup_sup_assn @ X @ bot_bot_assn )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1603_boolean__algebra_Odisj__zero__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1604_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ bot_bo3957492148770167129t_unit )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1605_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ bot_bot_set_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1606_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1607_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ bot_bot_set_int )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1608_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( sup_su5209123915105501825_nat_o @ X @ bot_bo7824918357723582233_nat_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1609_prod__set__defs_I2_J,axiom,
( basic_2698842327037663054et_nat
= ( ^ [P5: produc3658429121746597890et_nat] : ( insert_set_nat @ ( produc8586169260539613262et_nat @ P5 ) @ bot_bot_set_set_nat ) ) ) ).
% prod_set_defs(2)
thf(fact_1610_prod__set__defs_I2_J,axiom,
( basic_snds_int_int
= ( ^ [P5: product_prod_int_int] : ( insert_int @ ( product_snd_int_int @ P5 ) @ bot_bot_set_int ) ) ) ).
% prod_set_defs(2)
thf(fact_1611_snds_Ocases,axiom,
! [A2: set_nat,P3: produc3658429121746597890et_nat] :
( ( member_set_nat @ A2 @ ( basic_2698842327037663054et_nat @ P3 ) )
=> ( A2
= ( produc8586169260539613262et_nat @ P3 ) ) ) ).
% snds.cases
thf(fact_1612_snds_Ocases,axiom,
! [A2: int,P3: product_prod_int_int] :
( ( member_int @ A2 @ ( basic_snds_int_int @ P3 ) )
=> ( A2
= ( product_snd_int_int @ P3 ) ) ) ).
% snds.cases
thf(fact_1613_snds_Osimps,axiom,
! [A2: set_nat,P3: produc3658429121746597890et_nat] :
( ( member_set_nat @ A2 @ ( basic_2698842327037663054et_nat @ P3 ) )
= ( A2
= ( produc8586169260539613262et_nat @ P3 ) ) ) ).
% snds.simps
thf(fact_1614_snds_Osimps,axiom,
! [A2: int,P3: product_prod_int_int] :
( ( member_int @ A2 @ ( basic_snds_int_int @ P3 ) )
= ( A2
= ( product_snd_int_int @ P3 ) ) ) ).
% snds.simps
thf(fact_1615_snds_Ointros,axiom,
! [P3: produc3658429121746597890et_nat] : ( member_set_nat @ ( produc8586169260539613262et_nat @ P3 ) @ ( basic_2698842327037663054et_nat @ P3 ) ) ).
% snds.intros
thf(fact_1616_snds_Ointros,axiom,
! [P3: product_prod_int_int] : ( member_int @ ( product_snd_int_int @ P3 ) @ ( basic_snds_int_int @ P3 ) ) ).
% snds.intros
thf(fact_1617_wand__raw_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( Y
= ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
& ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
( ( ( ( inf_inf_set_nat @ As2 @ As6 )
= bot_bot_set_nat )
& ( relH @ As2 @ H @ H5 )
& ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
& ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) )
=> ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_1618_wand__raw_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
=> ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
& ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
( ( ( ( inf_inf_set_nat @ As2 @ As7 )
= bot_bot_set_nat )
& ( relH @ As2 @ H @ H7 )
& ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
& ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_1619_wand__raw_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
=> ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
& ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
( ( ( ( inf_inf_set_nat @ As2 @ As4 )
= bot_bot_set_nat )
& ( relH @ As2 @ H @ H6 )
& ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
& ( X @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
=> ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_1620_le__numeral__extra_I4_J,axiom,
ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).
% le_numeral_extra(4)
thf(fact_1621_le__numeral__extra_I4_J,axiom,
ord_le1926595141338095240atural @ one_one_Code_natural @ one_one_Code_natural ).
% le_numeral_extra(4)
thf(fact_1622_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_1623_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1624_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1625_less__eq__assn__def,axiom,
( ord_less_eq_assn
= ( ^ [A6: assn,B6: assn] :
( A6
= ( inf_inf_assn @ A6 @ B6 ) ) ) ) ).
% less_eq_assn_def
thf(fact_1626_times__assn__raw_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
=> ? [As13: set_nat,As24: set_nat] :
( ( As2
= ( sup_sup_set_nat @ As13 @ As24 ) )
& ( ( inf_inf_set_nat @ As13 @ As24 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_1627_times__assn__raw_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
=> ~ ? [As12: set_nat,As23: set_nat] :
( ( As2
= ( sup_sup_set_nat @ As12 @ As23 ) )
& ( ( inf_inf_set_nat @ As12 @ As23 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_1628_times__assn__raw_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( Xb
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( Y
= ( ? [As1: set_nat,As22: set_nat] :
( ( As2
= ( sup_sup_set_nat @ As1 @ As22 ) )
& ( ( inf_inf_set_nat @ As1 @ As22 )
= bot_bot_set_nat )
& ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
& ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) )
=> ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_1629_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( order_7534304569044981540t_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1630_subset__Image1__Image1__iff,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1631_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( order_4859164337913492053on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1632_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( order_174704566861754596_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1633_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1634_Field__insert,axiom,
! [A2: product_unit,B: product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( field_Product_unit @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) )
= ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) ) ) ).
% Field_insert
thf(fact_1635_Field__insert,axiom,
! [A2: $o,B: $o,R2: set_Product_prod_o_o] :
( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) )
= ( sup_sup_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).
% Field_insert
thf(fact_1636_Field__insert,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
= ( sup_sup_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).
% Field_insert
thf(fact_1637_Field__insert,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( field_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
= ( sup_sup_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) ) ) ).
% Field_insert
thf(fact_1638_Field__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
( ( field_81821745747897882_nat_o @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) )
= ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ).
% Field_insert
thf(fact_1639_dom__ran__disj__comp,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) )
= bot_bo3957492148770167129t_unit )
=> ( ( relcom848357998638694028t_unit @ R3 @ R3 )
= bot_bo1654022762305692595t_unit ) ) ).
% dom_ran_disj_comp
thf(fact_1640_dom__ran__disj__comp,axiom,
! [R3: set_Product_prod_o_o] :
( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) )
= bot_bot_set_o )
=> ( ( relcomp_o_o_o @ R3 @ R3 )
= bot_bo7073875226086086771od_o_o ) ) ).
% dom_ran_disj_comp
thf(fact_1641_dom__ran__disj__comp,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) )
= bot_bot_set_nat )
=> ( ( relcomp_nat_nat_nat @ R3 @ R3 )
= bot_bo2099793752762293965at_nat ) ) ).
% dom_ran_disj_comp
thf(fact_1642_dom__ran__disj__comp,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( ( inf_inf_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) )
= bot_bot_set_int )
=> ( ( relcomp_int_int_int @ R3 @ R3 )
= bot_bo1796632182523588997nt_int ) ) ).
% dom_ran_disj_comp
thf(fact_1643_dom__ran__disj__comp,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( relcom6251035627269349452_nat_o @ R3 @ R3 )
= bot_bo3657187498359500723_nat_o ) ) ).
% dom_ran_disj_comp
thf(fact_1644_proj__iff,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ A )
=> ( ( ( equiv_5763582929720877514t_unit @ R2 @ X )
= ( equiv_5763582929720877514t_unit @ R2 @ Y ) )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_1645_proj__iff,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) @ A )
=> ( ( ( equiv_proj_o_o @ R2 @ X )
= ( equiv_proj_o_o @ R2 @ Y ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_1646_proj__iff,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( ord_less_eq_set_int @ ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) @ A )
=> ( ( ( equiv_proj_int_int @ R2 @ X )
= ( equiv_proj_int_int @ R2 @ Y ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_1647_proj__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) @ A )
=> ( ( ( equiv_3235788788526563786_nat_o @ R2 @ X )
= ( equiv_3235788788526563786_nat_o @ R2 @ Y ) )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_1648_proj__iff,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A )
=> ( ( ( equiv_proj_nat_nat @ R2 @ X )
= ( equiv_proj_nat_nat @ R2 @ Y ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_1649_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( antisym_Product_unit @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1650_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( antisym_o @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1651_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( antisym_nat @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1652_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( refl_on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( antisym_int @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1653_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( refl_o1075864025857821388_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( antisy9190549550370998639_nat_o @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
= ( A2 = B ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1654_equiv__class__nondisjoint,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,A2: product_unit,B: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member_Product_unit @ X @ ( inf_in4660618365625256667t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1655_equiv__class__nondisjoint,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,A2: $o,B: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_o @ X @ ( inf_inf_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1656_equiv__class__nondisjoint,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,A2: nat,B: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_nat @ X @ ( inf_inf_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1657_equiv__class__nondisjoint,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,A2: int,B: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_int @ X @ ( inf_inf_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1658_equiv__class__nondisjoint,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member6576561426505652726_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1659_finite__reachable__advance,axiom,
! [E: set_Pr4811707699266497531nteger,V0: code_integer,V: code_integer] :
( ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ ( insert_Code_integer @ V0 @ bot_bo3990330152332043303nteger ) ) )
=> ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ V0 @ V ) @ ( transi3984306202085910426nteger @ E ) )
=> ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ ( insert_Code_integer @ V @ bot_bo3990330152332043303nteger ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1660_finite__reachable__advance,axiom,
! [E: set_Pr5094982260447487303t_unit,V0: product_unit,V: product_unit] :
( ( finite4290736615968046902t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V0 @ bot_bo3957492148770167129t_unit ) ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V0 @ V ) @ ( transi6481887417741755664t_unit @ E ) )
=> ( finite4290736615968046902t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1661_finite__reachable__advance,axiom,
! [E: set_Product_prod_o_o,V0: $o,V: $o] :
( ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V0 @ bot_bot_set_o ) ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V0 @ V ) @ ( transitive_rtrancl_o @ E ) )
=> ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1662_finite__reachable__advance,axiom,
! [E: set_Pr1261947904930325089at_nat,V0: nat,V: nat] :
( ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V0 @ bot_bot_set_nat ) ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V0 @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1663_finite__reachable__advance,axiom,
! [E: set_Pr958786334691620121nt_int,V0: int,V: int] :
( ( finite_finite_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V0 @ bot_bot_set_int ) ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V0 @ V ) @ ( transi2902850859426252137cl_int @ E ) )
=> ( finite_finite_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1664_finite__reachable__advance,axiom,
! [E: set_Pr2161125870931222855_nat_o,V0: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V0 @ bot_bo7824918357723582233_nat_o ) ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V0 @ V ) @ ( transi712980445759334096_nat_o @ E ) )
=> ( finite3252695134891459830_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_1665_finite__Field__eq__finite,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ ( field_nat @ R3 ) )
= ( finite6177210948735845034at_nat @ R3 ) ) ).
% finite_Field_eq_finite
thf(fact_1666_finite__Field__eq__finite,axiom,
! [R3: set_Pr4811707699266497531nteger] :
( ( finite6017078050557962740nteger @ ( field_Code_integer @ R3 ) )
= ( finite2154694888287295812nteger @ R3 ) ) ).
% finite_Field_eq_finite
thf(fact_1667_Field__empty,axiom,
( ( field_Product_unit @ bot_bo1654022762305692595t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Field_empty
thf(fact_1668_Field__empty,axiom,
( ( field_o @ bot_bo7073875226086086771od_o_o )
= bot_bot_set_o ) ).
% Field_empty
thf(fact_1669_Field__empty,axiom,
( ( field_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Field_empty
thf(fact_1670_Field__empty,axiom,
( ( field_int @ bot_bo1796632182523588997nt_int )
= bot_bot_set_int ) ).
% Field_empty
thf(fact_1671_Field__empty,axiom,
( ( field_81821745747897882_nat_o @ bot_bo3657187498359500723_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Field_empty
thf(fact_1672_Field__Un,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) )
= ( sup_sup_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S2 ) ) ) ).
% Field_Un
thf(fact_1673_finite__Field,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ R2 )
=> ( finite_finite_nat @ ( field_nat @ R2 ) ) ) ).
% finite_Field
thf(fact_1674_finite__Field,axiom,
! [R2: set_Pr4811707699266497531nteger] :
( ( finite2154694888287295812nteger @ R2 )
=> ( finite6017078050557962740nteger @ ( field_Code_integer @ R2 ) ) ) ).
% finite_Field
thf(fact_1675_relcompEpair,axiom,
! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
=> ~ ! [B2: heap_e7401611519738050253t_unit] :
( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ B2 ) @ R2 )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1676_relcompEpair,axiom,
! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
=> ~ ! [B2: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 )
=> ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1677_relcompEpair,axiom,
! [A2: nat,C: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
=> ~ ! [B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1678_relcompEpair,axiom,
! [A2: int,C: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
=> ~ ! [B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1679_relcompEpair,axiom,
! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 )
=> ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1680_relcompEpair,axiom,
! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 )
=> ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1681_relcompEpair,axiom,
! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 )
=> ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1682_relcompEpair,axiom,
! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 )
=> ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1683_relcompEpair,axiom,
! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 )
=> ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1684_relcompEpair,axiom,
! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 )
=> ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B2 @ C ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_1685_relcompE,axiom,
! [Xz: produc3658429121746597890et_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ Xz @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: heap_e7401611519738050253t_unit,Z3: set_nat] :
( ( Xz
= ( produc7507926704131184380et_nat @ X3 @ Z3 ) )
=> ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ X3 @ Y2 ) @ R2 )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1686_relcompE,axiom,
! [Xz: produc3658429121746597890et_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
( ( member6260224972018164377et_nat @ Xz @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
( ( Xz
= ( produc7507926704131184380et_nat @ X3 @ Z3 ) )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
=> ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1687_relcompE,axiom,
! [Xz: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
=> ~ ! [X3: nat,Y2: nat,Z3: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X3 @ Z3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1688_relcompE,axiom,
! [Xz: product_prod_int_int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ Xz @ ( relcomp_int_int_int @ R2 @ S2 ) )
=> ~ ! [X3: int,Y2: int,Z3: int] :
( ( Xz
= ( product_Pair_int_int @ X3 @ Z3 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1689_relcompE,axiom,
! [Xz: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ Xz @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat] :
( ( Xz
= ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
=> ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1690_relcompE,axiom,
! [Xz: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
( ( member1996754912294343701et_nat @ Xz @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
( ( Xz
= ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
=> ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1691_relcompE,axiom,
! [Xz: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
( ( member1996754912294343701et_nat @ Xz @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3658429121746597890et_nat] :
( ( Xz
= ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
=> ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1692_relcompE,axiom,
! [Xz: produc2732055786443039994et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ Xz @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3925858234332021118et_nat] :
( ( Xz
= ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
=> ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1693_relcompE,axiom,
! [Xz: produc2732055786443039994et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
( ( member6124377750444531601et_nat @ Xz @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3925858234332021118et_nat] :
( ( Xz
= ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
=> ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1694_relcompE,axiom,
! [Xz: produc2732055786443039994et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
( ( member6124377750444531601et_nat @ Xz @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
( ( Xz
= ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
=> ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_1695_relcomp_OrelcompI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: heap_e7401611519738050253t_unit,R2: set_Pr1776176089890104647t_unit,C: set_nat,S2: set_Pr3948176798113811640et_nat] :
( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ B ) @ R2 )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ C ) @ S2 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1696_relcomp_OrelcompI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o,C: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ C ) @ S2 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1697_relcomp_OrelcompI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o,C: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ C ) @ S2 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1698_relcomp_OrelcompI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat,C: set_nat,S2: set_Pr5488025237498180813et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
=> ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B @ C ) @ S2 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1699_relcomp_OrelcompI,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat,S2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1700_relcomp_OrelcompI,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int,S2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ S2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( relcomp_int_int_int @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1701_relcomp_OrelcompI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,C: produc3658429121746597890et_nat,S2: set_Pr719794911490849221et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
=> ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ C ) @ S2 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1702_relcomp_OrelcompI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,C: produc3925858234332021118et_nat,S2: set_Pr8807100098903490369et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
=> ( ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B @ C ) @ S2 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1703_relcomp_OrelcompI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,C: produc3658429121746597890et_nat,S2: set_Pr3721596426557623617et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
=> ( ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B @ C ) @ S2 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1704_relcomp_OrelcompI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,C: produc3925858234332021118et_nat,S2: set_Pr7928877670098842301et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
=> ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ C ) @ S2 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_1705_relcomp_Osimps,axiom,
! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
= ( ? [A6: heap_e7401611519738050253t_unit,B6: heap_e7401611519738050253t_unit,C4: set_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A6 @ B6 ) @ R2 )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1706_relcomp_Osimps,axiom,
! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
= ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat,C4: set_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 )
& ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1707_relcomp_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
= ( ? [A6: nat,B6: nat,C4: nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1708_relcomp_Osimps,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
= ( ? [A6: int,B6: int,C4: int] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1709_relcomp_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C4: produc3658429121746597890et_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A6 @ B6 ) @ R2 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1710_relcomp_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat,C4: produc3658429121746597890et_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 )
& ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1711_relcomp_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat,C4: produc3658429121746597890et_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 )
& ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1712_relcomp_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C4: produc3925858234332021118et_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A6 @ B6 ) @ R2 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1713_relcomp_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat,C4: produc3925858234332021118et_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 )
& ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1714_relcomp_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat,C4: produc3925858234332021118et_nat] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 )
& ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_1715_relcomp_Ocases,axiom,
! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
=> ~ ! [B2: heap_e7401611519738050253t_unit] :
( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A1 @ B2 ) @ R2 )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1716_relcomp_Ocases,axiom,
! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
=> ~ ! [B2: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ B2 ) @ R2 )
=> ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1717_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
=> ~ ! [B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1718_relcomp_Ocases,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
=> ~ ! [B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1719_relcomp_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A1 @ B2 ) @ R2 )
=> ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1720_relcomp_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ B2 ) @ R2 )
=> ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1721_relcomp_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ B2 ) @ R2 )
=> ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1722_relcomp_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A1 @ B2 ) @ R2 )
=> ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1723_relcomp_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ B2 ) @ R2 )
=> ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1724_relcomp_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
=> ~ ! [B2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ B2 ) @ R2 )
=> ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_1725_FieldI2,axiom,
! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
=> ( member6576561426505652726_nat_o @ J @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% FieldI2
thf(fact_1726_FieldI2,axiom,
! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
=> ( member_o @ J @ ( field_o @ R3 ) ) ) ).
% FieldI2
thf(fact_1727_FieldI2,axiom,
! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
=> ( member_Product_unit @ J @ ( field_Product_unit @ R3 ) ) ) ).
% FieldI2
thf(fact_1728_FieldI2,axiom,
! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
=> ( member_nat @ J @ ( field_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_1729_FieldI2,axiom,
! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
=> ( member_int @ J @ ( field_int @ R3 ) ) ) ).
% FieldI2
thf(fact_1730_FieldI1,axiom,
! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
=> ( member6576561426505652726_nat_o @ I2 @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% FieldI1
thf(fact_1731_FieldI1,axiom,
! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
=> ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).
% FieldI1
thf(fact_1732_FieldI1,axiom,
! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
=> ( member_Product_unit @ I2 @ ( field_Product_unit @ R3 ) ) ) ).
% FieldI1
thf(fact_1733_FieldI1,axiom,
! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
=> ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_1734_FieldI1,axiom,
! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
=> ( member_int @ I2 @ ( field_int @ R3 ) ) ) ).
% FieldI1
thf(fact_1735_mono__Field,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S2 ) ) ) ).
% mono_Field
thf(fact_1736_equiv__class__self,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member_Product_unit @ A2 @ A )
=> ( member_Product_unit @ A2 @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% equiv_class_self
thf(fact_1737_equiv__class__self,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_o @ A2 @ A )
=> ( member_o @ A2 @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ).
% equiv_class_self
thf(fact_1738_equiv__class__self,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ A2 @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_self
thf(fact_1739_equiv__class__self,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_int @ A2 @ A )
=> ( member_int @ A2 @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ).
% equiv_class_self
thf(fact_1740_equiv__class__self,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( member6576561426505652726_nat_o @ A2 @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% equiv_class_self
thf(fact_1741_Field__def,axiom,
( field_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] : ( sup_sup_set_nat @ ( domain_nat_nat @ R4 ) @ ( range_nat_nat @ R4 ) ) ) ) ).
% Field_def
thf(fact_1742_equiv__class__eq__iff,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
= ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) )
& ( member_Product_unit @ X @ A )
& ( member_Product_unit @ Y @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1743_equiv__class__eq__iff,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
= ( ( ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) )
= ( image_o_o @ R2 @ ( insert_o @ Y @ bot_bot_set_o ) ) )
& ( member_o @ X @ A )
& ( member_o @ Y @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1744_equiv__class__eq__iff,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
= ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
& ( member_nat @ X @ A )
& ( member_nat @ Y @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1745_equiv__class__eq__iff,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
= ( ( ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) )
= ( image_int_int @ R2 @ ( insert_int @ Y @ bot_bot_set_int ) ) )
& ( member_int @ X @ A )
& ( member_int @ Y @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1746_equiv__class__eq__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
= ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) )
& ( member6576561426505652726_nat_o @ X @ A )
& ( member6576561426505652726_nat_o @ Y @ A ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1747_eq__equiv__class__iff,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member_Product_unit @ X @ A )
=> ( ( member_Product_unit @ Y @ A )
=> ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1748_eq__equiv__class__iff,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_o @ X @ A )
=> ( ( member_o @ Y @ A )
=> ( ( ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) )
= ( image_o_o @ R2 @ ( insert_o @ Y @ bot_bot_set_o ) ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1749_eq__equiv__class__iff,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1750_eq__equiv__class__iff,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) )
= ( image_int_int @ R2 @ ( insert_int @ Y @ bot_bot_set_int ) ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1751_eq__equiv__class__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1752_equiv__class__eq,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
=> ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1753_equiv__class__eq,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1754_equiv__class__eq,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1755_equiv__class__eq,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1756_equiv__class__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1757_eq__equiv__class,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,A: set_Product_unit] :
( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
=> ( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member_Product_unit @ B @ A )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_1758_eq__equiv__class,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o,A: set_o] :
( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
=> ( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_o @ B @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_1759_eq__equiv__class,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,A: set_nat] :
( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
=> ( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_nat @ B @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_1760_eq__equiv__class,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,A: set_int] :
( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
=> ( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_int @ B @ A )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_1761_eq__equiv__class,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
=> ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member6576561426505652726_nat_o @ B @ A )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_1762_rtrancl__Image__in__Field,axiom,
! [R3: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ V2 ) @ ( sup_sup_set_nat @ ( field_nat @ R3 ) @ V2 ) ) ).
% rtrancl_Image_in_Field
thf(fact_1763_refines__equiv__class__eq2,axiom,
! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
( ( ord_le930853871177014503t_unit @ R3 @ S )
=> ( ( equiv_4189356295871601603t_unit @ A @ R3 )
=> ( ( equiv_4189356295871601603t_unit @ A @ S )
=> ( ( image_8431784415008483091t_unit @ S @ ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1764_refines__equiv__class__eq2,axiom,
! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
( ( ord_le192195778054999463od_o_o @ R3 @ S )
=> ( ( equiv_equiv_o @ A @ R3 )
=> ( ( equiv_equiv_o @ A @ S )
=> ( ( image_o_o @ S @ ( image_o_o @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1765_refines__equiv__class__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ S )
=> ( ( equiv_equiv_nat @ A @ R3 )
=> ( ( equiv_equiv_nat @ A @ S )
=> ( ( image_nat_nat @ S @ ( image_nat_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1766_refines__equiv__class__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
( ( ord_le2843351958646193337nt_int @ R3 @ S )
=> ( ( equiv_equiv_int @ A @ R3 )
=> ( ( equiv_equiv_int @ A @ S )
=> ( ( image_int_int @ S @ ( image_int_int @ R3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1767_refines__equiv__class__eq2,axiom,
! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( ord_le6659818918159358183_nat_o @ R3 @ S )
=> ( ( equiv_6726761681920839555_nat_o @ A @ R3 )
=> ( ( equiv_6726761681920839555_nat_o @ A @ S )
=> ( ( image_2232984160439939859_nat_o @ S @ ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
= ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_1768_refines__equiv__class__eq,axiom,
! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
( ( ord_le930853871177014503t_unit @ R3 @ S )
=> ( ( equiv_4189356295871601603t_unit @ A @ R3 )
=> ( ( equiv_4189356295871601603t_unit @ A @ S )
=> ( ( image_8431784415008483091t_unit @ R3 @ ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1769_refines__equiv__class__eq,axiom,
! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
( ( ord_le192195778054999463od_o_o @ R3 @ S )
=> ( ( equiv_equiv_o @ A @ R3 )
=> ( ( equiv_equiv_o @ A @ S )
=> ( ( image_o_o @ R3 @ ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1770_refines__equiv__class__eq,axiom,
! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ S )
=> ( ( equiv_equiv_nat @ A @ R3 )
=> ( ( equiv_equiv_nat @ A @ S )
=> ( ( image_nat_nat @ R3 @ ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1771_refines__equiv__class__eq,axiom,
! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
( ( ord_le2843351958646193337nt_int @ R3 @ S )
=> ( ( equiv_equiv_int @ A @ R3 )
=> ( ( equiv_equiv_int @ A @ S )
=> ( ( image_int_int @ R3 @ ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1772_refines__equiv__class__eq,axiom,
! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( ord_le6659818918159358183_nat_o @ R3 @ S )
=> ( ( equiv_6726761681920839555_nat_o @ A @ R3 )
=> ( ( equiv_6726761681920839555_nat_o @ A @ S )
=> ( ( image_2232984160439939859_nat_o @ R3 @ ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
= ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_1773_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( order_2505591618835473410t_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( A2 = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1774_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( order_8538205470752950541r_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( A2 = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1775_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( order_5251275573222108571on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( A2 = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1776_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( order_5248785102713058295on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( A2 = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1777_Partial__order__eq__Image1__Image1__iff,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( order_3690095022144090562_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
= ( A2 = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1778_finite__reachable__restrictedI,axiom,
! [Q2: set_Code_integer,I: set_Code_integer,E: set_Pr4811707699266497531nteger] :
( ( finite6017078050557962740nteger @ Q2 )
=> ( ( ord_le7084787975880047091nteger @ I @ Q2 )
=> ( ( ord_le7084787975880047091nteger @ ( range_1907724884897625393nteger @ E ) @ Q2 )
=> ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ I ) ) ) ) ) ).
% finite_reachable_restrictedI
thf(fact_1779_finite__reachable__restrictedI,axiom,
! [Q2: set_nat,I: set_nat,E: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ Q2 )
=> ( ( ord_less_eq_set_nat @ I @ Q2 )
=> ( ( ord_less_eq_set_nat @ ( range_nat_nat @ E ) @ Q2 )
=> ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ I ) ) ) ) ) ).
% finite_reachable_restrictedI
thf(fact_1780_subset__Image__Image__iff,axiom,
! [R2: set_Pr958786334691620121nt_int,A: set_int,B4: set_int] :
( ( order_4859164337913492053on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( ord_less_eq_set_int @ A @ ( field_int @ R2 ) )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ A ) @ ( image_int_int @ R2 @ B4 ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Y3: int] :
( ( member_int @ Y3 @ B4 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1781_subset__Image__Image__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( ord_less_eq_set_nat @ A @ ( field_nat @ R2 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A ) @ ( image_nat_nat @ R2 @ B4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1782_equiv__class__subset,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
=> ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1783_equiv__class__subset,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1784_equiv__class__subset,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1785_equiv__class__subset,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1786_equiv__class__subset,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1787_subset__equiv__class,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,B: product_unit,A2: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( ( member_Product_unit @ B @ A )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_1788_subset__equiv__class,axiom,
! [A: set_o,R2: set_Product_prod_o_o,B: $o,A2: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
=> ( ( member_o @ B @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_1789_subset__equiv__class,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,B: int,A2: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
=> ( ( member_int @ B @ A )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_1790_subset__equiv__class,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
=> ( ( member6576561426505652726_nat_o @ B @ A )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_1791_subset__equiv__class,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B: nat,A2: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ B @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_1792_finite__finite__vimage__IntI,axiom,
! [F2: set_Code_integer,H2: code_integer > code_integer,A: set_Code_integer] :
( ( finite6017078050557962740nteger @ F2 )
=> ( ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ F2 )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ H2 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1793_finite__finite__vimage__IntI,axiom,
! [F2: set_Code_integer,H2: nat > code_integer,A: set_nat] :
( ( finite6017078050557962740nteger @ F2 )
=> ( ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ H2 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1794_finite__finite__vimage__IntI,axiom,
! [F2: set_Product_unit,H2: code_integer > product_unit,A: set_Code_integer] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ F2 )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ H2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1795_finite__finite__vimage__IntI,axiom,
! [F2: set_Product_unit,H2: nat > product_unit,A: set_nat] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ H2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1796_finite__finite__vimage__IntI,axiom,
! [F2: set_o,H2: code_integer > $o,A: set_Code_integer] :
( ( finite_finite_o @ F2 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ F2 )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ H2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1797_finite__finite__vimage__IntI,axiom,
! [F2: set_o,H2: nat > $o,A: set_nat] :
( ( finite_finite_o @ F2 )
=> ( ! [Y2: $o] :
( ( member_o @ Y2 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1798_finite__finite__vimage__IntI,axiom,
! [F2: set_nat,H2: code_integer > nat,A: set_Code_integer] :
( ( finite_finite_nat @ F2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ F2 )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ H2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1799_finite__finite__vimage__IntI,axiom,
! [F2: set_nat,H2: nat > nat,A: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1800_finite__finite__vimage__IntI,axiom,
! [F2: set_int,H2: code_integer > int,A: set_Code_integer] :
( ( finite_finite_int @ F2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ F2 )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ H2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1801_finite__finite__vimage__IntI,axiom,
! [F2: set_int,H2: nat > int,A: set_nat] :
( ( finite_finite_int @ F2 )
=> ( ! [Y2: int] :
( ( member_int @ Y2 @ F2 )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H2 @ F2 ) @ A ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1802_finite__insert,axiom,
! [A2: int,A: set_int] :
( ( finite_finite_int @ ( insert_int @ A2 @ A ) )
= ( finite_finite_int @ A ) ) ).
% finite_insert
thf(fact_1803_finite__insert,axiom,
! [A2: $o,A: set_o] :
( ( finite_finite_o @ ( insert_o @ A2 @ A ) )
= ( finite_finite_o @ A ) ) ).
% finite_insert
thf(fact_1804_finite__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( insert_Product_unit @ A2 @ A ) )
= ( finite4290736615968046902t_unit @ A ) ) ).
% finite_insert
thf(fact_1805_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1806_finite__insert,axiom,
! [A2: code_integer,A: set_Code_integer] :
( ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) )
= ( finite6017078050557962740nteger @ A ) ) ).
% finite_insert
thf(fact_1807_finite__subset__induct_H,axiom,
! [F2: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ F2 )
=> ( ( ord_le7084787975880047091nteger @ F2 @ A )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [A4: code_integer,F3: set_Code_integer] :
( ( finite6017078050557962740nteger @ F3 )
=> ( ( member_Code_integer @ A4 @ A )
=> ( ( ord_le7084787975880047091nteger @ F3 @ A )
=> ( ~ ( member_Code_integer @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Code_integer @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1808_finite__subset__induct_H,axiom,
! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( ord_le3507040750410214029t_unit @ F2 @ A )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [A4: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( member_Product_unit @ A4 @ A )
=> ( ( ord_le3507040750410214029t_unit @ F3 @ A )
=> ( ~ ( member_Product_unit @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Product_unit @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1809_finite__subset__induct_H,axiom,
! [F2: set_o,A: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( ord_less_eq_set_o @ F2 @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A4: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( member_o @ A4 @ A )
=> ( ( ord_less_eq_set_o @ F3 @ A )
=> ( ~ ( member_o @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1810_finite__subset__induct_H,axiom,
! [F2: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( ord_less_eq_set_int @ F2 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ( member_int @ A4 @ A )
=> ( ( ord_less_eq_set_int @ F3 @ A )
=> ( ~ ( member_int @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1811_finite__subset__induct_H,axiom,
! [F2: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F2 )
=> ( ( ord_le2965882846123202637_nat_o @ F2 @ A )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [A4: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F3 )
=> ( ( member6576561426505652726_nat_o @ A4 @ A )
=> ( ( ord_le2965882846123202637_nat_o @ F3 @ A )
=> ( ~ ( member6576561426505652726_nat_o @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert5175938949040314269_nat_o @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1812_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1813_finite__subset__induct,axiom,
! [F2: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ F2 )
=> ( ( ord_le7084787975880047091nteger @ F2 @ A )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [A4: code_integer,F3: set_Code_integer] :
( ( finite6017078050557962740nteger @ F3 )
=> ( ( member_Code_integer @ A4 @ A )
=> ( ~ ( member_Code_integer @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Code_integer @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1814_finite__subset__induct,axiom,
! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( ord_le3507040750410214029t_unit @ F2 @ A )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [A4: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( member_Product_unit @ A4 @ A )
=> ( ~ ( member_Product_unit @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Product_unit @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1815_finite__subset__induct,axiom,
! [F2: set_o,A: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( ord_less_eq_set_o @ F2 @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A4: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( member_o @ A4 @ A )
=> ( ~ ( member_o @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1816_finite__subset__induct,axiom,
! [F2: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( ord_less_eq_set_int @ F2 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ( member_int @ A4 @ A )
=> ( ~ ( member_int @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1817_finite__subset__induct,axiom,
! [F2: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F2 )
=> ( ( ord_le2965882846123202637_nat_o @ F2 @ A )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [A4: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F3 )
=> ( ( member6576561426505652726_nat_o @ A4 @ A )
=> ( ~ ( member6576561426505652726_nat_o @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert5175938949040314269_nat_o @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1818_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1819_finite__ranking__induct,axiom,
! [S: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
( ( finite6017078050557962740nteger @ S )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer,S5: set_Code_integer] :
( ( finite6017078050557962740nteger @ S5 )
=> ( ! [Y5: code_integer] :
( ( member_Code_integer @ Y5 @ S5 )
=> ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1820_finite__ranking__induct,axiom,
! [S: set_Product_unit,P: set_Product_unit > $o,F: product_unit > rat] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,S5: set_Product_unit] :
( ( finite4290736615968046902t_unit @ S5 )
=> ( ! [Y5: product_unit] :
( ( member_Product_unit @ Y5 @ S5 )
=> ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_Product_unit @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1821_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F: $o > rat] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,S5: set_o] :
( ( finite_finite_o @ S5 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S5 )
=> ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_o @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1822_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S5: set_nat] :
( ( finite_finite_nat @ S5 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
=> ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_nat @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1823_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F: int > rat] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,S5: set_int] :
( ( finite_finite_int @ S5 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S5 )
=> ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_int @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1824_finite__ranking__induct,axiom,
! [S: set_Code_integer,P: set_Code_integer > $o,F: code_integer > nat] :
( ( finite6017078050557962740nteger @ S )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer,S5: set_Code_integer] :
( ( finite6017078050557962740nteger @ S5 )
=> ( ! [Y5: code_integer] :
( ( member_Code_integer @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1825_finite__ranking__induct,axiom,
! [S: set_Product_unit,P: set_Product_unit > $o,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,S5: set_Product_unit] :
( ( finite4290736615968046902t_unit @ S5 )
=> ( ! [Y5: product_unit] :
( ( member_Product_unit @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_Product_unit @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1826_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,S5: set_o] :
( ( finite_finite_o @ S5 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_o @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1827_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S5: set_nat] :
( ( finite_finite_nat @ S5 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_nat @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1828_finite__ranking__induct,axiom,
! [S: set_int,P: set_int > $o,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,S5: set_int] :
( ( finite_finite_int @ S5 )
=> ( ! [Y5: int] :
( ( member_int @ Y5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S5 )
=> ( P @ ( insert_int @ X3 @ S5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1829_infinite__finite__induct,axiom,
! [P: set_Code_integer > $o,A: set_Code_integer] :
( ! [A9: set_Code_integer] :
( ~ ( finite6017078050557962740nteger @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer,F3: set_Code_integer] :
( ( finite6017078050557962740nteger @ F3 )
=> ( ~ ( member_Code_integer @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1830_infinite__finite__induct,axiom,
! [P: set_Product_unit > $o,A: set_Product_unit] :
( ! [A9: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ~ ( member_Product_unit @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1831_infinite__finite__induct,axiom,
! [P: set_o > $o,A: set_o] :
( ! [A9: set_o] :
( ~ ( finite_finite_o @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ~ ( member_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1832_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A9: set_nat] :
( ~ ( finite_finite_nat @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1833_infinite__finite__induct,axiom,
! [P: set_int > $o,A: set_int] :
( ! [A9: set_int] :
( ~ ( finite_finite_int @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ~ ( member_int @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1834_infinite__finite__induct,axiom,
! [P: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
( ! [A9: set_Pr4532377907799695533_nat_o] :
( ~ ( finite3252695134891459830_nat_o @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F3 )
=> ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1835_finite__ne__induct,axiom,
! [F2: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ F2 )
=> ( ( F2 != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer] : ( P @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) )
=> ( ! [X3: code_integer,F3: set_Code_integer] :
( ( finite6017078050557962740nteger @ F3 )
=> ( ( F3 != bot_bo3990330152332043303nteger )
=> ( ~ ( member_Code_integer @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1836_finite__ne__induct,axiom,
! [F2: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( F2 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] : ( P @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) )
=> ( ! [X3: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ( F3 != bot_bo3957492148770167129t_unit )
=> ( ~ ( member_Product_unit @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1837_finite__ne__induct,axiom,
! [F2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( F2 != bot_bot_set_o )
=> ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
=> ( ! [X3: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( F3 != bot_bot_set_o )
=> ( ~ ( member_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1838_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1839_finite__ne__induct,axiom,
! [F2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( F2 != bot_bot_set_int )
=> ( ! [X3: int] : ( P @ ( insert_int @ X3 @ bot_bot_set_int ) )
=> ( ! [X3: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ( F3 != bot_bot_set_int )
=> ( ~ ( member_int @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1840_finite__ne__induct,axiom,
! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F2 )
=> ( ( F2 != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o] : ( P @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) )
=> ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F3 )
=> ( ( F3 != bot_bo7824918357723582233_nat_o )
=> ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1841_finite__induct,axiom,
! [F2: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ F2 )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer,F3: set_Code_integer] :
( ( finite6017078050557962740nteger @ F3 )
=> ( ~ ( member_Code_integer @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1842_finite__induct,axiom,
! [F2: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ F2 )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,F3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ F3 )
=> ( ~ ( member_Product_unit @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1843_finite__induct,axiom,
! [F2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ~ ( member_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1844_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1845_finite__induct,axiom,
! [F2: set_int,P: set_int > $o] :
( ( finite_finite_int @ F2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X3: int,F3: set_int] :
( ( finite_finite_int @ F3 )
=> ( ~ ( member_int @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_int @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1846_finite__induct,axiom,
! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ F2 )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ F3 )
=> ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1847_finite_Osimps,axiom,
( finite6017078050557962740nteger
= ( ^ [A6: set_Code_integer] :
( ( A6 = bot_bo3990330152332043303nteger )
| ? [A3: set_Code_integer,B6: code_integer] :
( ( A6
= ( insert_Code_integer @ B6 @ A3 ) )
& ( finite6017078050557962740nteger @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1848_finite_Osimps,axiom,
( finite4290736615968046902t_unit
= ( ^ [A6: set_Product_unit] :
( ( A6 = bot_bo3957492148770167129t_unit )
| ? [A3: set_Product_unit,B6: product_unit] :
( ( A6
= ( insert_Product_unit @ B6 @ A3 ) )
& ( finite4290736615968046902t_unit @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1849_finite_Osimps,axiom,
( finite_finite_o
= ( ^ [A6: set_o] :
( ( A6 = bot_bot_set_o )
| ? [A3: set_o,B6: $o] :
( ( A6
= ( insert_o @ B6 @ A3 ) )
& ( finite_finite_o @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1850_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
( ( A6 = bot_bot_set_nat )
| ? [A3: set_nat,B6: nat] :
( ( A6
= ( insert_nat @ B6 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1851_finite_Osimps,axiom,
( finite_finite_int
= ( ^ [A6: set_int] :
( ( A6 = bot_bot_set_int )
| ? [A3: set_int,B6: int] :
( ( A6
= ( insert_int @ B6 @ A3 ) )
& ( finite_finite_int @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1852_finite_Osimps,axiom,
( finite3252695134891459830_nat_o
= ( ^ [A6: set_Pr4532377907799695533_nat_o] :
( ( A6 = bot_bo7824918357723582233_nat_o )
| ? [A3: set_Pr4532377907799695533_nat_o,B6: produc3658429121746597890et_nat > $o] :
( ( A6
= ( insert5175938949040314269_nat_o @ B6 @ A3 ) )
& ( finite3252695134891459830_nat_o @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1853_infinite__imp__nonempty,axiom,
! [S: set_Code_integer] :
( ~ ( finite6017078050557962740nteger @ S )
=> ( S != bot_bo3990330152332043303nteger ) ) ).
% infinite_imp_nonempty
thf(fact_1854_infinite__imp__nonempty,axiom,
! [S: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ S )
=> ( S != bot_bo3957492148770167129t_unit ) ) ).
% infinite_imp_nonempty
thf(fact_1855_infinite__imp__nonempty,axiom,
! [S: set_o] :
( ~ ( finite_finite_o @ S )
=> ( S != bot_bot_set_o ) ) ).
% infinite_imp_nonempty
thf(fact_1856_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1857_infinite__imp__nonempty,axiom,
! [S: set_int] :
( ~ ( finite_finite_int @ S )
=> ( S != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_1858_infinite__imp__nonempty,axiom,
! [S: set_Pr4532377907799695533_nat_o] :
( ~ ( finite3252695134891459830_nat_o @ S )
=> ( S != bot_bo7824918357723582233_nat_o ) ) ).
% infinite_imp_nonempty
thf(fact_1859_finite_OemptyI,axiom,
finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).
% finite.emptyI
thf(fact_1860_finite_OemptyI,axiom,
finite4290736615968046902t_unit @ bot_bo3957492148770167129t_unit ).
% finite.emptyI
thf(fact_1861_finite_OemptyI,axiom,
finite_finite_o @ bot_bot_set_o ).
% finite.emptyI
thf(fact_1862_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1863_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_1864_finite_OemptyI,axiom,
finite3252695134891459830_nat_o @ bot_bo7824918357723582233_nat_o ).
% finite.emptyI
thf(fact_1865_finite_OinsertI,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( insert_int @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1866_finite_OinsertI,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( finite_finite_o @ ( insert_o @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1867_finite_OinsertI,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( finite4290736615968046902t_unit @ ( insert_Product_unit @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1868_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1869_finite_OinsertI,axiom,
! [A: set_Code_integer,A2: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1870_finite__has__minimal,axiom,
! [A: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
& ! [Xa2: code_integer] :
( ( member_Code_integer @ Xa2 @ A )
=> ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1871_finite__has__minimal,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ! [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ A )
=> ( ( ord_le3221252021190050221t_unit @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1872_finite__has__minimal,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A )
=> ( ( ord_less_eq_o @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1873_finite__has__minimal,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ? [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
& ! [Xa2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Xa2 @ A )
=> ( ( ord_le729326519192465773_nat_o @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1874_finite__has__minimal,axiom,
! [A: set_filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ? [X3: filter_nat] :
( ( member_filter_nat @ X3 @ A )
& ! [Xa2: filter_nat] :
( ( member_filter_nat @ Xa2 @ A )
=> ( ( ord_le2510731241096832064er_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1875_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A )
=> ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1876_finite__has__minimal,axiom,
! [A: set_rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ! [Xa2: rat] :
( ( member_rat @ Xa2 @ A )
=> ( ( ord_less_eq_rat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1877_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1878_finite__has__minimal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A )
=> ( ( ord_less_eq_int @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1879_finite__has__maximal,axiom,
! [A: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
& ! [Xa2: code_integer] :
( ( member_Code_integer @ Xa2 @ A )
=> ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1880_finite__has__maximal,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
& ! [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ A )
=> ( ( ord_le3221252021190050221t_unit @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1881_finite__has__maximal,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ! [Xa2: $o] :
( ( member_o @ Xa2 @ A )
=> ( ( ord_less_eq_o @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1882_finite__has__maximal,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ? [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
& ! [Xa2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Xa2 @ A )
=> ( ( ord_le729326519192465773_nat_o @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1883_finite__has__maximal,axiom,
! [A: set_filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ? [X3: filter_nat] :
( ( member_filter_nat @ X3 @ A )
& ! [Xa2: filter_nat] :
( ( member_filter_nat @ Xa2 @ A )
=> ( ( ord_le2510731241096832064er_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1884_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1885_finite__has__maximal,axiom,
! [A: set_rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A )
& ! [Xa2: rat] :
( ( member_rat @ Xa2 @ A )
=> ( ( ord_less_eq_rat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1886_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1887_finite__has__maximal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ A )
=> ( ( ord_less_eq_int @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1888_finite_Ocases,axiom,
! [A2: set_Code_integer] :
( ( finite6017078050557962740nteger @ A2 )
=> ( ( A2 != bot_bo3990330152332043303nteger )
=> ~ ! [A9: set_Code_integer] :
( ? [A4: code_integer] :
( A2
= ( insert_Code_integer @ A4 @ A9 ) )
=> ~ ( finite6017078050557962740nteger @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1889_finite_Ocases,axiom,
! [A2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A2 )
=> ( ( A2 != bot_bo3957492148770167129t_unit )
=> ~ ! [A9: set_Product_unit] :
( ? [A4: product_unit] :
( A2
= ( insert_Product_unit @ A4 @ A9 ) )
=> ~ ( finite4290736615968046902t_unit @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1890_finite_Ocases,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ~ ! [A9: set_o] :
( ? [A4: $o] :
( A2
= ( insert_o @ A4 @ A9 ) )
=> ~ ( finite_finite_o @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1891_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A9: set_nat] :
( ? [A4: nat] :
( A2
= ( insert_nat @ A4 @ A9 ) )
=> ~ ( finite_finite_nat @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1892_finite_Ocases,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ~ ! [A9: set_int] :
( ? [A4: int] :
( A2
= ( insert_int @ A4 @ A9 ) )
=> ~ ( finite_finite_int @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1893_finite_Ocases,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A2 )
=> ( ( A2 != bot_bo7824918357723582233_nat_o )
=> ~ ! [A9: set_Pr4532377907799695533_nat_o] :
( ? [A4: produc3658429121746597890et_nat > $o] :
( A2
= ( insert5175938949040314269_nat_o @ A4 @ A9 ) )
=> ~ ( finite3252695134891459830_nat_o @ A9 ) ) ) ) ).
% finite.cases
thf(fact_1894_min__ext__compat,axiom,
! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
( ( ord_le930853871177014503t_unit @ ( relcom848357998638694028t_unit @ R3 @ S ) @ R3 )
=> ( ord_le8072594200992201959t_unit @ ( relcom7108816970757792620t_unit @ ( min_ext_Product_unit @ R3 ) @ ( sup_su3573767163274163995t_unit @ ( min_ext_Product_unit @ S ) @ ( insert6793119152790066487t_unit @ ( produc7269801343787132631t_unit @ bot_bo3957492148770167129t_unit @ bot_bo3957492148770167129t_unit ) @ bot_bo5113747506523071411t_unit ) ) ) @ ( min_ext_Product_unit @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1895_min__ext__compat,axiom,
! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S ) @ R3 )
=> ( ord_le2885120122233735591_set_o @ ( relcom7534720040753365047_set_o @ ( min_ext_o @ R3 ) @ ( sup_su7766833992073560027_set_o @ ( min_ext_o @ S ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) @ bot_bo8143257439550059635_set_o ) ) ) @ ( min_ext_o @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1896_min__ext__compat,axiom,
! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( min_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( min_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( min_ext_nat @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1897_min__ext__compat,axiom,
! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( relcomp_int_int_int @ R3 @ S ) @ R3 )
=> ( ord_le1436784781161120549et_int @ ( relcom4034950705519254697et_int @ ( min_ext_int @ R3 ) @ ( sup_su1286422330746572633et_int @ ( min_ext_int @ S ) @ ( insert7328952640415156725et_int @ ( produc6363374080413544029et_int @ bot_bot_set_int @ bot_bot_set_int ) @ bot_bo1987282549146081265et_int ) ) ) @ ( min_ext_int @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1898_min__ext__compat,axiom,
! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o] :
( ( ord_le6659818918159358183_nat_o @ ( relcom6251035627269349452_nat_o @ R3 @ S ) @ R3 )
=> ( ord_le8001614231985977575_nat_o @ ( relcom7236094467932586796_nat_o @ ( min_ex3730736537596016502_nat_o @ R3 ) @ ( sup_su4863159966000489243_nat_o @ ( min_ex3730736537596016502_nat_o @ S ) @ ( insert3244108804717745463_nat_o @ ( produc5149316574908490455_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo7824918357723582233_nat_o ) @ bot_bo7535334947429593011_nat_o ) ) ) @ ( min_ex3730736537596016502_nat_o @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1899_arg__min__least,axiom,
! [S: set_Code_integer,Y: code_integer,F: code_integer > rat] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ( ( member_Code_integer @ Y @ S )
=> ( ord_less_eq_rat @ ( F @ ( lattic5439806495466278992er_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1900_arg__min__least,axiom,
! [S: set_Product_unit,Y: product_unit,F: product_unit > rat] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( S != bot_bo3957492148770167129t_unit )
=> ( ( member_Product_unit @ Y @ S )
=> ( ord_less_eq_rat @ ( F @ ( lattic725702332050428812it_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1901_arg__min__least,axiom,
! [S: set_o,Y: $o,F: $o > rat] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ( ( member_o @ Y @ S )
=> ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1902_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > rat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1903_arg__min__least,axiom,
! [S: set_int,Y: int,F: int > rat] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( member_int @ Y @ S )
=> ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1904_arg__min__least,axiom,
! [S: set_Code_integer,Y: code_integer,F: code_integer > nat] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ( ( member_Code_integer @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6074936555552774728er_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1905_arg__min__least,axiom,
! [S: set_Product_unit,Y: product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( S != bot_bo3957492148770167129t_unit )
=> ( ( member_Product_unit @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic1360832392136924548it_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1906_arg__min__least,axiom,
! [S: set_o,Y: $o,F: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ( ( member_o @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1907_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1908_arg__min__least,axiom,
! [S: set_int,Y: int,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( member_int @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1909_Total__subset__Id,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( total_on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( ord_le3146513528884898305at_nat @ R2 @ id_nat )
=> ( ( R2 = bot_bo2099793752762293965at_nat )
| ? [A4: nat] :
( R2
= ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1910_Total__subset__Id,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( total_on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( ord_le2843351958646193337nt_int @ R2 @ id_int )
=> ( ( R2 = bot_bo1796632182523588997nt_int )
| ? [A4: int] :
( R2
= ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A4 @ A4 ) @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1911_finite__transitivity__chain,axiom,
! [A: set_Code_integer,R3: code_integer > code_integer > $o] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [X3: code_integer] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: code_integer,Y2: code_integer,Z3: code_integer] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ? [Y5: code_integer] :
( ( member_Code_integer @ Y5 @ A )
& ( R3 @ X3 @ Y5 ) ) )
=> ( A = bot_bo3990330152332043303nteger ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1912_finite__transitivity__chain,axiom,
! [A: set_Product_unit,R3: product_unit > product_unit > $o] :
( ( finite4290736615968046902t_unit @ A )
=> ( ! [X3: product_unit] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ? [Y5: product_unit] :
( ( member_Product_unit @ Y5 @ A )
& ( R3 @ X3 @ Y5 ) ) )
=> ( A = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1913_finite__transitivity__chain,axiom,
! [A: set_o,R3: $o > $o > $o] :
( ( finite_finite_o @ A )
=> ( ! [X3: $o] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: $o,Y2: $o,Z3: $o] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ? [Y5: $o] :
( ( member_o @ Y5 @ A )
& ( R3 @ X3 @ Y5 ) ) )
=> ( A = bot_bot_set_o ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1914_finite__transitivity__chain,axiom,
! [A: set_nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( R3 @ X3 @ Y5 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1915_finite__transitivity__chain,axiom,
! [A: set_int,R3: int > int > $o] :
( ( finite_finite_int @ A )
=> ( ! [X3: int] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: int,Y2: int,Z3: int] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ? [Y5: int] :
( ( member_int @ Y5 @ A )
& ( R3 @ X3 @ Y5 ) ) )
=> ( A = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1916_finite__transitivity__chain,axiom,
! [A: set_Pr4532377907799695533_nat_o,R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat > $o] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ? [Y5: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y5 @ A )
& ( R3 @ X3 @ Y5 ) ) )
=> ( A = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1917_disjnt__equiv__class,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( disjnt_Product_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1918_disjnt__equiv__class,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( disjnt_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1919_disjnt__equiv__class,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( disjnt_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1920_disjnt__equiv__class,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( disjnt_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1921_disjnt__equiv__class,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( disjnt7869448817988085652_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
= ( ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1922_Sup__fin_Oinsert,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1923_Sup__fin_Oinsert,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
= ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1924_Sup__fin_Oinsert,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1925_Sup__fin_Oinsert,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1926_Sup__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1927_Sup__fin_Oinsert,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
= ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1928_Sup__fin_Oinsert,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1929_Inf__fin_Oinsert,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1930_Inf__fin_Oinsert,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
= ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1931_Inf__fin_Oinsert,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1932_Inf__fin_Oinsert,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1933_Inf__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1934_Inf__fin_Oinsert,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
= ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1935_Inf__fin_Oinsert,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1936_in__quotient__imp__in__rel,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ X4 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1937_in__quotient__imp__in__rel,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) @ X4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1938_in__quotient__imp__in__rel,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( ord_less_eq_set_int @ ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) @ X4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1939_in__quotient__imp__in__rel,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) @ X4 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1940_in__quotient__imp__in__rel,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ X4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1941_prod__set__simps_I1_J,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( basic_7833836874476830220et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
= ( insert8563697753282147485t_unit @ X @ bot_bo3686245532977126937t_unit ) ) ).
% prod_set_simps(1)
thf(fact_1942_prod__set__simps_I1_J,axiom,
! [X: nat,Y: nat] :
( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% prod_set_simps(1)
thf(fact_1943_prod__set__simps_I1_J,axiom,
! [X: int,Y: int] :
( ( basic_fsts_int_int @ ( product_Pair_int_int @ X @ Y ) )
= ( insert_int @ X @ bot_bot_set_int ) ) ).
% prod_set_simps(1)
thf(fact_1944_prod__set__simps_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( basic_8654530971723826568et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% prod_set_simps(1)
thf(fact_1945_prod__set__simps_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( basic_3966130907827183364et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% prod_set_simps(1)
thf(fact_1946_rel__restrict__tranclI,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi4009562729418357614_nat_o @ E ) )
=> ( ~ ( member6576561426505652726_nat_o @ X @ R3 )
=> ( ~ ( member6576561426505652726_nat_o @ Y @ R3 )
=> ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ E @ R3 ) @ R3 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1947_rel__restrict__tranclI,axiom,
! [X: $o,Y: $o,E: set_Product_prod_o_o,R3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_trancl_o @ E ) )
=> ( ~ ( member_o @ X @ R3 )
=> ( ~ ( member_o @ Y @ R3 )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R3 ) @ R3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1948_rel__restrict__tranclI,axiom,
! [X: product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi430408012122777006t_unit @ E ) )
=> ( ~ ( member_Product_unit @ X @ R3 )
=> ( ~ ( member_Product_unit @ Y @ R3 )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ E @ R3 ) @ R3 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1949_rel__restrict__tranclI,axiom,
! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ E ) )
=> ( ~ ( member_int @ X @ R3 )
=> ( ~ ( member_int @ Y @ R3 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ E @ R3 ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1950_rel__restrict__tranclI,axiom,
! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ E ) )
=> ( ~ ( member_nat @ X @ R3 )
=> ( ~ ( member_nat @ Y @ R3 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R3 ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_1951_pair__in__Id__conv,axiom,
! [A2: nat,B: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ id_nat )
= ( A2 = B ) ) ).
% pair_in_Id_conv
thf(fact_1952_pair__in__Id__conv,axiom,
! [A2: int,B: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ id_int )
= ( A2 = B ) ) ).
% pair_in_Id_conv
thf(fact_1953_IdI,axiom,
! [A2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ id_nat ) ).
% IdI
thf(fact_1954_IdI,axiom,
! [A2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ id_int ) ).
% IdI
thf(fact_1955_quotient__empty,axiom,
! [R2: set_Pr5094982260447487303t_unit] :
( ( equiv_5172566880711946708t_unit @ bot_bo3957492148770167129t_unit @ R2 )
= bot_bo6284674557190688313t_unit ) ).
% quotient_empty
thf(fact_1956_quotient__empty,axiom,
! [R2: set_Product_prod_o_o] :
( ( equiv_quotient_o @ bot_bot_set_o @ R2 )
= bot_bot_set_set_o ) ).
% quotient_empty
thf(fact_1957_quotient__empty,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( equiv_quotient_nat @ bot_bot_set_nat @ R2 )
= bot_bot_set_set_nat ) ).
% quotient_empty
thf(fact_1958_quotient__empty,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( equiv_quotient_int @ bot_bot_set_int @ R2 )
= bot_bot_set_set_int ) ).
% quotient_empty
thf(fact_1959_quotient__empty,axiom,
! [R2: set_Pr2161125870931222855_nat_o] :
( ( equiv_7899955228204526484_nat_o @ bot_bo7824918357723582233_nat_o @ R2 )
= bot_bo3736544561982865913_nat_o ) ).
% quotient_empty
thf(fact_1960_quotient__is__empty,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( ( equiv_5172566880711946708t_unit @ A @ R2 )
= bot_bo6284674557190688313t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% quotient_is_empty
thf(fact_1961_quotient__is__empty,axiom,
! [A: set_o,R2: set_Product_prod_o_o] :
( ( ( equiv_quotient_o @ A @ R2 )
= bot_bot_set_set_o )
= ( A = bot_bot_set_o ) ) ).
% quotient_is_empty
thf(fact_1962_quotient__is__empty,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( ( equiv_quotient_nat @ A @ R2 )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% quotient_is_empty
thf(fact_1963_quotient__is__empty,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int] :
( ( ( equiv_quotient_int @ A @ R2 )
= bot_bot_set_set_int )
= ( A = bot_bot_set_int ) ) ).
% quotient_is_empty
thf(fact_1964_quotient__is__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
( ( ( equiv_7899955228204526484_nat_o @ A @ R2 )
= bot_bo3736544561982865913_nat_o )
= ( A = bot_bo7824918357723582233_nat_o ) ) ).
% quotient_is_empty
thf(fact_1965_quotient__is__empty2,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( bot_bo6284674557190688313t_unit
= ( equiv_5172566880711946708t_unit @ A @ R2 ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% quotient_is_empty2
thf(fact_1966_quotient__is__empty2,axiom,
! [A: set_o,R2: set_Product_prod_o_o] :
( ( bot_bot_set_set_o
= ( equiv_quotient_o @ A @ R2 ) )
= ( A = bot_bot_set_o ) ) ).
% quotient_is_empty2
thf(fact_1967_quotient__is__empty2,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( bot_bot_set_set_nat
= ( equiv_quotient_nat @ A @ R2 ) )
= ( A = bot_bot_set_nat ) ) ).
% quotient_is_empty2
thf(fact_1968_quotient__is__empty2,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int] :
( ( bot_bot_set_set_int
= ( equiv_quotient_int @ A @ R2 ) )
= ( A = bot_bot_set_int ) ) ).
% quotient_is_empty2
thf(fact_1969_quotient__is__empty2,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
( ( bot_bo3736544561982865913_nat_o
= ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
= ( A = bot_bo7824918357723582233_nat_o ) ) ).
% quotient_is_empty2
thf(fact_1970_disjnt__self__iff__empty,axiom,
! [S: set_Product_unit] :
( ( disjnt_Product_unit @ S @ S )
= ( S = bot_bo3957492148770167129t_unit ) ) ).
% disjnt_self_iff_empty
thf(fact_1971_disjnt__self__iff__empty,axiom,
! [S: set_o] :
( ( disjnt_o @ S @ S )
= ( S = bot_bot_set_o ) ) ).
% disjnt_self_iff_empty
thf(fact_1972_disjnt__self__iff__empty,axiom,
! [S: set_nat] :
( ( disjnt_nat @ S @ S )
= ( S = bot_bot_set_nat ) ) ).
% disjnt_self_iff_empty
thf(fact_1973_disjnt__self__iff__empty,axiom,
! [S: set_int] :
( ( disjnt_int @ S @ S )
= ( S = bot_bot_set_int ) ) ).
% disjnt_self_iff_empty
thf(fact_1974_disjnt__self__iff__empty,axiom,
! [S: set_Pr4532377907799695533_nat_o] :
( ( disjnt7869448817988085652_nat_o @ S @ S )
= ( S = bot_bo7824918357723582233_nat_o ) ) ).
% disjnt_self_iff_empty
thf(fact_1975_disjnt__insert1,axiom,
! [A2: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
( ( disjnt7869448817988085652_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) @ Y6 )
= ( ~ ( member6576561426505652726_nat_o @ A2 @ Y6 )
& ( disjnt7869448817988085652_nat_o @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1976_disjnt__insert1,axiom,
! [A2: nat,X4: set_nat,Y6: set_nat] :
( ( disjnt_nat @ ( insert_nat @ A2 @ X4 ) @ Y6 )
= ( ~ ( member_nat @ A2 @ Y6 )
& ( disjnt_nat @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1977_disjnt__insert1,axiom,
! [A2: int,X4: set_int,Y6: set_int] :
( ( disjnt_int @ ( insert_int @ A2 @ X4 ) @ Y6 )
= ( ~ ( member_int @ A2 @ Y6 )
& ( disjnt_int @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1978_disjnt__insert1,axiom,
! [A2: $o,X4: set_o,Y6: set_o] :
( ( disjnt_o @ ( insert_o @ A2 @ X4 ) @ Y6 )
= ( ~ ( member_o @ A2 @ Y6 )
& ( disjnt_o @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1979_disjnt__insert1,axiom,
! [A2: product_unit,X4: set_Product_unit,Y6: set_Product_unit] :
( ( disjnt_Product_unit @ ( insert_Product_unit @ A2 @ X4 ) @ Y6 )
= ( ~ ( member_Product_unit @ A2 @ Y6 )
& ( disjnt_Product_unit @ X4 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_1980_disjnt__insert2,axiom,
! [Y6: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
( ( disjnt7869448817988085652_nat_o @ Y6 @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= ( ~ ( member6576561426505652726_nat_o @ A2 @ Y6 )
& ( disjnt7869448817988085652_nat_o @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1981_disjnt__insert2,axiom,
! [Y6: set_nat,A2: nat,X4: set_nat] :
( ( disjnt_nat @ Y6 @ ( insert_nat @ A2 @ X4 ) )
= ( ~ ( member_nat @ A2 @ Y6 )
& ( disjnt_nat @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1982_disjnt__insert2,axiom,
! [Y6: set_int,A2: int,X4: set_int] :
( ( disjnt_int @ Y6 @ ( insert_int @ A2 @ X4 ) )
= ( ~ ( member_int @ A2 @ Y6 )
& ( disjnt_int @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1983_disjnt__insert2,axiom,
! [Y6: set_o,A2: $o,X4: set_o] :
( ( disjnt_o @ Y6 @ ( insert_o @ A2 @ X4 ) )
= ( ~ ( member_o @ A2 @ Y6 )
& ( disjnt_o @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1984_disjnt__insert2,axiom,
! [Y6: set_Product_unit,A2: product_unit,X4: set_Product_unit] :
( ( disjnt_Product_unit @ Y6 @ ( insert_Product_unit @ A2 @ X4 ) )
= ( ~ ( member_Product_unit @ A2 @ Y6 )
& ( disjnt_Product_unit @ Y6 @ X4 ) ) ) ).
% disjnt_insert2
thf(fact_1985_rel__restrict__empty,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ( rel_re9207996470947987439t_unit @ R3 @ bot_bo3957492148770167129t_unit )
= R3 ) ).
% rel_restrict_empty
thf(fact_1986_rel__restrict__empty,axiom,
! [R3: set_Product_prod_o_o] :
( ( rel_restrict_o @ R3 @ bot_bot_set_o )
= R3 ) ).
% rel_restrict_empty
thf(fact_1987_rel__restrict__empty,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( rel_restrict_nat @ R3 @ bot_bot_set_nat )
= R3 ) ).
% rel_restrict_empty
thf(fact_1988_rel__restrict__empty,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( rel_restrict_int @ R3 @ bot_bot_set_int )
= R3 ) ).
% rel_restrict_empty
thf(fact_1989_rel__restrict__empty,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ( rel_re3331361160397388719_nat_o @ R3 @ bot_bo7824918357723582233_nat_o )
= R3 ) ).
% rel_restrict_empty
thf(fact_1990_disjnt__Un1,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( disjnt_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
= ( ( disjnt_nat @ A @ C5 )
& ( disjnt_nat @ B4 @ C5 ) ) ) ).
% disjnt_Un1
thf(fact_1991_disjnt__Un2,axiom,
! [C5: set_nat,A: set_nat,B4: set_nat] :
( ( disjnt_nat @ C5 @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( disjnt_nat @ C5 @ A )
& ( disjnt_nat @ C5 @ B4 ) ) ) ).
% disjnt_Un2
thf(fact_1992_Sup__fin_Osingleton,axiom,
! [X: product_unit] :
( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1993_Sup__fin_Osingleton,axiom,
! [X: $o] :
( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1994_Sup__fin_Osingleton,axiom,
! [X: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1995_Sup__fin_Osingleton,axiom,
! [X: int] :
( ( lattic1091506334969745077in_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1996_Sup__fin_Osingleton,axiom,
! [X: produc3658429121746597890et_nat > $o] :
( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1997_Inf__fin_Osingleton,axiom,
! [X: product_unit] :
( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% Inf_fin.singleton
thf(fact_1998_Inf__fin_Osingleton,axiom,
! [X: $o] :
( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% Inf_fin.singleton
thf(fact_1999_Inf__fin_Osingleton,axiom,
! [X: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Inf_fin.singleton
thf(fact_2000_Inf__fin_Osingleton,axiom,
! [X: int] :
( ( lattic5235898064620869839in_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% Inf_fin.singleton
thf(fact_2001_Inf__fin_Osingleton,axiom,
! [X: produc3658429121746597890et_nat > $o] :
( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X ) ).
% Inf_fin.singleton
thf(fact_2002_disjnt__iff,axiom,
( disjnt7869448817988085652_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
! [X2: produc3658429121746597890et_nat > $o] :
~ ( ( member6576561426505652726_nat_o @ X2 @ A3 )
& ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_2003_disjnt__iff,axiom,
( disjnt_nat
= ( ^ [A3: set_nat,B7: set_nat] :
! [X2: nat] :
~ ( ( member_nat @ X2 @ A3 )
& ( member_nat @ X2 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_2004_disjnt__iff,axiom,
( disjnt_int
= ( ^ [A3: set_int,B7: set_int] :
! [X2: int] :
~ ( ( member_int @ X2 @ A3 )
& ( member_int @ X2 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_2005_disjnt__iff,axiom,
( disjnt_o
= ( ^ [A3: set_o,B7: set_o] :
! [X2: $o] :
~ ( ( member_o @ X2 @ A3 )
& ( member_o @ X2 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_2006_disjnt__iff,axiom,
( disjnt_Product_unit
= ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
! [X2: product_unit] :
~ ( ( member_Product_unit @ X2 @ A3 )
& ( member_Product_unit @ X2 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_2007_IdE,axiom,
! [P3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ P3 @ id_nat )
=> ~ ! [X3: nat] :
( P3
!= ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_2008_IdE,axiom,
! [P3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ P3 @ id_int )
=> ~ ! [X3: int] :
( P3
!= ( product_Pair_int_int @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_2009_disjnt__empty1,axiom,
! [A: set_Product_unit] : ( disjnt_Product_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% disjnt_empty1
thf(fact_2010_disjnt__empty1,axiom,
! [A: set_o] : ( disjnt_o @ bot_bot_set_o @ A ) ).
% disjnt_empty1
thf(fact_2011_disjnt__empty1,axiom,
! [A: set_nat] : ( disjnt_nat @ bot_bot_set_nat @ A ) ).
% disjnt_empty1
thf(fact_2012_disjnt__empty1,axiom,
! [A: set_int] : ( disjnt_int @ bot_bot_set_int @ A ) ).
% disjnt_empty1
thf(fact_2013_disjnt__empty1,axiom,
! [A: set_Pr4532377907799695533_nat_o] : ( disjnt7869448817988085652_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ).
% disjnt_empty1
thf(fact_2014_disjnt__empty2,axiom,
! [A: set_Product_unit] : ( disjnt_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% disjnt_empty2
thf(fact_2015_disjnt__empty2,axiom,
! [A: set_o] : ( disjnt_o @ A @ bot_bot_set_o ) ).
% disjnt_empty2
thf(fact_2016_disjnt__empty2,axiom,
! [A: set_nat] : ( disjnt_nat @ A @ bot_bot_set_nat ) ).
% disjnt_empty2
thf(fact_2017_disjnt__empty2,axiom,
! [A: set_int] : ( disjnt_int @ A @ bot_bot_set_int ) ).
% disjnt_empty2
thf(fact_2018_disjnt__empty2,axiom,
! [A: set_Pr4532377907799695533_nat_o] : ( disjnt7869448817988085652_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% disjnt_empty2
thf(fact_2019_disjnt__subset1,axiom,
! [X4: set_nat,Y6: set_nat,Z5: set_nat] :
( ( disjnt_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_set_nat @ Z5 @ X4 )
=> ( disjnt_nat @ Z5 @ Y6 ) ) ) ).
% disjnt_subset1
thf(fact_2020_disjnt__subset2,axiom,
! [X4: set_nat,Y6: set_nat,Z5: set_nat] :
( ( disjnt_nat @ X4 @ Y6 )
=> ( ( ord_less_eq_set_nat @ Z5 @ Y6 )
=> ( disjnt_nat @ X4 @ Z5 ) ) ) ).
% disjnt_subset2
thf(fact_2021_disjnt__insert,axiom,
! [X: produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o,M: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ N )
=> ( ( disjnt7869448817988085652_nat_o @ M @ N )
=> ( disjnt7869448817988085652_nat_o @ ( insert5175938949040314269_nat_o @ X @ M ) @ N ) ) ) ).
% disjnt_insert
thf(fact_2022_disjnt__insert,axiom,
! [X: nat,N: set_nat,M: set_nat] :
( ~ ( member_nat @ X @ N )
=> ( ( disjnt_nat @ M @ N )
=> ( disjnt_nat @ ( insert_nat @ X @ M ) @ N ) ) ) ).
% disjnt_insert
thf(fact_2023_disjnt__insert,axiom,
! [X: int,N: set_int,M: set_int] :
( ~ ( member_int @ X @ N )
=> ( ( disjnt_int @ M @ N )
=> ( disjnt_int @ ( insert_int @ X @ M ) @ N ) ) ) ).
% disjnt_insert
thf(fact_2024_disjnt__insert,axiom,
! [X: $o,N: set_o,M: set_o] :
( ~ ( member_o @ X @ N )
=> ( ( disjnt_o @ M @ N )
=> ( disjnt_o @ ( insert_o @ X @ M ) @ N ) ) ) ).
% disjnt_insert
thf(fact_2025_disjnt__insert,axiom,
! [X: product_unit,N: set_Product_unit,M: set_Product_unit] :
( ~ ( member_Product_unit @ X @ N )
=> ( ( disjnt_Product_unit @ M @ N )
=> ( disjnt_Product_unit @ ( insert_Product_unit @ X @ M ) @ N ) ) ) ).
% disjnt_insert
thf(fact_2026_rel__restrict__notR_I2_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A @ R3 ) )
=> ~ ( member6576561426505652726_nat_o @ Y @ R3 ) ) ).
% rel_restrict_notR(2)
thf(fact_2027_rel__restrict__notR_I2_J,axiom,
! [X: $o,Y: $o,A: set_Product_prod_o_o,R3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ A @ R3 ) )
=> ~ ( member_o @ Y @ R3 ) ) ).
% rel_restrict_notR(2)
thf(fact_2028_rel__restrict__notR_I2_J,axiom,
! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ A @ R3 ) )
=> ~ ( member_Product_unit @ Y @ R3 ) ) ).
% rel_restrict_notR(2)
thf(fact_2029_rel__restrict__notR_I2_J,axiom,
! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A @ R3 ) )
=> ~ ( member_nat @ Y @ R3 ) ) ).
% rel_restrict_notR(2)
thf(fact_2030_rel__restrict__notR_I2_J,axiom,
! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A @ R3 ) )
=> ~ ( member_int @ Y @ R3 ) ) ).
% rel_restrict_notR(2)
thf(fact_2031_rel__restrict__notR_I1_J,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A @ R3 ) )
=> ~ ( member6576561426505652726_nat_o @ X @ R3 ) ) ).
% rel_restrict_notR(1)
thf(fact_2032_rel__restrict__notR_I1_J,axiom,
! [X: $o,Y: $o,A: set_Product_prod_o_o,R3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ A @ R3 ) )
=> ~ ( member_o @ X @ R3 ) ) ).
% rel_restrict_notR(1)
thf(fact_2033_rel__restrict__notR_I1_J,axiom,
! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ A @ R3 ) )
=> ~ ( member_Product_unit @ X @ R3 ) ) ).
% rel_restrict_notR(1)
thf(fact_2034_rel__restrict__notR_I1_J,axiom,
! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A @ R3 ) )
=> ~ ( member_nat @ X @ R3 ) ) ).
% rel_restrict_notR(1)
thf(fact_2035_rel__restrict__notR_I1_J,axiom,
! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A @ R3 ) )
=> ~ ( member_int @ X @ R3 ) ) ).
% rel_restrict_notR(1)
thf(fact_2036_rel__restrictI,axiom,
! [X: produc3658429121746597890et_nat > $o,R3: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ R3 )
=> ( ~ ( member6576561426505652726_nat_o @ Y @ R3 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ E )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ).
% rel_restrictI
thf(fact_2037_rel__restrictI,axiom,
! [X: $o,R3: set_o,Y: $o,E: set_Product_prod_o_o] :
( ~ ( member_o @ X @ R3 )
=> ( ~ ( member_o @ Y @ R3 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ E )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ).
% rel_restrictI
thf(fact_2038_rel__restrictI,axiom,
! [X: product_unit,R3: set_Product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit] :
( ~ ( member_Product_unit @ X @ R3 )
=> ( ~ ( member_Product_unit @ Y @ R3 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ E )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ).
% rel_restrictI
thf(fact_2039_rel__restrictI,axiom,
! [X: nat,R3: set_nat,Y: nat,E: set_Pr1261947904930325089at_nat] :
( ~ ( member_nat @ X @ R3 )
=> ( ~ ( member_nat @ Y @ R3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ).
% rel_restrictI
thf(fact_2040_rel__restrictI,axiom,
! [X: int,R3: set_int,Y: int,E: set_Pr958786334691620121nt_int] :
( ~ ( member_int @ X @ R3 )
=> ( ~ ( member_int @ Y @ R3 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ).
% rel_restrictI
thf(fact_2041_rel__restrict__lift,axiom,
! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E ) ) ).
% rel_restrict_lift
thf(fact_2042_rel__restrict__lift,axiom,
! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R3 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E ) ) ).
% rel_restrict_lift
thf(fact_2043_rel__restrict__union,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
( ( rel_restrict_nat @ R3 @ ( sup_sup_set_nat @ A @ B4 ) )
= ( rel_restrict_nat @ ( rel_restrict_nat @ R3 @ A ) @ B4 ) ) ).
% rel_restrict_union
thf(fact_2044_Inf__fin__le__Sup__fin,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2045_Inf__fin__le__Sup__fin,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2046_Inf__fin__le__Sup__fin,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ord_le729326519192465773_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2047_Inf__fin__le__Sup__fin,axiom,
! [A: set_filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ord_le2510731241096832064er_nat @ ( lattic9128708996678779395er_nat @ A ) @ ( lattic5930898082463196905er_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2048_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2049_Inf__fin__le__Sup__fin,axiom,
! [A: set_rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ord_less_eq_rat @ ( lattic4603258475043424379in_rat @ A ) @ ( lattic458866745392299617in_rat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2050_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2051_Inf__fin__le__Sup__fin,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic1091506334969745077in_int @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_2052_disjnt__Sigma__iff,axiom,
! [A: set_nat,C5: nat > set_Product_unit,B4: set_nat] :
( ( disjnt595863415355979947t_unit @ ( produc5506539147108004339t_unit @ A @ C5 ) @ ( produc5506539147108004339t_unit @ B4 @ C5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( C5 @ X2 )
= bot_bo3957492148770167129t_unit ) )
| ( disjnt_nat @ A @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_2053_disjnt__Sigma__iff,axiom,
! [A: set_nat,C5: nat > set_o,B4: set_nat] :
( ( disjnt5652777605541633334_nat_o @ ( product_Sigma_nat_o @ A @ C5 ) @ ( product_Sigma_nat_o @ B4 @ C5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( C5 @ X2 )
= bot_bot_set_o ) )
| ( disjnt_nat @ A @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_2054_disjnt__Sigma__iff,axiom,
! [A: set_nat,C5: nat > set_nat,B4: set_nat] :
( ( disjnt2713546992526078024at_nat @ ( produc457027306803732586at_nat @ A @ C5 ) @ ( produc457027306803732586at_nat @ B4 @ C5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( C5 @ X2 )
= bot_bot_set_nat ) )
| ( disjnt_nat @ A @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_2055_disjnt__Sigma__iff,axiom,
! [A: set_nat,C5: nat > set_int,B4: set_nat] :
( ( disjnt7759068009871657124at_int @ ( produc454536836294682310at_int @ A @ C5 ) @ ( produc454536836294682310at_int @ B4 @ C5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( C5 @ X2 )
= bot_bot_set_int ) )
| ( disjnt_nat @ A @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_2056_disjnt__Sigma__iff,axiom,
! [A: set_nat,C5: nat > set_Pr4532377907799695533_nat_o,B4: set_nat] :
( ( disjnt4476176636800211563_nat_o @ ( produc5650067098820117939_nat_o @ A @ C5 ) @ ( produc5650067098820117939_nat_o @ B4 @ C5 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( C5 @ X2 )
= bot_bo7824918357723582233_nat_o ) )
| ( disjnt_nat @ A @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_2057_disjnt__def,axiom,
( disjnt_Product_unit
= ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A3 @ B7 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% disjnt_def
thf(fact_2058_disjnt__def,axiom,
( disjnt_o
= ( ^ [A3: set_o,B7: set_o] :
( ( inf_inf_set_o @ A3 @ B7 )
= bot_bot_set_o ) ) ) ).
% disjnt_def
thf(fact_2059_disjnt__def,axiom,
( disjnt_nat
= ( ^ [A3: set_nat,B7: set_nat] :
( ( inf_inf_set_nat @ A3 @ B7 )
= bot_bot_set_nat ) ) ) ).
% disjnt_def
thf(fact_2060_disjnt__def,axiom,
( disjnt_int
= ( ^ [A3: set_int,B7: set_int] :
( ( inf_inf_set_int @ A3 @ B7 )
= bot_bot_set_int ) ) ) ).
% disjnt_def
thf(fact_2061_disjnt__def,axiom,
( disjnt7869448817988085652_nat_o
= ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ A3 @ B7 )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% disjnt_def
thf(fact_2062_quotient__eqI,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ X @ X4 )
=> ( ( member6576561426505652726_nat_o @ Y @ Y6 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_2063_quotient__eqI,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( member_o @ X @ X4 )
=> ( ( member_o @ Y @ Y6 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_2064_quotient__eqI,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( member_Product_unit @ X @ X4 )
=> ( ( member_Product_unit @ Y @ Y6 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_2065_quotient__eqI,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( member_nat @ X @ X4 )
=> ( ( member_nat @ Y @ Y6 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_2066_quotient__eqI,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( member_int @ X @ X4 )
=> ( ( member_int @ Y @ Y6 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( X4 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_2067_quotient__eq__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ X @ X4 )
=> ( ( member6576561426505652726_nat_o @ Y @ Y6 )
=> ( ( X4 = Y6 )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_2068_quotient__eq__iff,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( member_o @ X @ X4 )
=> ( ( member_o @ Y @ Y6 )
=> ( ( X4 = Y6 )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_2069_quotient__eq__iff,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( member_Product_unit @ X @ X4 )
=> ( ( member_Product_unit @ Y @ Y6 )
=> ( ( X4 = Y6 )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_2070_quotient__eq__iff,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( member_nat @ X @ X4 )
=> ( ( member_nat @ Y @ Y6 )
=> ( ( X4 = Y6 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_2071_quotient__eq__iff,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( member_int @ X @ X4 )
=> ( ( member_int @ Y @ Y6 )
=> ( ( X4 = Y6 )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_2072_in__quotient__imp__closed,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ X @ X4 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
=> ( member6576561426505652726_nat_o @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_2073_in__quotient__imp__closed,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( member_o @ X @ X4 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
=> ( member_o @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_2074_in__quotient__imp__closed,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( member_Product_unit @ X @ X4 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
=> ( member_Product_unit @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_2075_in__quotient__imp__closed,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( member_nat @ X @ X4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( member_nat @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_2076_in__quotient__imp__closed,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( member_int @ X @ X4 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( member_int @ Y @ X4 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_2077_in__quotient__imp__non__empty,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( X4 != bot_bo3957492148770167129t_unit ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_2078_in__quotient__imp__non__empty,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X4: set_o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( X4 != bot_bot_set_o ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_2079_in__quotient__imp__non__empty,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( X4 != bot_bot_set_nat ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_2080_in__quotient__imp__non__empty,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( X4 != bot_bot_set_int ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_2081_in__quotient__imp__non__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( X4 != bot_bo7824918357723582233_nat_o ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_2082_rel__restrict__trancl__notR_I2_J,axiom,
! [V: produc3658429121746597890et_nat > $o,W: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V @ W ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) )
=> ~ ( member6576561426505652726_nat_o @ W @ R3 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_2083_rel__restrict__trancl__notR_I2_J,axiom,
! [V: $o,W: $o,E: set_Product_prod_o_o,R3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) )
=> ~ ( member_o @ W @ R3 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_2084_rel__restrict__trancl__notR_I2_J,axiom,
! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V @ W ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) )
=> ~ ( member_Product_unit @ W @ R3 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_2085_rel__restrict__trancl__notR_I2_J,axiom,
! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) )
=> ~ ( member_nat @ W @ R3 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_2086_rel__restrict__trancl__notR_I2_J,axiom,
! [V: int,W: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) )
=> ~ ( member_int @ W @ R3 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_2087_rel__restrict__trancl__notR_I1_J,axiom,
! [V: produc3658429121746597890et_nat > $o,W: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V @ W ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) )
=> ~ ( member6576561426505652726_nat_o @ V @ R3 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_2088_rel__restrict__trancl__notR_I1_J,axiom,
! [V: $o,W: $o,E: set_Product_prod_o_o,R3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) )
=> ~ ( member_o @ V @ R3 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_2089_rel__restrict__trancl__notR_I1_J,axiom,
! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V @ W ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) )
=> ~ ( member_Product_unit @ V @ R3 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_2090_rel__restrict__trancl__notR_I1_J,axiom,
! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) )
=> ~ ( member_nat @ V @ R3 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_2091_rel__restrict__trancl__notR_I1_J,axiom,
! [V: int,W: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) )
=> ~ ( member_int @ V @ R3 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_2092_rel__restrict__trancl__mem,axiom,
! [A2: nat,B: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ A @ R3 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( rel_restrict_nat @ ( transi6264000038957366511cl_nat @ A ) @ R3 ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_2093_rel__restrict__trancl__mem,axiom,
! [A2: int,B: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ A @ R3 ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( rel_restrict_int @ ( transi6261509568448316235cl_int @ A ) @ R3 ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_2094_rel__restrict__mono2,axiom,
! [R3: set_nat,S: set_nat,A: set_Pr1261947904930325089at_nat] :
( ( ord_less_eq_set_nat @ R3 @ S )
=> ( ord_le3146513528884898305at_nat @ ( rel_restrict_nat @ A @ S ) @ ( rel_restrict_nat @ A @ R3 ) ) ) ).
% rel_restrict_mono2
thf(fact_2095_singleton__quotient,axiom,
! [X: product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ R2 )
= ( insert8986028188087627965t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ bot_bo6284674557190688313t_unit ) ) ).
% singleton_quotient
thf(fact_2096_singleton__quotient,axiom,
! [X: $o,R2: set_Product_prod_o_o] :
( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R2 )
= ( insert_set_o @ ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ bot_bot_set_set_o ) ) ).
% singleton_quotient
thf(fact_2097_singleton__quotient,axiom,
! [X: nat,R2: set_Pr1261947904930325089at_nat] :
( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
= ( insert_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat ) ) ).
% singleton_quotient
thf(fact_2098_singleton__quotient,axiom,
! [X: int,R2: set_Pr958786334691620121nt_int] :
( ( equiv_quotient_int @ ( insert_int @ X @ bot_bot_set_int ) @ R2 )
= ( insert_set_int @ ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ bot_bot_set_set_int ) ) ).
% singleton_quotient
thf(fact_2099_singleton__quotient,axiom,
! [X: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
( ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ R2 )
= ( insert2507855216428400253_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ bot_bo3736544561982865913_nat_o ) ) ).
% singleton_quotient
thf(fact_2100_quotientI,axiom,
! [X: product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( member_Product_unit @ X @ A )
=> ( member5877623283571906838t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( equiv_5172566880711946708t_unit @ A @ R2 ) ) ) ).
% quotientI
thf(fact_2101_quotientI,axiom,
! [X: $o,A: set_o,R2: set_Product_prod_o_o] :
( ( member_o @ X @ A )
=> ( member_set_o @ ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( equiv_quotient_o @ A @ R2 ) ) ) ).
% quotientI
thf(fact_2102_quotientI,axiom,
! [X: nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( equiv_quotient_nat @ A @ R2 ) ) ) ).
% quotientI
thf(fact_2103_quotientI,axiom,
! [X: int,A: set_int,R2: set_Pr958786334691620121nt_int] :
( ( member_int @ X @ A )
=> ( member_set_int @ ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( equiv_quotient_int @ A @ R2 ) ) ) ).
% quotientI
thf(fact_2104_quotientI,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
( ( member6576561426505652726_nat_o @ X @ A )
=> ( member1452989533377600726_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) ) ) ).
% quotientI
thf(fact_2105_quotientE,axiom,
! [X4: set_Product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ~ ! [X3: product_unit] :
( ( X4
= ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) )
=> ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% quotientE
thf(fact_2106_quotientE,axiom,
! [X4: set_o,A: set_o,R2: set_Product_prod_o_o] :
( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ~ ! [X3: $o] :
( ( X4
= ( image_o_o @ R2 @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% quotientE
thf(fact_2107_quotientE,axiom,
! [X4: set_nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ~ ! [X3: nat] :
( ( X4
= ( image_nat_nat @ R2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% quotientE
thf(fact_2108_quotientE,axiom,
! [X4: set_int,A: set_int,R2: set_Pr958786334691620121nt_int] :
( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ~ ! [X3: int] :
( ( X4
= ( image_int_int @ R2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) )
=> ~ ( member_int @ X3 @ A ) ) ) ).
% quotientE
thf(fact_2109_quotientE,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
( ( X4
= ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) )
=> ~ ( member6576561426505652726_nat_o @ X3 @ A ) ) ) ).
% quotientE
thf(fact_2110_Inf__fin_OboundedE,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) )
=> ! [A10: product_unit] :
( ( member_Product_unit @ A10 @ A )
=> ( ord_le3221252021190050221t_unit @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2111_Inf__fin_OboundedE,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
=> ! [A10: $o] :
( ( member_o @ A10 @ A )
=> ( ord_less_eq_o @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2112_Inf__fin_OboundedE,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
=> ! [A10: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ A10 @ A )
=> ( ord_le729326519192465773_nat_o @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2113_Inf__fin_OboundedE,axiom,
! [A: set_filter_nat,X: filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) )
=> ! [A10: filter_nat] :
( ( member_filter_nat @ A10 @ A )
=> ( ord_le2510731241096832064er_nat @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2114_Inf__fin_OboundedE,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
=> ! [A10: set_nat] :
( ( member_set_nat @ A10 @ A )
=> ( ord_less_eq_set_nat @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2115_Inf__fin_OboundedE,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) )
=> ! [A10: rat] :
( ( member_rat @ A10 @ A )
=> ( ord_less_eq_rat @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2116_Inf__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A )
=> ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2117_Inf__fin_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
=> ! [A10: int] :
( ( member_int @ A10 @ A )
=> ( ord_less_eq_int @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_2118_Inf__fin_OboundedI,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ord_le3221252021190050221t_unit @ X @ A4 ) )
=> ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2119_Inf__fin_OboundedI,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_o @ X @ A4 ) )
=> ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2120_Inf__fin_OboundedI,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [A4: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ A4 @ A )
=> ( ord_le729326519192465773_nat_o @ X @ A4 ) )
=> ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2121_Inf__fin_OboundedI,axiom,
! [A: set_filter_nat,X: filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ! [A4: filter_nat] :
( ( member_filter_nat @ A4 @ A )
=> ( ord_le2510731241096832064er_nat @ X @ A4 ) )
=> ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2122_Inf__fin_OboundedI,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
=> ( ord_less_eq_set_nat @ X @ A4 ) )
=> ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2123_Inf__fin_OboundedI,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ! [A4: rat] :
( ( member_rat @ A4 @ A )
=> ( ord_less_eq_rat @ X @ A4 ) )
=> ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2124_Inf__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2125_Inf__fin_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ X @ A4 ) )
=> ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_2126_Sup__fin_OboundedE,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X )
=> ! [A10: product_unit] :
( ( member_Product_unit @ A10 @ A )
=> ( ord_le3221252021190050221t_unit @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2127_Sup__fin_OboundedE,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
=> ! [A10: $o] :
( ( member_o @ A10 @ A )
=> ( ord_less_eq_o @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2128_Sup__fin_OboundedE,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
=> ! [A10: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ A10 @ A )
=> ( ord_le729326519192465773_nat_o @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2129_Sup__fin_OboundedE,axiom,
! [A: set_filter_nat,X: filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X )
=> ! [A10: filter_nat] :
( ( member_filter_nat @ A10 @ A )
=> ( ord_le2510731241096832064er_nat @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2130_Sup__fin_OboundedE,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
=> ! [A10: set_nat] :
( ( member_set_nat @ A10 @ A )
=> ( ord_less_eq_set_nat @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2131_Sup__fin_OboundedE,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X )
=> ! [A10: rat] :
( ( member_rat @ A10 @ A )
=> ( ord_less_eq_rat @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2132_Sup__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A )
=> ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2133_Sup__fin_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
=> ! [A10: int] :
( ( member_int @ A10 @ A )
=> ( ord_less_eq_int @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_2134_Sup__fin_OboundedI,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ord_le3221252021190050221t_unit @ A4 @ X ) )
=> ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2135_Sup__fin_OboundedI,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_o @ A4 @ X ) )
=> ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2136_Sup__fin_OboundedI,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [A4: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ A4 @ A )
=> ( ord_le729326519192465773_nat_o @ A4 @ X ) )
=> ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2137_Sup__fin_OboundedI,axiom,
! [A: set_filter_nat,X: filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ! [A4: filter_nat] :
( ( member_filter_nat @ A4 @ A )
=> ( ord_le2510731241096832064er_nat @ A4 @ X ) )
=> ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2138_Sup__fin_OboundedI,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
=> ( ord_less_eq_set_nat @ A4 @ X ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2139_Sup__fin_OboundedI,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ! [A4: rat] :
( ( member_rat @ A4 @ A )
=> ( ord_less_eq_rat @ A4 @ X ) )
=> ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2140_Sup__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2141_Sup__fin_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ A4 @ X ) )
=> ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_2142_Inf__fin_Obounded__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2143_Inf__fin_Obounded__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2144_Inf__fin_Obounded__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
=> ( ord_le729326519192465773_nat_o @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2145_Inf__fin_Obounded__iff,axiom,
! [A: set_filter_nat,X: filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) )
= ( ! [X2: filter_nat] :
( ( member_filter_nat @ X2 @ A )
=> ( ord_le2510731241096832064er_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2146_Inf__fin_Obounded__iff,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2147_Inf__fin_Obounded__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) )
= ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2148_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2149_Inf__fin_Obounded__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_2150_Sup__fin_Obounded__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2151_Sup__fin_Obounded__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2152_Sup__fin_Obounded__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
=> ( ord_le729326519192465773_nat_o @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2153_Sup__fin_Obounded__iff,axiom,
! [A: set_filter_nat,X: filter_nat] :
( ( finite2119507909894593271er_nat @ A )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X )
= ( ! [X2: filter_nat] :
( ( member_filter_nat @ X2 @ A )
=> ( ord_le2510731241096832064er_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2154_Sup__fin_Obounded__iff,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2155_Sup__fin_Obounded__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X )
= ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2156_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2157_Sup__fin_Obounded__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_2158_quotient__disj,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
=> ( ( X4 = Y6 )
| ( ( inf_in4660618365625256667t_unit @ X4 @ Y6 )
= bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% quotient_disj
thf(fact_2159_quotient__disj,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
=> ( ( X4 = Y6 )
| ( ( inf_inf_set_o @ X4 @ Y6 )
= bot_bot_set_o ) ) ) ) ) ).
% quotient_disj
thf(fact_2160_quotient__disj,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
=> ( ( X4 = Y6 )
| ( ( inf_inf_set_nat @ X4 @ Y6 )
= bot_bot_set_nat ) ) ) ) ) ).
% quotient_disj
thf(fact_2161_quotient__disj,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
=> ( ( X4 = Y6 )
| ( ( inf_inf_set_int @ X4 @ Y6 )
= bot_bot_set_int ) ) ) ) ) ).
% quotient_disj
thf(fact_2162_quotient__disj,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
=> ( ( X4 = Y6 )
| ( ( inf_in1906310914598751387_nat_o @ X4 @ Y6 )
= bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% quotient_disj
thf(fact_2163_rel__restrict__Int__empty,axiom,
! [A: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ ( field_Product_unit @ R3 ) )
= bot_bo3957492148770167129t_unit )
=> ( ( rel_re9207996470947987439t_unit @ R3 @ A )
= R3 ) ) ).
% rel_restrict_Int_empty
thf(fact_2164_rel__restrict__Int__empty,axiom,
! [A: set_o,R3: set_Product_prod_o_o] :
( ( ( inf_inf_set_o @ A @ ( field_o @ R3 ) )
= bot_bot_set_o )
=> ( ( rel_restrict_o @ R3 @ A )
= R3 ) ) ).
% rel_restrict_Int_empty
thf(fact_2165_rel__restrict__Int__empty,axiom,
! [A: set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( ( inf_inf_set_nat @ A @ ( field_nat @ R3 ) )
= bot_bot_set_nat )
=> ( ( rel_restrict_nat @ R3 @ A )
= R3 ) ) ).
% rel_restrict_Int_empty
thf(fact_2166_rel__restrict__Int__empty,axiom,
! [A: set_int,R3: set_Pr958786334691620121nt_int] :
( ( ( inf_inf_set_int @ A @ ( field_int @ R3 ) )
= bot_bot_set_int )
=> ( ( rel_restrict_int @ R3 @ A )
= R3 ) ) ).
% rel_restrict_Int_empty
thf(fact_2167_rel__restrict__Int__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A @ ( field_81821745747897882_nat_o @ R3 ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( rel_re3331361160397388719_nat_o @ R3 @ A )
= R3 ) ) ).
% rel_restrict_Int_empty
thf(fact_2168_eq__equiv__class__iff2,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
( ( equiv_4189356295871601603t_unit @ A @ R2 )
=> ( ( member_Product_unit @ X @ A )
=> ( ( member_Product_unit @ Y @ A )
=> ( ( ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ R2 )
= ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) @ R2 ) )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_2169_eq__equiv__class__iff2,axiom,
! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
( ( equiv_equiv_o @ A @ R2 )
=> ( ( member_o @ X @ A )
=> ( ( member_o @ Y @ A )
=> ( ( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R2 )
= ( equiv_quotient_o @ ( insert_o @ Y @ bot_bot_set_o ) @ R2 ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_2170_eq__equiv__class__iff2,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( equiv_equiv_nat @ A @ R2 )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
= ( equiv_quotient_nat @ ( insert_nat @ Y @ bot_bot_set_nat ) @ R2 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_2171_eq__equiv__class__iff2,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( equiv_equiv_int @ A @ R2 )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ( equiv_quotient_int @ ( insert_int @ X @ bot_bot_set_int ) @ R2 )
= ( equiv_quotient_int @ ( insert_int @ Y @ bot_bot_set_int ) @ R2 ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_2172_eq__equiv__class__iff2,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( equiv_6726761681920839555_nat_o @ A @ R2 )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ( ( ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ R2 )
= ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) @ R2 ) )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_2173_Inf__fin_Osubset__imp,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2174_Inf__fin_Osubset__imp,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ B4 ) @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2175_Inf__fin_Osubset__imp,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( finite3252695134891459830_nat_o @ B4 )
=> ( ord_le729326519192465773_nat_o @ ( lattic956194824204696298_nat_o @ B4 ) @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2176_Inf__fin_Osubset__imp,axiom,
! [A: set_filter_nat,B4: set_filter_nat] :
( ( ord_le2426478655948331894er_nat @ A @ B4 )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ( finite2119507909894593271er_nat @ B4 )
=> ( ord_le2510731241096832064er_nat @ ( lattic9128708996678779395er_nat @ B4 ) @ ( lattic9128708996678779395er_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2177_Inf__fin_Osubset__imp,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2178_Inf__fin_Osubset__imp,axiom,
! [A: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A @ B4 )
=> ( ( A != bot_bot_set_rat )
=> ( ( finite_finite_rat @ B4 )
=> ( ord_less_eq_rat @ ( lattic4603258475043424379in_rat @ B4 ) @ ( lattic4603258475043424379in_rat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2179_Inf__fin_Osubset__imp,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ B4 ) @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2180_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_2181_Sup__fin_Osubset__imp,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2182_Sup__fin_Osubset__imp,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2183_Sup__fin_Osubset__imp,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( finite3252695134891459830_nat_o @ B4 )
=> ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2184_Sup__fin_Osubset__imp,axiom,
! [A: set_filter_nat,B4: set_filter_nat] :
( ( ord_le2426478655948331894er_nat @ A @ B4 )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ( finite2119507909894593271er_nat @ B4 )
=> ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ ( lattic5930898082463196905er_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2185_Sup__fin_Osubset__imp,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2186_Sup__fin_Osubset__imp,axiom,
! [A: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A @ B4 )
=> ( ( A != bot_bot_set_rat )
=> ( ( finite_finite_rat @ B4 )
=> ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ ( lattic458866745392299617in_rat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2187_Sup__fin_Osubset__imp,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2188_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_2189_Inf__fin_Osubset,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( B4 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A ) )
= ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2190_Inf__fin_Osubset,axiom,
! [A: set_assn,B4: set_assn] :
( ( finite_finite_assn @ A )
=> ( ( B4 != bot_bot_set_assn )
=> ( ( ord_less_eq_set_assn @ B4 @ A )
=> ( ( inf_inf_assn @ ( lattic47131356835913163n_assn @ B4 ) @ ( lattic47131356835913163n_assn @ A ) )
= ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2191_Inf__fin_Osubset,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ A )
=> ( ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A ) )
= ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2192_Inf__fin_Osubset,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( B4 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ B4 @ A )
=> ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B4 ) @ ( lattic4107685809792843317_fin_o @ A ) )
= ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2193_Inf__fin_Osubset,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( B4 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ B4 @ A )
=> ( ( inf_inf_int @ ( lattic5235898064620869839in_int @ B4 ) @ ( lattic5235898064620869839in_int @ A ) )
= ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2194_Inf__fin_Osubset,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ A )
=> ( ( inf_in1318976480646536635_nat_o @ ( lattic956194824204696298_nat_o @ B4 ) @ ( lattic956194824204696298_nat_o @ A ) )
= ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2195_Inf__fin_Osubset,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_2196_Sup__fin_Osubset,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( B4 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B4 ) @ ( lattic3835124923745554447et_nat @ A ) )
= ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2197_Sup__fin_Osubset,axiom,
! [A: set_assn,B4: set_assn] :
( ( finite_finite_assn @ A )
=> ( ( B4 != bot_bot_set_assn )
=> ( ( ord_less_eq_set_assn @ B4 @ A )
=> ( ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ B4 ) @ ( lattic2150320897289308081n_assn @ A ) )
= ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2198_Sup__fin_Osubset,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ A )
=> ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B4 ) @ ( lattic5294303975357428420t_unit @ A ) )
= ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2199_Sup__fin_Osubset,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( B4 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ B4 @ A )
=> ( ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ B4 ) @ ( lattic1508158080041050831_fin_o @ A ) )
= ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2200_Sup__fin_Osubset,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( B4 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ B4 @ A )
=> ( ( sup_sup_int @ ( lattic1091506334969745077in_int @ B4 ) @ ( lattic1091506334969745077in_int @ A ) )
= ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2201_Sup__fin_Osubset,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ A )
=> ( ( sup_su5453871518329203617_nat_o @ ( lattic7320199455484906628_nat_o @ B4 ) @ ( lattic7320199455484906628_nat_o @ A ) )
= ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2202_Sup__fin_Osubset,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B4 ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_2203_Inf__fin_Oclosed,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y2: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2204_Inf__fin_Oclosed,axiom,
! [A: set_assn] :
( ( finite_finite_assn @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ! [X3: assn,Y2: assn] : ( member_assn @ ( inf_inf_assn @ X3 @ Y2 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
=> ( member_assn @ ( lattic47131356835913163n_assn @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2205_Inf__fin_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2206_Inf__fin_Oclosed,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] : ( member_o @ ( inf_inf_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
=> ( member_o @ ( lattic4107685809792843317_fin_o @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2207_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2208_Inf__fin_Oclosed,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [X3: int,Y2: int] : ( member_int @ ( inf_inf_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
=> ( member_int @ ( lattic5235898064620869839in_int @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2209_Inf__fin_Oclosed,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
=> ( member6576561426505652726_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_2210_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ~ ( member_set_nat @ X @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2211_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ~ ( member_assn @ X @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
= ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2212_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2213_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2214_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2215_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
= ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2216_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_2217_Sup__fin_Oclosed,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y2: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2218_Sup__fin_Oclosed,axiom,
! [A: set_assn] :
( ( finite_finite_assn @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ! [X3: assn,Y2: assn] : ( member_assn @ ( sup_sup_assn @ X3 @ Y2 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
=> ( member_assn @ ( lattic2150320897289308081n_assn @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2219_Sup__fin_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2220_Sup__fin_Oclosed,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] : ( member_o @ ( sup_sup_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
=> ( member_o @ ( lattic1508158080041050831_fin_o @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2221_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2222_Sup__fin_Oclosed,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [X3: int,Y2: int] : ( member_int @ ( sup_sup_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
=> ( member_int @ ( lattic1091506334969745077in_int @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2223_Sup__fin_Oclosed,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
=> ( member6576561426505652726_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_2224_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ~ ( member_set_nat @ X @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2225_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ~ ( member_assn @ X @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
= ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2226_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2227_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2228_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2229_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
= ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2230_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_2231_Inf__fin_Ounion,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ( B4 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3014633134055518761et_nat @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2232_Inf__fin_Ounion,axiom,
! [A: set_assn,B4: set_assn] :
( ( finite_finite_assn @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ( finite_finite_assn @ B4 )
=> ( ( B4 != bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ ( sup_sup_set_assn @ A @ B4 ) )
= ( inf_inf_assn @ ( lattic47131356835913163n_assn @ A ) @ ( lattic47131356835913163n_assn @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2233_Inf__fin_Ounion,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic1263872656861969706t_unit @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2234_Inf__fin_Ounion,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ( B4 != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A @ B4 ) )
= ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic4107685809792843317_fin_o @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2235_Inf__fin_Ounion,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2236_Inf__fin_Ounion,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ( B4 != bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ ( sup_sup_set_int @ A @ B4 ) )
= ( inf_inf_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic5235898064620869839in_int @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2237_Inf__fin_Ounion,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( finite3252695134891459830_nat_o @ B4 )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( inf_in1318976480646536635_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ ( lattic956194824204696298_nat_o @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_2238_Sup__fin_Ounion,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ( B4 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2239_Sup__fin_Ounion,axiom,
! [A: set_assn,B4: set_assn] :
( ( finite_finite_assn @ A )
=> ( ( A != bot_bot_set_assn )
=> ( ( finite_finite_assn @ B4 )
=> ( ( B4 != bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ ( sup_sup_set_assn @ A @ B4 ) )
= ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ A ) @ ( lattic2150320897289308081n_assn @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2240_Sup__fin_Ounion,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2241_Sup__fin_Ounion,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ( B4 != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ ( sup_sup_set_o @ A @ B4 ) )
= ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2242_Sup__fin_Ounion,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2243_Sup__fin_Ounion,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ( B4 != bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ ( sup_sup_set_int @ A @ B4 ) )
= ( sup_sup_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2244_Sup__fin_Ounion,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( finite3252695134891459830_nat_o @ B4 )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( sup_su5453871518329203617_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_2245_refl__on__reflcl__Image,axiom,
! [B4: set_nat,A: set_Pr1261947904930325089at_nat,C5: set_nat] :
( ( refl_on_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ C5 @ B4 )
=> ( ( image_nat_nat @ ( sup_su6327502436637775413at_nat @ A @ id_nat ) @ C5 )
= ( image_nat_nat @ A @ C5 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_2246_E__closed__restr__reach__cases,axiom,
! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ ( transi712980445759334096_nat_o @ E ) )
=> ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ E @ R3 ) @ R3 )
=> ( ~ ( member6576561426505652726_nat_o @ V @ R3 )
=> ~ ( ~ ( member6576561426505652726_nat_o @ U @ R3 )
=> ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ ( transi712980445759334096_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_2247_E__closed__restr__reach__cases,axiom,
! [U: $o,V: $o,E: set_Product_prod_o_o,R3: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ E ) )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R3 ) @ R3 )
=> ( ~ ( member_o @ V @ R3 )
=> ~ ( ~ ( member_o @ U @ R3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_2248_E__closed__restr__reach__cases,axiom,
! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ ( transi6481887417741755664t_unit @ E ) )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ E @ R3 ) @ R3 )
=> ( ~ ( member_Product_unit @ V @ R3 )
=> ~ ( ~ ( member_Product_unit @ U @ R3 )
=> ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ ( transi6481887417741755664t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_2249_E__closed__restr__reach__cases,axiom,
! [U: int,V: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ E ) )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ E @ R3 ) @ R3 )
=> ( ~ ( member_int @ V @ R3 )
=> ~ ( ~ ( member_int @ U @ R3 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_2250_E__closed__restr__reach__cases,axiom,
! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R3 ) @ R3 )
=> ( ~ ( member_nat @ V @ R3 )
=> ~ ( ~ ( member_nat @ U @ R3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_2251_max__ext__compat,axiom,
! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
( ( ord_le930853871177014503t_unit @ ( relcom848357998638694028t_unit @ R3 @ S ) @ R3 )
=> ( ord_le8072594200992201959t_unit @ ( relcom7108816970757792620t_unit @ ( max_ext_Product_unit @ R3 ) @ ( sup_su3573767163274163995t_unit @ ( max_ext_Product_unit @ S ) @ ( insert6793119152790066487t_unit @ ( produc7269801343787132631t_unit @ bot_bo3957492148770167129t_unit @ bot_bo3957492148770167129t_unit ) @ bot_bo5113747506523071411t_unit ) ) ) @ ( max_ext_Product_unit @ R3 ) ) ) ).
% max_ext_compat
thf(fact_2252_max__ext__compat,axiom,
! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o] :
( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S ) @ R3 )
=> ( ord_le2885120122233735591_set_o @ ( relcom7534720040753365047_set_o @ ( max_ext_o @ R3 ) @ ( sup_su7766833992073560027_set_o @ ( max_ext_o @ S ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) @ bot_bo8143257439550059635_set_o ) ) ) @ ( max_ext_o @ R3 ) ) ) ).
% max_ext_compat
thf(fact_2253_max__ext__compat,axiom,
! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
=> ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( max_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( max_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( max_ext_nat @ R3 ) ) ) ).
% max_ext_compat
thf(fact_2254_max__ext__compat,axiom,
! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( relcomp_int_int_int @ R3 @ S ) @ R3 )
=> ( ord_le1436784781161120549et_int @ ( relcom4034950705519254697et_int @ ( max_ext_int @ R3 ) @ ( sup_su1286422330746572633et_int @ ( max_ext_int @ S ) @ ( insert7328952640415156725et_int @ ( produc6363374080413544029et_int @ bot_bot_set_int @ bot_bot_set_int ) @ bot_bo1987282549146081265et_int ) ) ) @ ( max_ext_int @ R3 ) ) ) ).
% max_ext_compat
thf(fact_2255_max__ext__compat,axiom,
! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o] :
( ( ord_le6659818918159358183_nat_o @ ( relcom6251035627269349452_nat_o @ R3 @ S ) @ R3 )
=> ( ord_le8001614231985977575_nat_o @ ( relcom7236094467932586796_nat_o @ ( max_ex5265246208886859912_nat_o @ R3 ) @ ( sup_su4863159966000489243_nat_o @ ( max_ex5265246208886859912_nat_o @ S ) @ ( insert3244108804717745463_nat_o @ ( produc5149316574908490455_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo7824918357723582233_nat_o ) @ bot_bo7535334947429593011_nat_o ) ) ) @ ( max_ex5265246208886859912_nat_o @ R3 ) ) ) ).
% max_ext_compat
thf(fact_2256_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A2: nat,B: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ id_nat )
=> ( A2 = B ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_2257_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A2: int,B: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ id_int )
=> ( A2 = B ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_2258_Sup__fin_Oremove,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ X @ A )
=> ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2259_Sup__fin_Oremove,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( member_assn @ X @ A )
=> ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
= bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ A )
= X ) )
& ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
!= bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ A )
= ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2260_Sup__fin_Oremove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2261_Sup__fin_Oremove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( lattic1508158080041050831_fin_o @ A )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2262_Sup__fin_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ A )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ A )
= ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2263_Sup__fin_Oremove,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ A )
= X ) )
& ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
!= bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ A )
= ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2264_Sup__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_2265_Sup__fin_Oinsert__remove,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2266_Sup__fin_Oinsert__remove,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
= bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
!= bot_bot_set_assn )
=> ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
= ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2267_Sup__fin_Oinsert__remove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2268_Sup__fin_Oinsert__remove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2269_Sup__fin_Oinsert__remove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
= ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2270_Sup__fin_Oinsert__remove,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= X ) )
& ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
!= bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2271_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_2272_Inf__fin_Oremove,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ X @ A )
=> ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2273_Inf__fin_Oremove,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( member_assn @ X @ A )
=> ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
= bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ A )
= X ) )
& ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
!= bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ A )
= ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2274_Inf__fin_Oremove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2275_Inf__fin_Oremove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( lattic4107685809792843317_fin_o @ A )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2276_Inf__fin_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ A )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ A )
= ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2277_Inf__fin_Oremove,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ A )
= X ) )
& ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
!= bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ A )
= ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2278_Inf__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_2279_Inf__fin_Oinsert__remove,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2280_Inf__fin_Oinsert__remove,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
= bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
!= bot_bot_set_assn )
=> ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
= ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2281_Inf__fin_Oinsert__remove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2282_Inf__fin_Oinsert__remove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2283_Inf__fin_Oinsert__remove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
= ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2284_Inf__fin_Oinsert__remove,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= X ) )
& ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
!= bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2285_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_2286_acyclic__insert,axiom,
! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
( ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) )
= ( ( transi3139058337892597825ic_nat @ R2 )
& ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_2287_acyclic__insert,axiom,
! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
( ( transi3136567867383547549ic_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ Y @ X ) @ R2 ) )
= ( ( transi3136567867383547549ic_int @ R2 )
& ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_2288_irrefl__tranclI,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat] :
( ( ( inf_in2572325071724192079at_nat @ ( converse_nat_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
= bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_2289_irrefl__tranclI,axiom,
! [R2: set_Pr958786334691620121nt_int,X: int] :
( ( ( inf_in2269163501485487111nt_int @ ( converse_int_int @ R2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
= bot_bo1796632182523588997nt_int )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_2290_inf__img__fin__dom_H,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2291_inf__img__fin__dom_H,axiom,
! [F: nat > code_integer,A: set_nat] :
( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2292_inf__img__fin__dom_H,axiom,
! [F: code_integer > product_unit,A: set_Code_integer] :
( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( image_7182656606676592809t_unit @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2293_inf__img__fin__dom_H,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( image_8730104196221521654t_unit @ F @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2294_inf__img__fin__dom_H,axiom,
! [F: code_integer > $o,A: set_Code_integer] :
( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ ( image_Code_integer_o @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2295_inf__img__fin__dom_H,axiom,
! [F: nat > $o,A: set_nat] :
( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ ( image_nat_o2 @ F @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2296_inf__img__fin__dom_H,axiom,
! [F: code_integer > nat,A: set_Code_integer] :
( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( image_951025933927791156er_nat @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2297_inf__img__fin__dom_H,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2298_inf__img__fin__dom_H,axiom,
! [F: code_integer > int,A: set_Code_integer] :
( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: int] :
( ( member_int @ X3 @ ( image_948535463418740880er_int @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2299_inf__img__fin__dom_H,axiom,
! [F: nat > int,A: set_nat] :
( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: int] :
( ( member_int @ X3 @ ( image_nat_int2 @ F @ A ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_2300_inf__img__fin__domE_H,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2301_inf__img__fin__domE_H,axiom,
! [F: nat > code_integer,A: set_nat] :
( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2302_inf__img__fin__domE_H,axiom,
! [F: code_integer > product_unit,A: set_Code_integer] :
( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( image_7182656606676592809t_unit @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2303_inf__img__fin__domE_H,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2304_inf__img__fin__domE_H,axiom,
! [F: code_integer > $o,A: set_Code_integer] :
( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: $o] :
( ( member_o @ Y2 @ ( image_Code_integer_o @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2305_inf__img__fin__domE_H,axiom,
! [F: nat > $o,A: set_nat] :
( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: $o] :
( ( member_o @ Y2 @ ( image_nat_o2 @ F @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2306_inf__img__fin__domE_H,axiom,
! [F: code_integer > nat,A: set_Code_integer] :
( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_951025933927791156er_nat @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2307_inf__img__fin__domE_H,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_nat_nat2 @ F @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2308_inf__img__fin__domE_H,axiom,
! [F: code_integer > int,A: set_Code_integer] :
( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ ( image_948535463418740880er_int @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2309_inf__img__fin__domE_H,axiom,
! [F: nat > int,A: set_nat] :
( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ ( image_nat_int2 @ F @ A ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_2310_image__eqI,axiom,
! [B: code_integer,F: code_integer > code_integer,X: code_integer,A: set_Code_integer] :
( ( B
= ( F @ X ) )
=> ( ( member_Code_integer @ X @ A )
=> ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2311_image__eqI,axiom,
! [B: code_integer,F: nat > code_integer,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2312_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2313_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_int @ B @ ( image_nat_int2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2314_image__eqI,axiom,
! [B: $o,F: nat > $o,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2315_image__eqI,axiom,
! [B: product_unit,F: nat > product_unit,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_Product_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2316_image__eqI,axiom,
! [B: nat,F: int > nat,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_nat @ B @ ( image_int_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2317_image__eqI,axiom,
! [B: int,F: int > int,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_int @ B @ ( image_int_int2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2318_image__eqI,axiom,
! [B: $o,F: int > $o,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_o @ B @ ( image_int_o2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2319_image__eqI,axiom,
! [B: product_unit,F: int > product_unit,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_Product_unit @ B @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_2320_Diff__idemp,axiom,
! [A: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) ).
% Diff_idemp
thf(fact_2321_Diff__iff,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
= ( ( member6576561426505652726_nat_o @ C @ A )
& ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_2322_Diff__iff,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
= ( ( member_int @ C @ A )
& ~ ( member_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_2323_Diff__iff,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
= ( ( member_o @ C @ A )
& ~ ( member_o @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_2324_Diff__iff,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
= ( ( member_Product_unit @ C @ A )
& ~ ( member_Product_unit @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_2325_Diff__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_2326_DiffI,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ A )
=> ( ~ ( member6576561426505652726_nat_o @ C @ B4 )
=> ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_2327_DiffI,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ A )
=> ( ~ ( member_int @ C @ B4 )
=> ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_2328_DiffI,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ A )
=> ( ~ ( member_o @ C @ B4 )
=> ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_2329_DiffI,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ A )
=> ( ~ ( member_Product_unit @ C @ B4 )
=> ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_2330_DiffI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_2331_image__is__empty,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( ( image_4470545334726330049nteger @ F @ A )
= bot_bo3990330152332043303nteger )
= ( A = bot_bo3990330152332043303nteger ) ) ).
% image_is_empty
thf(fact_2332_image__is__empty,axiom,
! [F: nat > code_integer,A: set_nat] :
( ( ( image_1215581382706833972nteger @ F @ A )
= bot_bo3990330152332043303nteger )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_2333_image__is__empty,axiom,
! [F: product_unit > product_unit,A: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% image_is_empty
thf(fact_2334_image__is__empty,axiom,
! [F: $o > product_unit,A: set_o] :
( ( ( image_o_Product_unit2 @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_2335_image__is__empty,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( ( image_8730104196221521654t_unit @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_2336_image__is__empty,axiom,
! [F: int > product_unit,A: set_int] :
( ( ( image_5846526143423701914t_unit @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_2337_image__is__empty,axiom,
! [F: product_unit > $o,A: set_Product_unit] :
( ( ( image_Product_unit_o2 @ F @ A )
= bot_bot_set_o )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% image_is_empty
thf(fact_2338_image__is__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o2 @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_2339_image__is__empty,axiom,
! [F: nat > $o,A: set_nat] :
( ( ( image_nat_o2 @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_2340_image__is__empty,axiom,
! [F: int > $o,A: set_int] :
( ( ( image_int_o2 @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_2341_empty__is__image,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( bot_bo3990330152332043303nteger
= ( image_4470545334726330049nteger @ F @ A ) )
= ( A = bot_bo3990330152332043303nteger ) ) ).
% empty_is_image
thf(fact_2342_empty__is__image,axiom,
! [F: nat > code_integer,A: set_nat] :
( ( bot_bo3990330152332043303nteger
= ( image_1215581382706833972nteger @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_2343_empty__is__image,axiom,
! [F: product_unit > product_unit,A: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( image_405062704495631173t_unit @ F @ A ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% empty_is_image
thf(fact_2344_empty__is__image,axiom,
! [F: $o > product_unit,A: set_o] :
( ( bot_bo3957492148770167129t_unit
= ( image_o_Product_unit2 @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_2345_empty__is__image,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( bot_bo3957492148770167129t_unit
= ( image_8730104196221521654t_unit @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_2346_empty__is__image,axiom,
! [F: int > product_unit,A: set_int] :
( ( bot_bo3957492148770167129t_unit
= ( image_5846526143423701914t_unit @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_2347_empty__is__image,axiom,
! [F: product_unit > $o,A: set_Product_unit] :
( ( bot_bot_set_o
= ( image_Product_unit_o2 @ F @ A ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% empty_is_image
thf(fact_2348_empty__is__image,axiom,
! [F: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o2 @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_2349_empty__is__image,axiom,
! [F: nat > $o,A: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o2 @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_2350_empty__is__image,axiom,
! [F: int > $o,A: set_int] :
( ( bot_bot_set_o
= ( image_int_o2 @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_2351_image__empty,axiom,
! [F: code_integer > code_integer] :
( ( image_4470545334726330049nteger @ F @ bot_bo3990330152332043303nteger )
= bot_bo3990330152332043303nteger ) ).
% image_empty
thf(fact_2352_image__empty,axiom,
! [F: product_unit > product_unit] :
( ( image_405062704495631173t_unit @ F @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% image_empty
thf(fact_2353_image__empty,axiom,
! [F: product_unit > $o] :
( ( image_Product_unit_o2 @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_o ) ).
% image_empty
thf(fact_2354_image__empty,axiom,
! [F: product_unit > nat] :
( ( image_875570014554754200it_nat @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_2355_image__empty,axiom,
! [F: product_unit > int] :
( ( image_873079544045703924it_int @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_int ) ).
% image_empty
thf(fact_2356_image__empty,axiom,
! [F: $o > product_unit] :
( ( image_o_Product_unit2 @ F @ bot_bot_set_o )
= bot_bo3957492148770167129t_unit ) ).
% image_empty
thf(fact_2357_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o2 @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_2358_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat2 @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_2359_image__empty,axiom,
! [F: $o > int] :
( ( image_o_int2 @ F @ bot_bot_set_o )
= bot_bot_set_int ) ).
% image_empty
thf(fact_2360_image__empty,axiom,
! [F: nat > code_integer] :
( ( image_1215581382706833972nteger @ F @ bot_bot_set_nat )
= bot_bo3990330152332043303nteger ) ).
% image_empty
thf(fact_2361_insert__image,axiom,
! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
( ( member_Code_integer @ X @ A )
=> ( ( insert_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ A ) )
= ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% insert_image
thf(fact_2362_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > code_integer] :
( ( member_nat @ X @ A )
=> ( ( insert_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ A ) )
= ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% insert_image
thf(fact_2363_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat2 @ F @ A ) )
= ( image_nat_nat2 @ F @ A ) ) ) ).
% insert_image
thf(fact_2364_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( insert_int @ ( F @ X ) @ ( image_nat_int2 @ F @ A ) )
= ( image_nat_int2 @ F @ A ) ) ) ).
% insert_image
thf(fact_2365_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( ( insert_o @ ( F @ X ) @ ( image_nat_o2 @ F @ A ) )
= ( image_nat_o2 @ F @ A ) ) ) ).
% insert_image
thf(fact_2366_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > product_unit] :
( ( member_nat @ X @ A )
=> ( ( insert_Product_unit @ ( F @ X ) @ ( image_8730104196221521654t_unit @ F @ A ) )
= ( image_8730104196221521654t_unit @ F @ A ) ) ) ).
% insert_image
thf(fact_2367_insert__image,axiom,
! [X: int,A: set_int,F: int > nat] :
( ( member_int @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_int_nat2 @ F @ A ) )
= ( image_int_nat2 @ F @ A ) ) ) ).
% insert_image
thf(fact_2368_insert__image,axiom,
! [X: int,A: set_int,F: int > int] :
( ( member_int @ X @ A )
=> ( ( insert_int @ ( F @ X ) @ ( image_int_int2 @ F @ A ) )
= ( image_int_int2 @ F @ A ) ) ) ).
% insert_image
thf(fact_2369_insert__image,axiom,
! [X: int,A: set_int,F: int > $o] :
( ( member_int @ X @ A )
=> ( ( insert_o @ ( F @ X ) @ ( image_int_o2 @ F @ A ) )
= ( image_int_o2 @ F @ A ) ) ) ).
% insert_image
thf(fact_2370_insert__image,axiom,
! [X: int,A: set_int,F: int > product_unit] :
( ( member_int @ X @ A )
=> ( ( insert_Product_unit @ ( F @ X ) @ ( image_5846526143423701914t_unit @ F @ A ) )
= ( image_5846526143423701914t_unit @ F @ A ) ) ) ).
% insert_image
thf(fact_2371_image__insert,axiom,
! [F: code_integer > code_integer,A2: code_integer,B4: set_Code_integer] :
( ( image_4470545334726330049nteger @ F @ ( insert_Code_integer @ A2 @ B4 ) )
= ( insert_Code_integer @ ( F @ A2 ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2372_image__insert,axiom,
! [F: nat > code_integer,A2: nat,B4: set_nat] :
( ( image_1215581382706833972nteger @ F @ ( insert_nat @ A2 @ B4 ) )
= ( insert_Code_integer @ ( F @ A2 ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2373_image__insert,axiom,
! [F: nat > nat,A2: nat,B4: set_nat] :
( ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2374_image__insert,axiom,
! [F: nat > int,A2: nat,B4: set_nat] :
( ( image_nat_int2 @ F @ ( insert_nat @ A2 @ B4 ) )
= ( insert_int @ ( F @ A2 ) @ ( image_nat_int2 @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2375_image__insert,axiom,
! [F: nat > $o,A2: nat,B4: set_nat] :
( ( image_nat_o2 @ F @ ( insert_nat @ A2 @ B4 ) )
= ( insert_o @ ( F @ A2 ) @ ( image_nat_o2 @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2376_image__insert,axiom,
! [F: nat > product_unit,A2: nat,B4: set_nat] :
( ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ B4 ) )
= ( insert_Product_unit @ ( F @ A2 ) @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2377_image__insert,axiom,
! [F: int > nat,A2: int,B4: set_int] :
( ( image_int_nat2 @ F @ ( insert_int @ A2 @ B4 ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_int_nat2 @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2378_image__insert,axiom,
! [F: int > int,A2: int,B4: set_int] :
( ( image_int_int2 @ F @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ ( F @ A2 ) @ ( image_int_int2 @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2379_image__insert,axiom,
! [F: int > $o,A2: int,B4: set_int] :
( ( image_int_o2 @ F @ ( insert_int @ A2 @ B4 ) )
= ( insert_o @ ( F @ A2 ) @ ( image_int_o2 @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2380_image__insert,axiom,
! [F: int > product_unit,A2: int,B4: set_int] :
( ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ B4 ) )
= ( insert_Product_unit @ ( F @ A2 ) @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2381_img__snd,axiom,
! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
=> ( member_nat @ B @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) ) ) ).
% img_snd
thf(fact_2382_img__snd,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
=> ( member6260224972018164377et_nat @ B @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) ) ) ).
% img_snd
thf(fact_2383_img__snd,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
=> ( member1996754912294343701et_nat @ B @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) ) ) ).
% img_snd
thf(fact_2384_img__snd,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
=> ( member_set_nat @ B @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) ) ) ).
% img_snd
thf(fact_2385_img__snd,axiom,
! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
=> ( member_int @ B @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) ) ) ).
% img_snd
thf(fact_2386_Diff__cancel,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ A )
= bot_bo3957492148770167129t_unit ) ).
% Diff_cancel
thf(fact_2387_Diff__cancel,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ A )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_2388_Diff__cancel,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ A @ A )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_2389_Diff__cancel,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( minus_1801376950450012436_nat_o @ A @ A )
= bot_bo7824918357723582233_nat_o ) ).
% Diff_cancel
thf(fact_2390_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_2391_empty__Diff,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ bot_bo3957492148770167129t_unit @ A )
= bot_bo3957492148770167129t_unit ) ).
% empty_Diff
thf(fact_2392_empty__Diff,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_2393_empty__Diff,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_2394_empty__Diff,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( minus_1801376950450012436_nat_o @ bot_bo7824918357723582233_nat_o @ A )
= bot_bo7824918357723582233_nat_o ) ).
% empty_Diff
thf(fact_2395_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_2396_Diff__empty,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ bot_bo3957492148770167129t_unit )
= A ) ).
% Diff_empty
thf(fact_2397_Diff__empty,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ bot_bot_set_o )
= A ) ).
% Diff_empty
thf(fact_2398_Diff__empty,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ A @ bot_bot_set_int )
= A ) ).
% Diff_empty
thf(fact_2399_Diff__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( minus_1801376950450012436_nat_o @ A @ bot_bo7824918357723582233_nat_o )
= A ) ).
% Diff_empty
thf(fact_2400_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_2401_insert__Diff1,axiom,
! [X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ B4 )
=> ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
= ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_2402_insert__Diff1,axiom,
! [X: int,B4: set_int,A: set_int] :
( ( member_int @ X @ B4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( minus_minus_set_int @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_2403_insert__Diff1,axiom,
! [X: $o,B4: set_o,A: set_o] :
( ( member_o @ X @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
= ( minus_minus_set_o @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_2404_insert__Diff1,axiom,
! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X @ B4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
= ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_2405_insert__Diff1,axiom,
! [X: nat,B4: set_nat,A: set_nat] :
( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_2406_Diff__insert0,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
= ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_2407_Diff__insert0,axiom,
! [X: int,A: set_int,B4: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( minus_minus_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( minus_minus_set_int @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_2408_Diff__insert0,axiom,
! [X: $o,A: set_o,B4: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X @ B4 ) )
= ( minus_minus_set_o @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_2409_Diff__insert0,axiom,
! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A )
=> ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
= ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_2410_Diff__insert0,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_2411_Un__Diff__cancel2,axiom,
! [B4: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B4 @ A ) @ A )
= ( sup_sup_set_nat @ B4 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_2412_Un__Diff__cancel,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
= ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_2413_converse__iff,axiom,
! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ ( conver7272282955447398104et_nat @ R2 ) )
= ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2414_converse__iff,axiom,
! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ ( conver4003969491454187092et_nat @ R2 ) )
= ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2415_converse__iff,axiom,
! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ ( conver3577387295554834384et_nat @ R2 ) )
= ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2416_converse__iff,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( conver2236330673913229642t_unit @ R2 ) )
= ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2417_converse__iff,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( converse_nat_nat @ R2 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2418_converse__iff,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( converse_int_int @ R2 ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2419_converse__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( conver1831365297310299342_nat_o @ R2 ) )
= ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2420_converse__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( conver5006497266169685586_nat_o @ R2 ) )
= ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ R2 ) ) ).
% converse_iff
thf(fact_2421_Diff__eq__empty__iff,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_2422_Diff__eq__empty__iff,axiom,
! [A: set_o,B4: set_o] :
( ( ( minus_minus_set_o @ A @ B4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_2423_Diff__eq__empty__iff,axiom,
! [A: set_int,B4: set_int] :
( ( ( minus_minus_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_2424_Diff__eq__empty__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
= ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_2425_Diff__eq__empty__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_2426_insert__Diff__single,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( insert_Product_unit @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( insert_Product_unit @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_2427_insert__Diff__single,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( insert_o @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_2428_insert__Diff__single,axiom,
! [A2: int,A: set_int] :
( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( insert_int @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_2429_insert__Diff__single,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
= ( insert5175938949040314269_nat_o @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_2430_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_2431_finite__Diff__insert,axiom,
! [A: set_int,A2: int,B4: set_int] :
( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) ) )
= ( finite_finite_int @ ( minus_minus_set_int @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_2432_finite__Diff__insert,axiom,
! [A: set_o,A2: $o,B4: set_o] :
( ( finite_finite_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) ) )
= ( finite_finite_o @ ( minus_minus_set_o @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_2433_finite__Diff__insert,axiom,
! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) ) )
= ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_2434_finite__Diff__insert,axiom,
! [A: set_Code_integer,A2: code_integer,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ B4 ) ) )
= ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_2435_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_2436_Diff__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
= bot_bo3957492148770167129t_unit ) ).
% Diff_disjoint
thf(fact_2437_Diff__disjoint,axiom,
! [A: set_o,B4: set_o] :
( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B4 @ A ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_2438_Diff__disjoint,axiom,
! [A: set_int,B4: set_int] :
( ( inf_inf_set_int @ A @ ( minus_minus_set_int @ B4 @ A ) )
= bot_bot_set_int ) ).
% Diff_disjoint
thf(fact_2439_Diff__disjoint,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ A ) )
= bot_bo7824918357723582233_nat_o ) ).
% Diff_disjoint
thf(fact_2440_Diff__disjoint,axiom,
! [A: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_2441_image__diff__subset,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( minus_2355218937544613996nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ).
% image_diff_subset
thf(fact_2442_image__diff__subset,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% image_diff_subset
thf(fact_2443_image__diff__subset,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_nat] : ( ord_le7084787975880047091nteger @ ( minus_2355218937544613996nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% image_diff_subset
thf(fact_2444_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% image_diff_subset
thf(fact_2445_rev__image__eqI,axiom,
! [X: code_integer,A: set_Code_integer,B: code_integer,F: code_integer > code_integer] :
( ( member_Code_integer @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2446_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: code_integer,F: nat > code_integer] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2447_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2448_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2449_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2450_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: product_unit,F: nat > product_unit] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_Product_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2451_rev__image__eqI,axiom,
! [X: int,A: set_int,B: nat,F: int > nat] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_int_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2452_rev__image__eqI,axiom,
! [X: int,A: set_int,B: int,F: int > int] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2453_rev__image__eqI,axiom,
! [X: int,A: set_int,B: $o,F: int > $o] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_int_o2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2454_rev__image__eqI,axiom,
! [X: int,A: set_int,B: product_unit,F: int > product_unit] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_Product_unit @ B @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_2455_ball__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_2456_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_2457_ball__imageD,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,P: code_integer > $o] :
( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X6: code_integer] :
( ( member_Code_integer @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_2458_ball__imageD,axiom,
! [F: nat > code_integer,A: set_nat,P: code_integer > $o] :
( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_2459_image__cong,axiom,
! [M: set_Code_integer,N: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
( ( M = N )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4470545334726330049nteger @ F @ M )
= ( image_4470545334726330049nteger @ G @ N ) ) ) ) ).
% image_cong
thf(fact_2460_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_set_nat @ F @ M )
= ( image_nat_set_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_2461_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat2 @ F @ M )
= ( image_nat_nat2 @ G @ N ) ) ) ) ).
% image_cong
thf(fact_2462_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > code_integer,G: nat > code_integer] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1215581382706833972nteger @ F @ M )
= ( image_1215581382706833972nteger @ G @ N ) ) ) ) ).
% image_cong
thf(fact_2463_bex__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_2464_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat2 @ F @ A ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_2465_bex__imageD,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,P: code_integer > $o] :
( ? [X6: code_integer] :
( ( member_Code_integer @ X6 @ ( image_4470545334726330049nteger @ F @ A ) )
& ( P @ X6 ) )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_2466_bex__imageD,axiom,
! [F: nat > code_integer,A: set_nat,P: code_integer > $o] :
( ? [X6: code_integer] :
( ( member_Code_integer @ X6 @ ( image_1215581382706833972nteger @ F @ A ) )
& ( P @ X6 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_2467_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_2468_image__iff,axiom,
! [Z: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
( ( member_Code_integer @ Z @ ( image_4470545334726330049nteger @ F @ A ) )
= ( ? [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_2469_image__iff,axiom,
! [Z: code_integer,F: nat > code_integer,A: set_nat] :
( ( member_Code_integer @ Z @ ( image_1215581382706833972nteger @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_2470_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat2 @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_2471_imageI,axiom,
! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
( ( member_Code_integer @ X @ A )
=> ( member_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% imageI
thf(fact_2472_imageI,axiom,
! [X: nat,A: set_nat,F: nat > code_integer] :
( ( member_nat @ X @ A )
=> ( member_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% imageI
thf(fact_2473_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_2474_imageI,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_nat_int2 @ F @ A ) ) ) ).
% imageI
thf(fact_2475_imageI,axiom,
! [X: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_nat_o2 @ F @ A ) ) ) ).
% imageI
thf(fact_2476_imageI,axiom,
! [X: nat,A: set_nat,F: nat > product_unit] :
( ( member_nat @ X @ A )
=> ( member_Product_unit @ ( F @ X ) @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ).
% imageI
thf(fact_2477_imageI,axiom,
! [X: int,A: set_int,F: int > nat] :
( ( member_int @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_int_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_2478_imageI,axiom,
! [X: int,A: set_int,F: int > int] :
( ( member_int @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_int_int2 @ F @ A ) ) ) ).
% imageI
thf(fact_2479_imageI,axiom,
! [X: int,A: set_int,F: int > $o] :
( ( member_int @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_int_o2 @ F @ A ) ) ) ).
% imageI
thf(fact_2480_imageI,axiom,
! [X: int,A: set_int,F: int > product_unit] :
( ( member_int @ X @ A )
=> ( member_Product_unit @ ( F @ X ) @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ).
% imageI
thf(fact_2481_DiffD2,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
=> ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ).
% DiffD2
thf(fact_2482_DiffD2,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
=> ~ ( member_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_2483_DiffD2,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
=> ~ ( member_o @ C @ B4 ) ) ).
% DiffD2
thf(fact_2484_DiffD2,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
=> ~ ( member_Product_unit @ C @ B4 ) ) ).
% DiffD2
thf(fact_2485_DiffD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_2486_DiffD1,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
=> ( member6576561426505652726_nat_o @ C @ A ) ) ).
% DiffD1
thf(fact_2487_DiffD1,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
=> ( member_int @ C @ A ) ) ).
% DiffD1
thf(fact_2488_DiffD1,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
=> ( member_o @ C @ A ) ) ).
% DiffD1
thf(fact_2489_DiffD1,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
=> ( member_Product_unit @ C @ A ) ) ).
% DiffD1
thf(fact_2490_DiffD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_2491_DiffE,axiom,
! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
=> ~ ( ( member6576561426505652726_nat_o @ C @ A )
=> ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% DiffE
thf(fact_2492_DiffE,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
=> ~ ( ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_2493_DiffE,axiom,
! [C: $o,A: set_o,B4: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
=> ~ ( ( member_o @ C @ A )
=> ( member_o @ C @ B4 ) ) ) ).
% DiffE
thf(fact_2494_DiffE,axiom,
! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
=> ~ ( ( member_Product_unit @ C @ A )
=> ( member_Product_unit @ C @ B4 ) ) ) ).
% DiffE
thf(fact_2495_DiffE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_2496_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B )
= ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2497_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2498_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2499_diff__eq__diff__eq,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ( minus_minus_rat @ A2 @ B )
= ( minus_minus_rat @ C @ D2 ) )
=> ( ( A2 = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_2500_diff__eq__diff__eq,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A2 = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_2501_in__image__insert__iff,axiom,
! [B4: set_set_Product_unit,X: product_unit,A: set_Product_unit] :
( ! [C6: set_Product_unit] :
( ( member5877623283571906838t_unit @ C6 @ B4 )
=> ~ ( member_Product_unit @ X @ C6 ) )
=> ( ( member5877623283571906838t_unit @ A @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ X ) @ B4 ) )
= ( ( member_Product_unit @ X @ A )
& ( member5877623283571906838t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2502_in__image__insert__iff,axiom,
! [B4: set_set_o,X: $o,A: set_o] :
( ! [C6: set_o] :
( ( member_set_o @ C6 @ B4 )
=> ~ ( member_o @ X @ C6 ) )
=> ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X ) @ B4 ) )
= ( ( member_o @ X @ A )
& ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2503_in__image__insert__iff,axiom,
! [B4: set_set_int,X: int,A: set_int] :
( ! [C6: set_int] :
( ( member_set_int @ C6 @ B4 )
=> ~ ( member_int @ X @ C6 ) )
=> ( ( member_set_int @ A @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B4 ) )
= ( ( member_int @ X @ A )
& ( member_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2504_in__image__insert__iff,axiom,
! [B4: set_se1353816476878259085_nat_o,X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ! [C6: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ C6 @ B4 )
=> ~ ( member6576561426505652726_nat_o @ X @ C6 ) )
=> ( ( member1452989533377600726_nat_o @ A @ ( image_8925030434418351941_nat_o @ ( insert5175938949040314269_nat_o @ X ) @ B4 ) )
= ( ( member6576561426505652726_nat_o @ X @ A )
& ( member1452989533377600726_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2505_in__image__insert__iff,axiom,
! [B4: set_set_nat,X: nat,A: set_nat] :
( ! [C6: set_nat] :
( ( member_set_nat @ C6 @ B4 )
=> ~ ( member_nat @ X @ C6 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B4 ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2506_converse_Ocases,axiom,
! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A1 @ A22 ) @ ( conver7272282955447398104et_nat @ R2 ) )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2507_converse_Ocases,axiom,
! [A1: produc3658429121746597890et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A1 @ A22 ) @ ( conver4003969491454187092et_nat @ R2 ) )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2508_converse_Ocases,axiom,
! [A1: produc3925858234332021118et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A1 @ A22 ) @ ( conver3577387295554834384et_nat @ R2 ) )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2509_converse_Ocases,axiom,
! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( conver2236330673913229642t_unit @ R2 ) )
=> ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2510_converse_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2511_converse_Ocases,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( converse_int_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2512_converse_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( conver1831365297310299342_nat_o @ R2 ) )
=> ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2513_converse_Ocases,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( conver5006497266169685586_nat_o @ R2 ) )
=> ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_2514_converse_Osimps,axiom,
! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A1 @ A22 ) @ ( conver7272282955447398104et_nat @ R2 ) )
= ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2515_converse_Osimps,axiom,
! [A1: produc3658429121746597890et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A1 @ A22 ) @ ( conver4003969491454187092et_nat @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2516_converse_Osimps,axiom,
! [A1: produc3925858234332021118et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A1 @ A22 ) @ ( conver3577387295554834384et_nat @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2517_converse_Osimps,axiom,
! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( conver2236330673913229642t_unit @ R2 ) )
= ( ? [A6: set_nat,B6: heap_e7401611519738050253t_unit] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2518_converse_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R2 ) )
= ( ? [A6: nat,B6: nat] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2519_converse_Osimps,axiom,
! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( converse_int_int @ R2 ) )
= ( ? [A6: int,B6: int] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2520_converse_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( conver1831365297310299342_nat_o @ R2 ) )
= ( ? [A6: produc3658429121746597890et_nat,B6: produc3658429121746597890et_nat > $o] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2521_converse_Osimps,axiom,
! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( conver5006497266169685586_nat_o @ R2 ) )
= ( ? [A6: produc3925858234332021118et_nat,B6: produc3658429121746597890et_nat > $o] :
( ( A1 = B6 )
& ( A22 = A6 )
& ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A6 @ B6 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_2522_converseD,axiom,
! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ ( conver7272282955447398104et_nat @ R2 ) )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2523_converseD,axiom,
! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ ( conver4003969491454187092et_nat @ R2 ) )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2524_converseD,axiom,
! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ ( conver3577387295554834384et_nat @ R2 ) )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2525_converseD,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( conver2236330673913229642t_unit @ R2 ) )
=> ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2526_converseD,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( converse_nat_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2527_converseD,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( converse_int_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2528_converseD,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( conver1831365297310299342_nat_o @ R2 ) )
=> ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2529_converseD,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( conver5006497266169685586_nat_o @ R2 ) )
=> ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ R2 ) ) ).
% converseD
thf(fact_2530_converseE,axiom,
! [Yx: produc7348995623955488188t_unit,R2: set_Pr3948176798113811640et_nat] :
( ( member727419437372278867t_unit @ Yx @ ( conver7272282955447398104et_nat @ R2 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( ( Yx
= ( produc2471974422597015918t_unit @ Y2 @ X3 ) )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2531_converseE,axiom,
! [Yx: produc6907119039986491712_nat_o,R2: set_Pr3286484037609594932et_nat] :
( ( member4978015717948814295_nat_o @ Yx @ ( conver4003969491454187092et_nat @ R2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( Yx
= ( produc2829238748666232050_nat_o @ Y2 @ X3 ) )
=> ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2532_converseE,axiom,
! [Yx: produc4962441903510821572_nat_o,R2: set_Pr8536935166611901872et_nat] :
( ( member8354763867512313179_nat_o @ Yx @ ( conver3577387295554834384et_nat @ R2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( ( Yx
= ( produc3674526432113299062_nat_o @ Y2 @ X3 ) )
=> ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2533_converseE,axiom,
! [Yx: produc3658429121746597890et_nat,R2: set_Pr4071100604818519410t_unit] :
( ( member6260224972018164377et_nat @ Yx @ ( conver2236330673913229642t_unit @ R2 ) )
=> ~ ! [X3: set_nat,Y2: heap_e7401611519738050253t_unit] :
( ( Yx
= ( produc7507926704131184380et_nat @ Y2 @ X3 ) )
=> ~ ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2534_converseE,axiom,
! [Yx: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Yx @ ( converse_nat_nat @ R2 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( Yx
= ( product_Pair_nat_nat @ Y2 @ X3 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2535_converseE,axiom,
! [Yx: product_prod_int_int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ Yx @ ( converse_int_int @ R2 ) )
=> ~ ! [X3: int,Y2: int] :
( ( Yx
= ( product_Pair_int_int @ Y2 @ X3 ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2536_converseE,axiom,
! [Yx: produc3925858234332021118et_nat,R2: set_Pr4208978187861016310_nat_o] :
( ( member1996754912294343701et_nat @ Yx @ ( conver1831365297310299342_nat_o @ R2 ) )
=> ~ ! [X3: produc3658429121746597890et_nat,Y2: produc3658429121746597890et_nat > $o] :
( ( Yx
= ( produc5001842942810119800et_nat @ Y2 @ X3 ) )
=> ~ ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2537_converseE,axiom,
! [Yx: produc2732055786443039994et_nat,R2: set_Pr7705059561495379578_nat_o] :
( ( member6124377750444531601et_nat @ Yx @ ( conver5006497266169685586_nat_o @ R2 ) )
=> ~ ! [X3: produc3925858234332021118et_nat,Y2: produc3658429121746597890et_nat > $o] :
( ( Yx
= ( produc2245416461498447860et_nat @ Y2 @ X3 ) )
=> ~ ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ X3 @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_2538_converseI,axiom,
! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr4071100604818519410t_unit] :
( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ R2 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ ( conver2236330673913229642t_unit @ R2 ) ) ) ).
% converseI
thf(fact_2539_converseI,axiom,
! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr4208978187861016310_nat_o] :
( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ R2 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ ( conver1831365297310299342_nat_o @ R2 ) ) ) ).
% converseI
thf(fact_2540_converseI,axiom,
! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr7705059561495379578_nat_o] :
( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ R2 )
=> ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ ( conver5006497266169685586_nat_o @ R2 ) ) ) ).
% converseI
thf(fact_2541_converseI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
=> ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ ( conver7272282955447398104et_nat @ R2 ) ) ) ).
% converseI
thf(fact_2542_converseI,axiom,
! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( converse_nat_nat @ R2 ) ) ) ).
% converseI
thf(fact_2543_converseI,axiom,
! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( converse_int_int @ R2 ) ) ) ).
% converseI
thf(fact_2544_converseI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
=> ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ ( conver4003969491454187092et_nat @ R2 ) ) ) ).
% converseI
thf(fact_2545_converseI,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
=> ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ ( conver3577387295554834384et_nat @ R2 ) ) ) ).
% converseI
thf(fact_2546_diff__eq__diff__less__eq,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ( minus_minus_rat @ A2 @ B )
= ( minus_minus_rat @ C @ D2 ) )
=> ( ( ord_less_eq_rat @ A2 @ B )
= ( ord_less_eq_rat @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_2547_diff__eq__diff__less__eq,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A2 @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_2548_diff__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_2549_diff__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_2550_diff__left__mono,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_2551_diff__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_2552_diff__mono,axiom,
! [A2: rat,B: rat,D2: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ D2 @ C )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_2553_diff__mono,axiom,
! [A2: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_2554_subset__image__iff,axiom,
! [B4: set_Code_integer,F: code_integer > code_integer,A: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A ) )
= ( ? [AA: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ AA @ A )
& ( B4
= ( image_4470545334726330049nteger @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_2555_subset__image__iff,axiom,
! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_2556_subset__image__iff,axiom,
! [B4: set_Code_integer,F: nat > code_integer,A: set_nat] :
( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_1215581382706833972nteger @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_2557_subset__image__iff,axiom,
! [B4: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat2 @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_2558_image__subset__iff,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_2559_image__subset__iff,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
= ( ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( member_Code_integer @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_2560_image__subset__iff,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_Code_integer @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_2561_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_2562_subset__imageE,axiom,
! [B4: set_Code_integer,F: code_integer > code_integer,A: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A ) )
=> ~ ! [C6: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ C6 @ A )
=> ( B4
!= ( image_4470545334726330049nteger @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_2563_subset__imageE,axiom,
! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A )
=> ( B4
!= ( image_nat_set_nat @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_2564_subset__imageE,axiom,
! [B4: set_Code_integer,F: nat > code_integer,A: set_nat] :
( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A )
=> ( B4
!= ( image_1215581382706833972nteger @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_2565_subset__imageE,axiom,
! [B4: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat2 @ F @ A ) )
=> ~ ! [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A )
=> ( B4
!= ( image_nat_nat2 @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_2566_image__subsetI,axiom,
! [A: set_Code_integer,F: code_integer > code_integer,B4: set_Code_integer] :
( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( member_Code_integer @ ( F @ X3 ) @ B4 ) )
=> ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2567_image__subsetI,axiom,
! [A: set_nat,F: nat > code_integer,B4: set_Code_integer] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_Code_integer @ ( F @ X3 ) @ B4 ) )
=> ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2568_image__subsetI,axiom,
! [A: set_nat,F: nat > int,B4: set_int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int2 @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2569_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B4: set_o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2570_image__subsetI,axiom,
! [A: set_nat,F: nat > product_unit,B4: set_Product_unit] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_Product_unit @ ( F @ X3 ) @ B4 ) )
=> ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2571_image__subsetI,axiom,
! [A: set_int,F: int > int,B4: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_int_int2 @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2572_image__subsetI,axiom,
! [A: set_int,F: int > $o,B4: set_o] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_int_o2 @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2573_image__subsetI,axiom,
! [A: set_int,F: int > product_unit,B4: set_Product_unit] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( member_Product_unit @ ( F @ X3 ) @ B4 ) )
=> ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2574_image__subsetI,axiom,
! [A: set_o,F: $o > int,B4: set_int] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_int @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_o_int2 @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2575_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B4: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_2576_image__mono,axiom,
! [A: set_Code_integer,B4: set_Code_integer,F: code_integer > code_integer] :
( ( ord_le7084787975880047091nteger @ A @ B4 )
=> ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% image_mono
thf(fact_2577_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_2578_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > code_integer] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% image_mono
thf(fact_2579_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% image_mono
thf(fact_2580_image__Un,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
( ( image_4470545334726330049nteger @ F @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( sup_su848401254843788991nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% image_Un
thf(fact_2581_image__Un,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% image_Un
thf(fact_2582_image__Un,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_nat] :
( ( image_1215581382706833972nteger @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_su848401254843788991nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% image_Un
thf(fact_2583_image__Un,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% image_Un
thf(fact_2584_double__diff,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C5 )
=> ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C5 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_2585_Diff__subset,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ A ) ).
% Diff_subset
thf(fact_2586_Diff__mono,axiom,
! [A: set_nat,C5: set_nat,D3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ C5 )
=> ( ( ord_less_eq_set_nat @ D3 @ B4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ C5 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_2587_insert__Diff__if,axiom,
! [X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( ( member6576561426505652726_nat_o @ X @ B4 )
=> ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
= ( minus_1801376950450012436_nat_o @ A @ B4 ) ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
=> ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
= ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2588_insert__Diff__if,axiom,
! [X: int,B4: set_int,A: set_int] :
( ( ( member_int @ X @ B4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( minus_minus_set_int @ A @ B4 ) ) )
& ( ~ ( member_int @ X @ B4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( insert_int @ X @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2589_insert__Diff__if,axiom,
! [X: $o,B4: set_o,A: set_o] :
( ( ( member_o @ X @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
= ( minus_minus_set_o @ A @ B4 ) ) )
& ( ~ ( member_o @ X @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
= ( insert_o @ X @ ( minus_minus_set_o @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2590_insert__Diff__if,axiom,
! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
( ( ( member_Product_unit @ X @ B4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
= ( minus_6452836326544984404t_unit @ A @ B4 ) ) )
& ( ~ ( member_Product_unit @ X @ B4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
= ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2591_insert__Diff__if,axiom,
! [X: nat,B4: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_2592_Diff__Int__distrib2,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_2593_Diff__Int__distrib,axiom,
! [C5: set_nat,A: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ C5 @ ( minus_minus_set_nat @ A @ B4 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C5 @ A ) @ ( inf_inf_set_nat @ C5 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_2594_Diff__Diff__Int,axiom,
! [A: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_2595_Diff__Int2,axiom,
! [A: set_nat,C5: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ ( inf_inf_set_nat @ B4 @ C5 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_2596_Int__Diff,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C5 ) ) ) ).
% Int_Diff
thf(fact_2597_set__diff__diff__left,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
= ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% set_diff_diff_left
thf(fact_2598_Un__Diff,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C5 ) @ ( minus_minus_set_nat @ B4 @ C5 ) ) ) ).
% Un_Diff
thf(fact_2599_vimage__Diff,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( vimage_nat_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) )
= ( minus_minus_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% vimage_Diff
thf(fact_2600_rtrancl__converseI,axiom,
! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_2601_rtrancl__converseI,axiom,
! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ ( converse_int_int @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_2602_rtrancl__converseD,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R2 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_2603_rtrancl__converseD,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ ( converse_int_int @ R2 ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_2604_trancl__converseI,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R2 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_2605_trancl__converseI,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( converse_int_int @ ( transi6261509568448316235cl_int @ R2 ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( converse_int_int @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_2606_trancl__converseD,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R2 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_2607_trancl__converseD,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( converse_int_int @ R2 ) ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( converse_int_int @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_2608_diff__shunt__var,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ X @ Y )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_2609_diff__shunt__var,axiom,
! [X: set_o,Y: set_o] :
( ( ( minus_minus_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_2610_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_2611_diff__shunt__var,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( ( minus_1801376950450012436_nat_o @ X @ Y )
= bot_bo7824918357723582233_nat_o )
= ( ord_le2965882846123202637_nat_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_2612_diff__shunt__var,axiom,
! [X: assn,Y: assn] :
( ( ( minus_minus_assn @ X @ Y )
= bot_bot_assn )
= ( ord_less_eq_assn @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_2613_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_2614_in__snd__imageE,axiom,
! [Y: nat,S: set_Pr1261947904930325089at_nat] :
( ( member_nat @ Y @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) )
=> ~ ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_2615_in__snd__imageE,axiom,
! [Y: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
( ( member6260224972018164377et_nat @ Y @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_2616_in__snd__imageE,axiom,
! [Y: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
( ( member1996754912294343701et_nat @ Y @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_2617_in__snd__imageE,axiom,
! [Y: set_nat,S: set_Pr3948176798113811640et_nat] :
( ( member_set_nat @ Y @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit] :
~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_2618_in__snd__imageE,axiom,
! [Y: int,S: set_Pr958786334691620121nt_int] :
( ( member_int @ Y @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) )
=> ~ ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_2619_image__Int__subset,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( inf_in1364745209274528805nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_2620_image__Int__subset,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_2621_image__Int__subset,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_nat] : ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_in1364745209274528805nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_2622_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_2623_subset__minus__empty,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( minus_6452836326544984404t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit ) ) ).
% subset_minus_empty
thf(fact_2624_subset__minus__empty,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( minus_minus_set_o @ A @ B4 )
= bot_bot_set_o ) ) ).
% subset_minus_empty
thf(fact_2625_subset__minus__empty,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( minus_minus_set_int @ A @ B4 )
= bot_bot_set_int ) ) ).
% subset_minus_empty
thf(fact_2626_subset__minus__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ( minus_1801376950450012436_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o ) ) ).
% subset_minus_empty
thf(fact_2627_subset__minus__empty,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( minus_minus_set_nat @ A @ B4 )
= bot_bot_set_nat ) ) ).
% subset_minus_empty
thf(fact_2628_image__subset__iff__subset__vimage,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
= ( ord_le7084787975880047091nteger @ A @ ( vimage1318793397690208119nteger @ F @ B4 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_2629_image__subset__iff__subset__vimage,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_set_nat @ F @ B4 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_2630_image__subset__iff__subset__vimage,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
= ( ord_less_eq_set_nat @ A @ ( vimage6104394758324456170nteger @ F @ B4 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_2631_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_2632_image__vimage__subset,axiom,
! [F: nat > set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( vimage_nat_set_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_2633_image__vimage__subset,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ ( vimage1318793397690208119nteger @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_2634_image__vimage__subset,axiom,
! [F: nat > code_integer,A: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ ( vimage6104394758324456170nteger @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_2635_image__vimage__subset,axiom,
! [F: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_2636_Diff__insert,axiom,
! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% Diff_insert
thf(fact_2637_Diff__insert,axiom,
! [A: set_o,A2: $o,B4: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B4 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_2638_Diff__insert,axiom,
! [A: set_int,A2: int,B4: set_int] :
( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B4 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% Diff_insert
thf(fact_2639_Diff__insert,axiom,
! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% Diff_insert
thf(fact_2640_Diff__insert,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_2641_insert__Diff,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ( insert_Product_unit @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= A ) ) ).
% insert_Diff
thf(fact_2642_insert__Diff,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_2643_insert__Diff,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= A ) ) ).
% insert_Diff
thf(fact_2644_insert__Diff,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_2645_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_2646_Diff__insert2,axiom,
! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_2647_Diff__insert2,axiom,
! [A: set_o,A2: $o,B4: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_2648_Diff__insert2,axiom,
! [A: set_int,A2: int,B4: set_int] :
( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_2649_Diff__insert2,axiom,
! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
= ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_2650_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_2651_insert__minus__eq,axiom,
! [X: product_unit,Y: product_unit,A: set_Product_unit] :
( ( X != Y )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) )
= ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2652_insert__minus__eq,axiom,
! [X: $o,Y: $o,A: set_o] :
( ( X != Y )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ Y @ bot_bot_set_o ) )
= ( insert_o @ X @ ( minus_minus_set_o @ A @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2653_insert__minus__eq,axiom,
! [X: int,Y: int,A: set_int] :
( ( X != Y )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ Y @ bot_bot_set_int ) )
= ( insert_int @ X @ ( minus_minus_set_int @ A @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2654_insert__minus__eq,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( X != Y )
=> ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) )
= ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2655_insert__minus__eq,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( X != Y )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2656_Diff__insert__absorb,axiom,
! [X: product_unit,A: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_2657_Diff__insert__absorb,axiom,
! [X: $o,A: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ X @ bot_bot_set_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_2658_Diff__insert__absorb,axiom,
! [X: int,A: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ X @ bot_bot_set_int ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_2659_Diff__insert__absorb,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_2660_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_2661_set__minus__singleton__eq,axiom,
! [X: product_unit,X4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ X4 )
=> ( ( minus_6452836326544984404t_unit @ X4 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X4 ) ) ).
% set_minus_singleton_eq
thf(fact_2662_set__minus__singleton__eq,axiom,
! [X: $o,X4: set_o] :
( ~ ( member_o @ X @ X4 )
=> ( ( minus_minus_set_o @ X4 @ ( insert_o @ X @ bot_bot_set_o ) )
= X4 ) ) ).
% set_minus_singleton_eq
thf(fact_2663_set__minus__singleton__eq,axiom,
! [X: int,X4: set_int] :
( ~ ( member_int @ X @ X4 )
=> ( ( minus_minus_set_int @ X4 @ ( insert_int @ X @ bot_bot_set_int ) )
= X4 ) ) ).
% set_minus_singleton_eq
thf(fact_2664_set__minus__singleton__eq,axiom,
! [X: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
( ~ ( member6576561426505652726_nat_o @ X @ X4 )
=> ( ( minus_1801376950450012436_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X4 ) ) ).
% set_minus_singleton_eq
thf(fact_2665_set__minus__singleton__eq,axiom,
! [X: nat,X4: set_nat] :
( ~ ( member_nat @ X @ X4 )
=> ( ( minus_minus_set_nat @ X4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X4 ) ) ).
% set_minus_singleton_eq
thf(fact_2666_subset__Diff__insert,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ ( insert5175938949040314269_nat_o @ X @ C5 ) ) )
= ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ C5 ) )
& ~ ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_2667_subset__Diff__insert,axiom,
! [A: set_int,B4: set_int,X: int,C5: set_int] :
( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ ( insert_int @ X @ C5 ) ) )
= ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ C5 ) )
& ~ ( member_int @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_2668_subset__Diff__insert,axiom,
! [A: set_o,B4: set_o,X: $o,C5: set_o] :
( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B4 @ ( insert_o @ X @ C5 ) ) )
= ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B4 @ C5 ) )
& ~ ( member_o @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_2669_subset__Diff__insert,axiom,
! [A: set_Product_unit,B4: set_Product_unit,X: product_unit,C5: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ ( insert_Product_unit @ X @ C5 ) ) )
= ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ C5 ) )
& ~ ( member_Product_unit @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_2670_subset__Diff__insert,axiom,
! [A: set_nat,B4: set_nat,X: nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C5 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C5 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_2671_Diff__triv,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( minus_6452836326544984404t_unit @ A @ B4 )
= A ) ) ).
% Diff_triv
thf(fact_2672_Diff__triv,axiom,
! [A: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ B4 )
= A ) ) ).
% Diff_triv
thf(fact_2673_Diff__triv,axiom,
! [A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
=> ( ( minus_minus_set_int @ A @ B4 )
= A ) ) ).
% Diff_triv
thf(fact_2674_Diff__triv,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
=> ( ( minus_1801376950450012436_nat_o @ A @ B4 )
= A ) ) ).
% Diff_triv
thf(fact_2675_Diff__triv,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B4 )
= A ) ) ).
% Diff_triv
thf(fact_2676_Int__Diff__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
= bot_bo3957492148770167129t_unit ) ).
% Int_Diff_disjoint
thf(fact_2677_Int__Diff__disjoint,axiom,
! [A: set_o,B4: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B4 ) @ ( minus_minus_set_o @ A @ B4 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_2678_Int__Diff__disjoint,axiom,
! [A: set_int,B4: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ A @ B4 ) @ ( minus_minus_set_int @ A @ B4 ) )
= bot_bot_set_int ) ).
% Int_Diff_disjoint
thf(fact_2679_Int__Diff__disjoint,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
= bot_bo7824918357723582233_nat_o ) ).
% Int_Diff_disjoint
thf(fact_2680_Int__Diff__disjoint,axiom,
! [A: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ B4 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_2681_disjoint__alt__simp1,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ A @ B4 )
= A )
= ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit ) ) ).
% disjoint_alt_simp1
thf(fact_2682_disjoint__alt__simp1,axiom,
! [A: set_o,B4: set_o] :
( ( ( minus_minus_set_o @ A @ B4 )
= A )
= ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o ) ) ).
% disjoint_alt_simp1
thf(fact_2683_disjoint__alt__simp1,axiom,
! [A: set_int,B4: set_int] :
( ( ( minus_minus_set_int @ A @ B4 )
= A )
= ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int ) ) ).
% disjoint_alt_simp1
thf(fact_2684_disjoint__alt__simp1,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
= A )
= ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o ) ) ).
% disjoint_alt_simp1
thf(fact_2685_disjoint__alt__simp1,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A @ B4 )
= A )
= ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat ) ) ).
% disjoint_alt_simp1
thf(fact_2686_disjoint__alt__simp2,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ A @ B4 )
!= A )
= ( ( inf_in4660618365625256667t_unit @ A @ B4 )
!= bot_bo3957492148770167129t_unit ) ) ).
% disjoint_alt_simp2
thf(fact_2687_disjoint__alt__simp2,axiom,
! [A: set_o,B4: set_o] :
( ( ( minus_minus_set_o @ A @ B4 )
!= A )
= ( ( inf_inf_set_o @ A @ B4 )
!= bot_bot_set_o ) ) ).
% disjoint_alt_simp2
thf(fact_2688_disjoint__alt__simp2,axiom,
! [A: set_int,B4: set_int] :
( ( ( minus_minus_set_int @ A @ B4 )
!= A )
= ( ( inf_inf_set_int @ A @ B4 )
!= bot_bot_set_int ) ) ).
% disjoint_alt_simp2
thf(fact_2689_disjoint__alt__simp2,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
!= A )
= ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
!= bot_bo7824918357723582233_nat_o ) ) ).
% disjoint_alt_simp2
thf(fact_2690_disjoint__alt__simp2,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A @ B4 )
!= A )
= ( ( inf_inf_set_nat @ A @ B4 )
!= bot_bot_set_nat ) ) ).
% disjoint_alt_simp2
thf(fact_2691_Diff__subset__conv,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_2692_Diff__partition,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
= B4 ) ) ).
% Diff_partition
thf(fact_2693_Un__Diff__Int,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ A @ B4 ) )
= A ) ).
% Un_Diff_Int
thf(fact_2694_Int__Diff__Un,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ B4 ) )
= A ) ).
% Int_Diff_Un
thf(fact_2695_Diff__Int,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ C5 ) ) ) ).
% Diff_Int
thf(fact_2696_Diff__Un,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ C5 ) ) ) ).
% Diff_Un
thf(fact_2697_Range__snd,axiom,
( range_9132311542094983838et_nat
= ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat ) ) ).
% Range_snd
thf(fact_2698_Range__snd,axiom,
( range_int_int
= ( image_5042161079198086560nt_int @ product_snd_int_int ) ) ).
% Range_snd
thf(fact_2699_snd__eq__Range,axiom,
! [R3: set_Pr3948176798113811640et_nat] :
( ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ R3 )
= ( range_9132311542094983838et_nat @ R3 ) ) ).
% snd_eq_Range
thf(fact_2700_snd__eq__Range,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( image_5042161079198086560nt_int @ product_snd_int_int @ R3 )
= ( range_int_int @ R3 ) ) ).
% snd_eq_Range
thf(fact_2701_the__elem__image__unique,axiom,
! [A: set_Code_integer,F: code_integer > code_integer,X: code_integer] :
( ( A != bot_bo3990330152332043303nteger )
=> ( ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_el4656377885804969768nteger @ ( image_4470545334726330049nteger @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_2702_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > set_nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_2703_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat2 @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_2704_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > code_integer,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_el4656377885804969768nteger @ ( image_1215581382706833972nteger @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_2705_cyclicE,axiom,
! [G: set_Pr1261947904930325089at_nat] :
( ~ ( transi3139058337892597825ic_nat @ G )
=> ~ ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ G ) ) ) ).
% cyclicE
thf(fact_2706_cyclicE,axiom,
! [G: set_Pr958786334691620121nt_int] :
( ~ ( transi3136567867383547549ic_int @ G )
=> ~ ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ G ) ) ) ).
% cyclicE
thf(fact_2707_acyclic__def,axiom,
( transi3139058337892597825ic_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ ( transi6264000038957366511cl_nat @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_2708_acyclic__def,axiom,
( transi3136567867383547549ic_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [X2: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ ( transi6261509568448316235cl_int @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_2709_acyclicI,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% acyclicI
thf(fact_2710_acyclicI,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
=> ( transi3136567867383547549ic_int @ R2 ) ) ).
% acyclicI
thf(fact_2711_snd__image__mp,axiom,
! [A: set_Pr3286484037609594932et_nat,B4: set_Pr3948176798113811640et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( ord_le8090042543978450200et_nat @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ A ) @ B4 )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
=> ( member6260224972018164377et_nat @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_2712_snd__image__mp,axiom,
! [A: set_Pr8536935166611901872et_nat,B4: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( ord_le5966269811547037844et_nat @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ A ) @ B4 )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
=> ( member1996754912294343701et_nat @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_2713_snd__image__mp,axiom,
! [A: set_Pr3948176798113811640et_nat,B4: set_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) @ B4 )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
=> ( member_set_nat @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_2714_snd__image__mp,axiom,
! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) @ B4 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
=> ( member_int @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_2715_snd__image__mp,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A ) @ B4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
=> ( member_nat @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_2716_snd__in__Field,axiom,
! [R3: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ R3 ) @ ( field_int @ R3 ) ) ).
% snd_in_Field
thf(fact_2717_snd__in__Field,axiom,
! [R3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ R3 ) @ ( field_nat @ R3 ) ) ).
% snd_in_Field
thf(fact_2718_Image__subset__snd__image,axiom,
! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit] : ( ord_le6893508408891458716et_nat @ ( image_3942343239045813184et_nat @ A @ B4 ) @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) ) ).
% Image_subset_snd_image
thf(fact_2719_Image__subset__snd__image,axiom,
! [A: set_Pr958786334691620121nt_int,B4: set_int] : ( ord_less_eq_set_int @ ( image_int_int @ A @ B4 ) @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) ) ).
% Image_subset_snd_image
thf(fact_2720_infinite__remove,axiom,
! [S: set_Code_integer,A2: code_integer] :
( ~ ( finite6017078050557962740nteger @ S )
=> ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ).
% infinite_remove
thf(fact_2721_infinite__remove,axiom,
! [S: set_Product_unit,A2: product_unit] :
( ~ ( finite4290736615968046902t_unit @ S )
=> ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% infinite_remove
thf(fact_2722_infinite__remove,axiom,
! [S: set_o,A2: $o] :
( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% infinite_remove
thf(fact_2723_infinite__remove,axiom,
! [S: set_int,A2: int] :
( ~ ( finite_finite_int @ S )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% infinite_remove
thf(fact_2724_infinite__remove,axiom,
! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ~ ( finite3252695134891459830_nat_o @ S )
=> ~ ( finite3252695134891459830_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% infinite_remove
thf(fact_2725_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_2726_infinite__coinduct,axiom,
! [X4: set_Code_integer > $o,A: set_Code_integer] :
( ( X4 @ A )
=> ( ! [A9: set_Code_integer] :
( ( X4 @ A9 )
=> ? [X6: code_integer] :
( ( member_Code_integer @ X6 @ A9 )
& ( ( X4 @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) )
| ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
=> ~ ( finite6017078050557962740nteger @ A ) ) ) ).
% infinite_coinduct
thf(fact_2727_infinite__coinduct,axiom,
! [X4: set_Product_unit > $o,A: set_Product_unit] :
( ( X4 @ A )
=> ( ! [A9: set_Product_unit] :
( ( X4 @ A9 )
=> ? [X6: product_unit] :
( ( member_Product_unit @ X6 @ A9 )
& ( ( X4 @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) )
| ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
=> ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% infinite_coinduct
thf(fact_2728_infinite__coinduct,axiom,
! [X4: set_o > $o,A: set_o] :
( ( X4 @ A )
=> ( ! [A9: set_o] :
( ( X4 @ A9 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A9 )
& ( ( X4 @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) )
| ~ ( finite_finite_o @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) ) ) )
=> ~ ( finite_finite_o @ A ) ) ) ).
% infinite_coinduct
thf(fact_2729_infinite__coinduct,axiom,
! [X4: set_int > $o,A: set_int] :
( ( X4 @ A )
=> ( ! [A9: set_int] :
( ( X4 @ A9 )
=> ? [X6: int] :
( ( member_int @ X6 @ A9 )
& ( ( X4 @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) )
| ~ ( finite_finite_int @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) ) ) )
=> ~ ( finite_finite_int @ A ) ) ) ).
% infinite_coinduct
thf(fact_2730_infinite__coinduct,axiom,
! [X4: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
( ( X4 @ A )
=> ( ! [A9: set_Pr4532377907799695533_nat_o] :
( ( X4 @ A9 )
=> ? [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ A9 )
& ( ( X4 @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) )
| ~ ( finite3252695134891459830_nat_o @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
=> ~ ( finite3252695134891459830_nat_o @ A ) ) ) ).
% infinite_coinduct
thf(fact_2731_infinite__coinduct,axiom,
! [X4: set_nat > $o,A: set_nat] :
( ( X4 @ A )
=> ( ! [A9: set_nat] :
( ( X4 @ A9 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A9 )
& ( ( X4 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_2732_finite__empty__induct,axiom,
! [A: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ A )
=> ( ( P @ A )
=> ( ! [A4: code_integer,A9: set_Code_integer] :
( ( finite6017078050557962740nteger @ A9 )
=> ( ( member_Code_integer @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ A4 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
=> ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).
% finite_empty_induct
thf(fact_2733_finite__empty__induct,axiom,
! [A: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( P @ A )
=> ( ! [A4: product_unit,A9: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A9 )
=> ( ( member_Product_unit @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
=> ( P @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% finite_empty_induct
thf(fact_2734_finite__empty__induct,axiom,
! [A: set_o,P: set_o > $o] :
( ( finite_finite_o @ A )
=> ( ( P @ A )
=> ( ! [A4: $o,A9: set_o] :
( ( finite_finite_o @ A9 )
=> ( ( member_o @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
=> ( P @ bot_bot_set_o ) ) ) ) ).
% finite_empty_induct
thf(fact_2735_finite__empty__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ A )
=> ( ! [A4: int,A9: set_int] :
( ( finite_finite_int @ A9 )
=> ( ( member_int @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
=> ( P @ bot_bot_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_2736_finite__empty__induct,axiom,
! [A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( P @ A )
=> ( ! [A4: produc3658429121746597890et_nat > $o,A9: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A9 )
=> ( ( member6576561426505652726_nat_o @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ A4 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
=> ( P @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% finite_empty_induct
thf(fact_2737_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ( member_nat @ A4 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_2738_subset__insert__iff,axiom,
! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
= ( ( ( member_Product_unit @ X @ A )
=> ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_2739_subset__insert__iff,axiom,
! [A: set_o,X: $o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) )
= ( ( ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
& ( ~ ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_2740_subset__insert__iff,axiom,
! [A: set_int,X: int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( ( ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
& ( ~ ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_2741_subset__insert__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
= ( ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_2742_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_2743_Diff__single__insert,axiom,
! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 )
=> ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_2744_Diff__single__insert,axiom,
! [A: set_o,X: $o,B4: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_2745_Diff__single__insert,axiom,
! [A: set_int,X: int,B4: set_int] :
( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 )
=> ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_2746_Diff__single__insert,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 )
=> ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_2747_Diff__single__insert,axiom,
! [A: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_2748_Field__rel__restrict,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( field_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( field_nat @ R3 ) @ A ) ) ).
% Field_rel_restrict
thf(fact_2749_Domain__rel__restrict,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( domain_nat_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( domain_nat_nat @ R3 ) @ A ) ) ).
% Domain_rel_restrict
thf(fact_2750_Range__rel__restrict,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( range_nat_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( range_nat_nat @ R3 ) @ A ) ) ).
% Range_rel_restrict
thf(fact_2751_inf__img__fin__domE,axiom,
! [F: nat > code_integer,A: set_nat] :
( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( finite_finite_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2752_inf__img__fin__domE,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: code_integer] :
( ( member_Code_integer @ Y2 @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2753_inf__img__fin__domE,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2754_inf__img__fin__domE,axiom,
! [F: code_integer > product_unit,A: set_Code_integer] :
( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: product_unit] :
( ( member_Product_unit @ Y2 @ ( image_7182656606676592809t_unit @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2755_inf__img__fin__domE,axiom,
! [F: nat > $o,A: set_nat] :
( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: $o] :
( ( member_o @ Y2 @ ( image_nat_o2 @ F @ A ) )
=> ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2756_inf__img__fin__domE,axiom,
! [F: code_integer > $o,A: set_Code_integer] :
( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: $o] :
( ( member_o @ Y2 @ ( image_Code_integer_o @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2757_inf__img__fin__domE,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_nat_nat2 @ F @ A ) )
=> ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2758_inf__img__fin__domE,axiom,
! [F: code_integer > nat,A: set_Code_integer] :
( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: nat] :
( ( member_nat @ Y2 @ ( image_951025933927791156er_nat @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2759_inf__img__fin__domE,axiom,
! [F: nat > int,A: set_nat] :
( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ ( image_nat_int2 @ F @ A ) )
=> ( finite_finite_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2760_inf__img__fin__domE,axiom,
! [F: code_integer > int,A: set_Code_integer] :
( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ~ ! [Y2: int] :
( ( member_int @ Y2 @ ( image_948535463418740880er_int @ F @ A ) )
=> ( finite6017078050557962740nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_2761_inf__img__fin__dom,axiom,
! [F: nat > code_integer,A: set_nat] :
( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
& ~ ( finite_finite_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2762_inf__img__fin__dom,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2763_inf__img__fin__dom,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( image_8730104196221521654t_unit @ F @ A ) )
& ~ ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2764_inf__img__fin__dom,axiom,
! [F: code_integer > product_unit,A: set_Code_integer] :
( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( image_7182656606676592809t_unit @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2765_inf__img__fin__dom,axiom,
! [F: nat > $o,A: set_nat] :
( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ ( image_nat_o2 @ F @ A ) )
& ~ ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2766_inf__img__fin__dom,axiom,
! [F: code_integer > $o,A: set_Code_integer] :
( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ ( image_Code_integer_o @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2767_inf__img__fin__dom,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
& ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2768_inf__img__fin__dom,axiom,
! [F: code_integer > nat,A: set_Code_integer] :
( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( image_951025933927791156er_nat @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2769_inf__img__fin__dom,axiom,
! [F: nat > int,A: set_nat] :
( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
=> ( ~ ( finite_finite_nat @ A )
=> ? [X3: int] :
( ( member_int @ X3 @ ( image_nat_int2 @ F @ A ) )
& ~ ( finite_finite_nat @ ( vimage_nat_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2770_inf__img__fin__dom,axiom,
! [F: code_integer > int,A: set_Code_integer] :
( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
=> ( ~ ( finite6017078050557962740nteger @ A )
=> ? [X3: int] :
( ( member_int @ X3 @ ( image_948535463418740880er_int @ F @ A ) )
& ~ ( finite6017078050557962740nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_2771_acyclic__insert__cyclic,axiom,
! [G: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( transi3139058337892597825ic_nat @ G )
=> ( ~ ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ G ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ G ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_2772_acyclic__insert__cyclic,axiom,
! [G: set_Pr958786334691620121nt_int,X: int,Y: int] :
( ( transi3136567867383547549ic_int @ G )
=> ( ~ ( transi3136567867383547549ic_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ Y ) @ G ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ G ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_2773_Inf__fin_Ohom__commute,axiom,
! [H2: set_nat > set_nat,N: set_set_nat] :
( ! [X3: set_nat,Y2: set_nat] :
( ( H2 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= ( inf_inf_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite1152437895449049373et_nat @ N )
=> ( ( N != bot_bot_set_set_nat )
=> ( ( H2 @ ( lattic3014633134055518761et_nat @ N ) )
= ( lattic3014633134055518761et_nat @ ( image_7916887816326733075et_nat @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2774_Inf__fin_Ohom__commute,axiom,
! [H2: assn > assn,N: set_assn] :
( ! [X3: assn,Y2: assn] :
( ( H2 @ ( inf_inf_assn @ X3 @ Y2 ) )
= ( inf_inf_assn @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_assn @ N )
=> ( ( N != bot_bot_set_assn )
=> ( ( H2 @ ( lattic47131356835913163n_assn @ N ) )
= ( lattic47131356835913163n_assn @ ( image_assn_assn @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2775_Inf__fin_Ohom__commute,axiom,
! [H2: product_unit > product_unit,N: set_Product_unit] :
( ! [X3: product_unit,Y2: product_unit] :
( ( H2 @ ( inf_inf_Product_unit @ X3 @ Y2 ) )
= ( inf_inf_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite4290736615968046902t_unit @ N )
=> ( ( N != bot_bo3957492148770167129t_unit )
=> ( ( H2 @ ( lattic1263872656861969706t_unit @ N ) )
= ( lattic1263872656861969706t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2776_Inf__fin_Ohom__commute,axiom,
! [H2: $o > $o,N: set_o] :
( ! [X3: $o,Y2: $o] :
( ( H2 @ ( inf_inf_o @ X3 @ Y2 ) )
= ( inf_inf_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_o @ N )
=> ( ( N != bot_bot_set_o )
=> ( ( H2 @ ( lattic4107685809792843317_fin_o @ N ) )
= ( lattic4107685809792843317_fin_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2777_Inf__fin_Ohom__commute,axiom,
! [H2: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H2 @ ( inf_inf_nat @ X3 @ Y2 ) )
= ( inf_inf_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H2 @ ( lattic5238388535129920115in_nat @ N ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2778_Inf__fin_Ohom__commute,axiom,
! [H2: int > int,N: set_int] :
( ! [X3: int,Y2: int] :
( ( H2 @ ( inf_inf_int @ X3 @ Y2 ) )
= ( inf_inf_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_int @ N )
=> ( ( N != bot_bot_set_int )
=> ( ( H2 @ ( lattic5235898064620869839in_int @ N ) )
= ( lattic5235898064620869839in_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2779_Inf__fin_Ohom__commute,axiom,
! [H2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( H2 @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) )
= ( inf_in1318976480646536635_nat_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite3252695134891459830_nat_o @ N )
=> ( ( N != bot_bo7824918357723582233_nat_o )
=> ( ( H2 @ ( lattic956194824204696298_nat_o @ N ) )
= ( lattic956194824204696298_nat_o @ ( image_3399141053782081349_nat_o @ H2 @ N ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2780_Sup__fin_Ohom__commute,axiom,
! [H2: set_nat > set_nat,N: set_set_nat] :
( ! [X3: set_nat,Y2: set_nat] :
( ( H2 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite1152437895449049373et_nat @ N )
=> ( ( N != bot_bot_set_set_nat )
=> ( ( H2 @ ( lattic3835124923745554447et_nat @ N ) )
= ( lattic3835124923745554447et_nat @ ( image_7916887816326733075et_nat @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2781_Sup__fin_Ohom__commute,axiom,
! [H2: assn > assn,N: set_assn] :
( ! [X3: assn,Y2: assn] :
( ( H2 @ ( sup_sup_assn @ X3 @ Y2 ) )
= ( sup_sup_assn @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_assn @ N )
=> ( ( N != bot_bot_set_assn )
=> ( ( H2 @ ( lattic2150320897289308081n_assn @ N ) )
= ( lattic2150320897289308081n_assn @ ( image_assn_assn @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2782_Sup__fin_Ohom__commute,axiom,
! [H2: product_unit > product_unit,N: set_Product_unit] :
( ! [X3: product_unit,Y2: product_unit] :
( ( H2 @ ( sup_sup_Product_unit @ X3 @ Y2 ) )
= ( sup_sup_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite4290736615968046902t_unit @ N )
=> ( ( N != bot_bo3957492148770167129t_unit )
=> ( ( H2 @ ( lattic5294303975357428420t_unit @ N ) )
= ( lattic5294303975357428420t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2783_Sup__fin_Ohom__commute,axiom,
! [H2: $o > $o,N: set_o] :
( ! [X3: $o,Y2: $o] :
( ( H2 @ ( sup_sup_o @ X3 @ Y2 ) )
= ( sup_sup_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_o @ N )
=> ( ( N != bot_bot_set_o )
=> ( ( H2 @ ( lattic1508158080041050831_fin_o @ N ) )
= ( lattic1508158080041050831_fin_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2784_Sup__fin_Ohom__commute,axiom,
! [H2: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H2 @ ( sup_sup_nat @ X3 @ Y2 ) )
= ( sup_sup_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H2 @ ( lattic1093996805478795353in_nat @ N ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2785_Sup__fin_Ohom__commute,axiom,
! [H2: int > int,N: set_int] :
( ! [X3: int,Y2: int] :
( ( H2 @ ( sup_sup_int @ X3 @ Y2 ) )
= ( sup_sup_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_int @ N )
=> ( ( N != bot_bot_set_int )
=> ( ( H2 @ ( lattic1091506334969745077in_int @ N ) )
= ( lattic1091506334969745077in_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2786_Sup__fin_Ohom__commute,axiom,
! [H2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( H2 @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) )
= ( sup_su5453871518329203617_nat_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite3252695134891459830_nat_o @ N )
=> ( ( N != bot_bo7824918357723582233_nat_o )
=> ( ( H2 @ ( lattic7320199455484906628_nat_o @ N ) )
= ( lattic7320199455484906628_nat_o @ ( image_3399141053782081349_nat_o @ H2 @ N ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2787_finite__remove__induct,axiom,
! [B4: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ B4 )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [A9: set_Code_integer] :
( ( finite6017078050557962740nteger @ A9 )
=> ( ( A9 != bot_bo3990330152332043303nteger )
=> ( ( ord_le7084787975880047091nteger @ A9 @ B4 )
=> ( ! [X6: code_integer] :
( ( member_Code_integer @ X6 @ A9 )
=> ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_2788_finite__remove__induct,axiom,
! [B4: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ B4 )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [A9: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A9 )
=> ( ( A9 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A9 @ B4 )
=> ( ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ A9 )
=> ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_2789_finite__remove__induct,axiom,
! [B4: set_o,P: set_o > $o] :
( ( finite_finite_o @ B4 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A9: set_o] :
( ( finite_finite_o @ A9 )
=> ( ( A9 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A9 @ B4 )
=> ( ! [X6: $o] :
( ( member_o @ X6 @ A9 )
=> ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_2790_finite__remove__induct,axiom,
! [B4: set_int,P: set_int > $o] :
( ( finite_finite_int @ B4 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A9: set_int] :
( ( finite_finite_int @ A9 )
=> ( ( A9 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A9 @ B4 )
=> ( ! [X6: int] :
( ( member_int @ X6 @ A9 )
=> ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_2791_finite__remove__induct,axiom,
! [B4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ B4 )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [A9: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A9 )
=> ( ( A9 != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le2965882846123202637_nat_o @ A9 @ B4 )
=> ( ! [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ A9 )
=> ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_2792_finite__remove__induct,axiom,
! [B4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ( A9 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A9 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A9 )
=> ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_2793_remove__induct,axiom,
! [P: set_Code_integer > $o,B4: set_Code_integer] :
( ( P @ bot_bo3990330152332043303nteger )
=> ( ( ~ ( finite6017078050557962740nteger @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_Code_integer] :
( ( finite6017078050557962740nteger @ A9 )
=> ( ( A9 != bot_bo3990330152332043303nteger )
=> ( ( ord_le7084787975880047091nteger @ A9 @ B4 )
=> ( ! [X6: code_integer] :
( ( member_Code_integer @ X6 @ A9 )
=> ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_2794_remove__induct,axiom,
! [P: set_Product_unit > $o,B4: set_Product_unit] :
( ( P @ bot_bo3957492148770167129t_unit )
=> ( ( ~ ( finite4290736615968046902t_unit @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A9 )
=> ( ( A9 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A9 @ B4 )
=> ( ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ A9 )
=> ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_2795_remove__induct,axiom,
! [P: set_o > $o,B4: set_o] :
( ( P @ bot_bot_set_o )
=> ( ( ~ ( finite_finite_o @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_o] :
( ( finite_finite_o @ A9 )
=> ( ( A9 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A9 @ B4 )
=> ( ! [X6: $o] :
( ( member_o @ X6 @ A9 )
=> ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_2796_remove__induct,axiom,
! [P: set_int > $o,B4: set_int] :
( ( P @ bot_bot_set_int )
=> ( ( ~ ( finite_finite_int @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_int] :
( ( finite_finite_int @ A9 )
=> ( ( A9 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A9 @ B4 )
=> ( ! [X6: int] :
( ( member_int @ X6 @ A9 )
=> ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_2797_remove__induct,axiom,
! [P: set_Pr4532377907799695533_nat_o > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ( ~ ( finite3252695134891459830_nat_o @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A9 )
=> ( ( A9 != bot_bo7824918357723582233_nat_o )
=> ( ( ord_le2965882846123202637_nat_o @ A9 @ B4 )
=> ( ! [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ A9 )
=> ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_2798_remove__induct,axiom,
! [P: set_nat > $o,B4: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ( A9 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A9 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A9 )
=> ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_2799_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
= ( ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2800_Linear__order__in__diff__Id,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
= ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2801_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
= ( ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2802_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
= ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2803_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
= ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2804_max__ext_Omax__extI,axiom,
! [X4: set_Code_integer,Y6: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
( ( finite6017078050557962740nteger @ X4 )
=> ( ( finite6017078050557962740nteger @ Y6 )
=> ( ( Y6 != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ X4 )
=> ? [Xa2: code_integer] :
( ( member_Code_integer @ Xa2 @ Y6 )
& ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ X4 @ Y6 ) @ ( max_ext_Code_integer @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2805_max__ext_Omax__extI,axiom,
! [X4: set_Product_unit,Y6: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( finite4290736615968046902t_unit @ Y6 )
=> ( ( Y6 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ? [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ Y6 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ X4 @ Y6 ) @ ( max_ext_Product_unit @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2806_max__ext_Omax__extI,axiom,
! [X4: set_o,Y6: set_o,R3: set_Product_prod_o_o] :
( ( finite_finite_o @ X4 )
=> ( ( finite_finite_o @ Y6 )
=> ( ( Y6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X4 )
=> ? [Xa2: $o] :
( ( member_o @ Xa2 @ Y6 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X4 @ Y6 ) @ ( max_ext_o @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2807_max__ext_Omax__extI,axiom,
! [X4: set_nat,Y6: set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( Y6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ Y6 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X4 @ Y6 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2808_max__ext_Omax__extI,axiom,
! [X4: set_int,Y6: set_int,R3: set_Pr958786334691620121nt_int] :
( ( finite_finite_int @ X4 )
=> ( ( finite_finite_int @ Y6 )
=> ( ( Y6 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ Y6 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ X4 @ Y6 ) @ ( max_ext_int @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2809_max__ext_Omax__extI,axiom,
! [X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
( ( finite3252695134891459830_nat_o @ X4 )
=> ( ( finite3252695134891459830_nat_o @ Y6 )
=> ( ( Y6 != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X4 )
=> ? [Xa2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Xa2 @ Y6 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ X4 @ Y6 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2810_max__ext_Osimps,axiom,
! [A1: set_Code_integer,A22: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A22 ) @ ( max_ext_Code_integer @ R3 ) )
= ( ( finite6017078050557962740nteger @ A1 )
& ( finite6017078050557962740nteger @ A22 )
& ( A22 != bot_bo3990330152332043303nteger )
& ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A1 )
=> ? [Y3: code_integer] :
( ( member_Code_integer @ Y3 @ A22 )
& ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2811_max__ext_Osimps,axiom,
! [A1: set_Product_unit,A22: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ A1 @ A22 ) @ ( max_ext_Product_unit @ R3 ) )
= ( ( finite4290736615968046902t_unit @ A1 )
& ( finite4290736615968046902t_unit @ A22 )
& ( A22 != bot_bo3957492148770167129t_unit )
& ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A1 )
=> ? [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ A22 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2812_max__ext_Osimps,axiom,
! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
= ( ( finite_finite_o @ A1 )
& ( finite_finite_o @ A22 )
& ( A22 != bot_bot_set_o )
& ! [X2: $o] :
( ( member_o @ X2 @ A1 )
=> ? [Y3: $o] :
( ( member_o @ Y3 @ A22 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2813_max__ext_Osimps,axiom,
! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
= ( ( finite_finite_nat @ A1 )
& ( finite_finite_nat @ A22 )
& ( A22 != bot_bot_set_nat )
& ! [X2: nat] :
( ( member_nat @ X2 @ A1 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2814_max__ext_Osimps,axiom,
! [A1: set_int,A22: set_int,R3: set_Pr958786334691620121nt_int] :
( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A22 ) @ ( max_ext_int @ R3 ) )
= ( ( finite_finite_int @ A1 )
& ( finite_finite_int @ A22 )
& ( A22 != bot_bot_set_int )
& ! [X2: int] :
( ( member_int @ X2 @ A1 )
=> ? [Y3: int] :
( ( member_int @ Y3 @ A22 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2815_max__ext_Osimps,axiom,
! [A1: set_Pr4532377907799695533_nat_o,A22: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
( ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ A1 @ A22 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) )
= ( ( finite3252695134891459830_nat_o @ A1 )
& ( finite3252695134891459830_nat_o @ A22 )
& ( A22 != bot_bo7824918357723582233_nat_o )
& ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A1 )
=> ? [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ A22 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2816_max__ext_Ocases,axiom,
! [A1: set_Code_integer,A22: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A22 ) @ ( max_ext_Code_integer @ R3 ) )
=> ~ ( ( finite6017078050557962740nteger @ A1 )
=> ( ( finite6017078050557962740nteger @ A22 )
=> ( ( A22 != bot_bo3990330152332043303nteger )
=> ~ ! [X6: code_integer] :
( ( member_Code_integer @ X6 @ A1 )
=> ? [Xa3: code_integer] :
( ( member_Code_integer @ Xa3 @ A22 )
& ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2817_max__ext_Ocases,axiom,
! [A1: set_Product_unit,A22: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ A1 @ A22 ) @ ( max_ext_Product_unit @ R3 ) )
=> ~ ( ( finite4290736615968046902t_unit @ A1 )
=> ( ( finite4290736615968046902t_unit @ A22 )
=> ( ( A22 != bot_bo3957492148770167129t_unit )
=> ~ ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ A1 )
=> ? [Xa3: product_unit] :
( ( member_Product_unit @ Xa3 @ A22 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2818_max__ext_Ocases,axiom,
! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
=> ~ ( ( finite_finite_o @ A1 )
=> ( ( finite_finite_o @ A22 )
=> ( ( A22 != bot_bot_set_o )
=> ~ ! [X6: $o] :
( ( member_o @ X6 @ A1 )
=> ? [Xa3: $o] :
( ( member_o @ Xa3 @ A22 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2819_max__ext_Ocases,axiom,
! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
=> ~ ( ( finite_finite_nat @ A1 )
=> ( ( finite_finite_nat @ A22 )
=> ( ( A22 != bot_bot_set_nat )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ A1 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ A22 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2820_max__ext_Ocases,axiom,
! [A1: set_int,A22: set_int,R3: set_Pr958786334691620121nt_int] :
( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A22 ) @ ( max_ext_int @ R3 ) )
=> ~ ( ( finite_finite_int @ A1 )
=> ( ( finite_finite_int @ A22 )
=> ( ( A22 != bot_bot_set_int )
=> ~ ! [X6: int] :
( ( member_int @ X6 @ A1 )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ A22 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2821_max__ext_Ocases,axiom,
! [A1: set_Pr4532377907799695533_nat_o,A22: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
( ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ A1 @ A22 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) )
=> ~ ( ( finite3252695134891459830_nat_o @ A1 )
=> ( ( finite3252695134891459830_nat_o @ A22 )
=> ( ( A22 != bot_bo7824918357723582233_nat_o )
=> ~ ! [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ A1 )
=> ? [Xa3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Xa3 @ A22 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2822_inf__period_I2_J,axiom,
! [P: rat > $o,D3: rat,Q2: rat > $o] :
( ! [X3: rat,K: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
=> ( ! [X3: rat,K: rat] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
=> ! [X6: rat,K2: rat] :
( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) )
| ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2823_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q2: int > $o] :
( ! [X3: int,K: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ! [X6: int,K2: int] :
( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) )
| ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_2824_inf__period_I1_J,axiom,
! [P: rat > $o,D3: rat,Q2: rat > $o] :
( ! [X3: rat,K: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
=> ( ! [X3: rat,K: rat] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
=> ! [X6: rat,K2: rat] :
( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) )
& ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2825_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q2: int > $o] :
( ! [X3: int,K: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ! [X6: int,K2: int] :
( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) )
& ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_2826_left__diff__distrib,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_2827_left__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_2828_right__diff__distrib,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_2829_right__diff__distrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib
thf(fact_2830_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A2: rat] :
( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A2 )
= ( minus_minus_rat @ ( times_times_rat @ B @ A2 ) @ ( times_times_rat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_2831_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_2832_left__diff__distrib_H,axiom,
! [B: int,C: int,A2: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
= ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_2833_right__diff__distrib_H,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_2834_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_2835_right__diff__distrib_H,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_2836_in__lex__prod,axiom,
! [A2: nat,B: int,A5: nat,B3: int,R2: set_Pr1261947904930325089at_nat,S2: set_Pr958786334691620121nt_int] :
( ( member4619661753630440592at_int @ ( produc886916002739626967at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( product_Pair_nat_int @ A5 @ B3 ) ) @ ( lex_prod_nat_int @ R2 @ S2 ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2837_in__lex__prod,axiom,
! [A2: int,B: nat,A5: int,B3: nat,R2: set_Pr958786334691620121nt_int,S2: set_Pr1261947904930325089at_nat] :
( ( member2930414080797523088nt_nat @ ( produc8921240378546987991nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( product_Pair_int_nat @ A5 @ B3 ) ) @ ( lex_prod_int_nat @ R2 @ S2 ) )
= ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2838_in__lex__prod,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr5488025237498180813et_nat] :
( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc7507926704131184380et_nat @ A5 @ B3 ) ) @ ( lex_pr538634069021852900et_nat @ R2 @ S2 ) )
= ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2839_in__lex__prod,axiom,
! [A2: nat,B: nat,A5: nat,B3: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( product_Pair_nat_nat @ A5 @ B3 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2840_in__lex__prod,axiom,
! [A2: int,B: int,A5: int,B3: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( product_Pair_int_int @ A5 @ B3 ) ) @ ( lex_prod_int_int @ R2 @ S2 ) )
= ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2841_in__lex__prod,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr719794911490849221et_nat] :
( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc5001842942810119800et_nat @ A5 @ B3 ) ) @ ( lex_pr5340572901959109728et_nat @ R2 @ S2 ) )
= ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2842_in__lex__prod,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr7928877670098842301et_nat] :
( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc2245416461498447860et_nat @ A5 @ B3 ) ) @ ( lex_pr4722427456421979612et_nat @ R2 @ S2 ) )
= ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A5 ) @ R2 )
| ( ( A2 = A5 )
& ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_2843_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr5094982260447487303t_unit] :
( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( wf_Product_unit @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) )
= ( ! [A3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R2 ) )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A3 )
& ! [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ A3 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2844_Linear__order__wf__diff__Id,axiom,
! [R2: set_Product_prod_o_o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) )
= ( ! [A3: set_o] :
( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
=> ( ( A3 != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2845_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( wf_int @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) )
= ( ! [A3: set_int] :
( ( ord_less_eq_set_int @ A3 @ ( field_int @ R2 ) )
=> ( ( A3 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ! [Y3: int] :
( ( member_int @ Y3 @ A3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2846_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr2161125870931222855_nat_o] :
( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( wf_Pro8787509627421085110_nat_o @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) )
= ( ! [A3: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( A3 != bot_bo7824918357723582233_nat_o )
=> ? [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A3 )
& ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ A3 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2847_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
= ( ! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2848_remove__def,axiom,
( remove_Product_unit
= ( ^ [X2: product_unit,A3: set_Product_unit] : ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% remove_def
thf(fact_2849_remove__def,axiom,
( remove_o
= ( ^ [X2: $o,A3: set_o] : ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_2850_remove__def,axiom,
( remove_int
= ( ^ [X2: int,A3: set_int] : ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% remove_def
thf(fact_2851_remove__def,axiom,
( remove4651630035290841522_nat_o
= ( ^ [X2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] : ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% remove_def
thf(fact_2852_remove__def,axiom,
( remove_nat
= ( ^ [X2: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_2853_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) )
=> ( ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) )
| ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) ) )
=> ( Phi @ A2 @ B ) )
=> ( ( ( A2 = B )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2854_wo__rel_Ocases__Total3,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: $o > $o > $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
=> ( ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) )
| ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) ) )
=> ( Phi @ A2 @ B ) )
=> ( ( ( A2 = B )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2855_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: int > int > $o] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) )
=> ( ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) ) )
=> ( Phi @ A2 @ B ) )
=> ( ( ( A2 = B )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2856_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,Phi: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) )
| ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) ) )
=> ( Phi @ A2 @ B ) )
=> ( ( ( A2 = B )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2857_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: nat > nat > $o] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
=> ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) )
=> ( Phi @ A2 @ B ) )
=> ( ( ( A2 = B )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2858_wf__max,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ( wf_Pro8787509627421085110_nat_o @ ( conver2764731894203590139_nat_o @ R3 ) )
=> ( ( R3 != bot_bo3657187498359500723_nat_o )
=> ~ ! [M2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ M2 @ ( minus_1801376950450012436_nat_o @ ( range_5289318983786108341_nat_o @ R3 ) @ ( domain4881241903163524940_nat_o @ R3 ) ) ) ) ) ).
% wf_max
thf(fact_2859_wf__max,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( wf_int @ ( converse_int_int @ R3 ) )
=> ( ( R3 != bot_bo1796632182523588997nt_int )
=> ~ ! [M2: int] :
~ ( member_int @ M2 @ ( minus_minus_set_int @ ( range_int_int @ R3 ) @ ( domain_int_int @ R3 ) ) ) ) ) ).
% wf_max
thf(fact_2860_wf__max,axiom,
! [R3: set_Product_prod_o_o] :
( ( wf_o @ ( converse_o_o @ R3 ) )
=> ( ( R3 != bot_bo7073875226086086771od_o_o )
=> ~ ! [M2: $o] :
~ ( member_o @ M2 @ ( minus_minus_set_o @ ( range_o_o @ R3 ) @ ( domain_o_o @ R3 ) ) ) ) ) ).
% wf_max
thf(fact_2861_wf__max,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ( wf_Product_unit @ ( conver2495244158383847931t_unit @ R3 ) )
=> ( ( R3 != bot_bo1654022762305692595t_unit )
=> ~ ! [M2: product_unit] :
~ ( member_Product_unit @ M2 @ ( minus_6452836326544984404t_unit @ ( range_6617724647562260917t_unit @ R3 ) @ ( domain4292024440186670924t_unit @ R3 ) ) ) ) ) ).
% wf_max
thf(fact_2862_wf__max,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ ( converse_nat_nat @ R3 ) )
=> ( ( R3 != bot_bo2099793752762293965at_nat )
=> ~ ! [M2: nat] :
~ ( member_nat @ M2 @ ( minus_minus_set_nat @ ( range_nat_nat @ R3 ) @ ( domain_nat_nat @ R3 ) ) ) ) ) ).
% wf_max
thf(fact_2863_wf__Un,axiom,
! [R2: set_Pr5094982260447487303t_unit,S2: set_Pr5094982260447487303t_unit] :
( ( wf_Product_unit @ R2 )
=> ( ( wf_Product_unit @ S2 )
=> ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R2 ) @ ( range_6617724647562260917t_unit @ S2 ) )
= bot_bo3957492148770167129t_unit )
=> ( wf_Product_unit @ ( sup_su6481247232241170203t_unit @ R2 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_2864_wf__Un,axiom,
! [R2: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
( ( wf_o @ R2 )
=> ( ( wf_o @ S2 )
=> ( ( ( inf_inf_set_o @ ( domain_o_o @ R2 ) @ ( range_o_o @ S2 ) )
= bot_bot_set_o )
=> ( wf_o @ ( sup_su5769328420594410459od_o_o @ R2 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_2865_wf__Un,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ R2 )
=> ( ( wf_nat @ S2 )
=> ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R2 ) @ ( range_nat_nat @ S2 ) )
= bot_bot_set_nat )
=> ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_2866_wf__Un,axiom,
! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ( wf_int @ R2 )
=> ( ( wf_int @ S2 )
=> ( ( ( inf_inf_set_int @ ( domain_int_int @ R2 ) @ ( range_int_int @ S2 ) )
= bot_bot_set_int )
=> ( wf_int @ ( sup_su6024340866399070445nt_int @ R2 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_2867_wf__Un,axiom,
! [R2: set_Pr2161125870931222855_nat_o,S2: set_Pr2161125870931222855_nat_o] :
( ( wf_Pro8787509627421085110_nat_o @ R2 )
=> ( ( wf_Pro8787509627421085110_nat_o @ S2 )
=> ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R2 ) @ ( range_5289318983786108341_nat_o @ S2 ) )
= bot_bo7824918357723582233_nat_o )
=> ( wf_Pro8787509627421085110_nat_o @ ( sup_su8328026650783933211_nat_o @ R2 @ S2 ) ) ) ) ) ).
% wf_Un
thf(fact_2868_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr5094982260447487303t_unit] :
( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( order_3164716736300062379t_unit @ ( field_Product_unit @ R2 ) @ R2 )
= ( ! [A3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R2 ) )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A3 )
& ! [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ A3 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_2869_Linear__order__Well__order__iff,axiom,
! [R2: set_Product_prod_o_o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
= ( ! [A3: set_o] :
( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
=> ( ( A3 != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_2870_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( order_2886507596567047182on_int @ ( field_int @ R2 ) @ R2 )
= ( ! [A3: set_int] :
( ( ord_less_eq_set_int @ A3 @ ( field_int @ R2 ) )
=> ( ( A3 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ! [Y3: int] :
( ( member_int @ Y3 @ A3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_2871_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr2161125870931222855_nat_o] :
( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( order_2118566763991480939_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
= ( ! [A3: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( A3 != bot_bo7824918357723582233_nat_o )
=> ? [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A3 )
& ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ A3 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_2872_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
= ( ! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_2873_underS__incr,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( trans_int @ R2 )
=> ( ( antisym_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( order_underS_int @ R2 @ B ) ) ) ) ) ).
% underS_incr
thf(fact_2874_underS__incr,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( trans_nat @ R2 )
=> ( ( antisym_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( order_underS_nat @ R2 @ B ) ) ) ) ) ).
% underS_incr
thf(fact_2875_member__remove,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ ( remove4651630035290841522_nat_o @ Y @ A ) )
= ( ( member6576561426505652726_nat_o @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_2876_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_2877_member__remove,axiom,
! [X: int,Y: int,A: set_int] :
( ( member_int @ X @ ( remove_int @ Y @ A ) )
= ( ( member_int @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_2878_member__remove,axiom,
! [X: $o,Y: $o,A: set_o] :
( ( member_o @ X @ ( remove_o @ Y @ A ) )
= ( ( member_o @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_2879_member__remove,axiom,
! [X: product_unit,Y: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X @ ( remove_Product_unit @ Y @ A ) )
= ( ( member_Product_unit @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_2880_wf__insert,axiom,
! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) )
= ( ( wf_nat @ R2 )
& ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% wf_insert
thf(fact_2881_wf__insert,axiom,
! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
( ( wf_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ Y @ X ) @ R2 ) )
= ( ( wf_int @ R2 )
& ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% wf_insert
thf(fact_2882_wo__rel_Owell__order__induct,axiom,
! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ! [X3: nat] :
( ! [Y5: nat] :
( ( ( Y5 != X3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_2883_wo__rel_Owell__order__induct,axiom,
! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ! [X3: int] :
( ! [Y5: int] :
( ( ( Y5 != X3 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_2884_wf__induct__rule,axiom,
! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
( ( wf_nat @ R2 )
=> ( ! [X3: nat] :
( ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_2885_wf__induct__rule,axiom,
! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
( ( wf_int @ R2 )
=> ( ! [X3: int] :
( ! [Y5: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_2886_wf__eq__minimal,axiom,
( wf_Pro8787509627421085110_nat_o
= ( ^ [R4: set_Pr2161125870931222855_nat_o] :
! [Q: set_Pr4532377907799695533_nat_o] :
( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ Q )
=> ? [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ Q )
& ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y3 @ X2 ) @ R4 )
=> ~ ( member6576561426505652726_nat_o @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2887_wf__eq__minimal,axiom,
( wf_o
= ( ^ [R4: set_Product_prod_o_o] :
! [Q: set_o] :
( ? [X2: $o] : ( member_o @ X2 @ Q )
=> ? [X2: $o] :
( ( member_o @ X2 @ Q )
& ! [Y3: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R4 )
=> ~ ( member_o @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2888_wf__eq__minimal,axiom,
( wf_Product_unit
= ( ^ [R4: set_Pr5094982260447487303t_unit] :
! [Q: set_Product_unit] :
( ? [X2: product_unit] : ( member_Product_unit @ X2 @ Q )
=> ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ Q )
& ! [Y3: product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X2 ) @ R4 )
=> ~ ( member_Product_unit @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2889_wf__eq__minimal,axiom,
( wf_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [Q: set_nat] :
( ? [X2: nat] : ( member_nat @ X2 @ Q )
=> ? [X2: nat] :
( ( member_nat @ X2 @ Q )
& ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
=> ~ ( member_nat @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2890_wf__eq__minimal,axiom,
( wf_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [Q: set_int] :
( ? [X2: int] : ( member_int @ X2 @ Q )
=> ? [X2: int] :
( ( member_int @ X2 @ Q )
& ! [Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
=> ~ ( member_int @ Y3 @ Q ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2891_wf__not__refl,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
( ( wf_nat @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_2892_wf__not__refl,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int] :
( ( wf_int @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_2893_wf__not__sym,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
( ( wf_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A2 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_2894_wf__not__sym,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
( ( wf_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ A2 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_2895_wf__irrefl,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
( ( wf_nat @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_2896_wf__irrefl,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int] :
( ( wf_int @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_2897_wf__induct,axiom,
! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
( ( wf_nat @ R2 )
=> ( ! [X3: nat] :
( ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% wf_induct
thf(fact_2898_wf__induct,axiom,
! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
( ( wf_int @ R2 )
=> ( ! [X3: int] :
( ! [Y5: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% wf_induct
thf(fact_2899_wf__asym,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
( ( wf_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A2 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_2900_wf__asym,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
( ( wf_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X ) @ R2 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ A2 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_2901_wfUNIVI,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ! [P4: nat > $o,X3: nat] :
( ! [Xa2: nat] :
( ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Xa2 ) @ R2 )
=> ( P4 @ Y2 ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf_nat @ R2 ) ) ).
% wfUNIVI
thf(fact_2902_wfUNIVI,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ! [P4: int > $o,X3: int] :
( ! [Xa2: int] :
( ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Xa2 ) @ R2 )
=> ( P4 @ Y2 ) )
=> ( P4 @ Xa2 ) )
=> ( P4 @ X3 ) )
=> ( wf_int @ R2 ) ) ).
% wfUNIVI
thf(fact_2903_wfI__min,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o,Q3: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X3 @ Q3 )
=> ? [Xa2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Xa2 @ Q3 )
& ! [Y2: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ Xa2 ) @ R3 )
=> ~ ( member6576561426505652726_nat_o @ Y2 @ Q3 ) ) ) )
=> ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% wfI_min
thf(fact_2904_wfI__min,axiom,
! [R3: set_Product_prod_o_o] :
( ! [X3: $o,Q3: set_o] :
( ( member_o @ X3 @ Q3 )
=> ? [Xa2: $o] :
( ( member_o @ Xa2 @ Q3 )
& ! [Y2: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ Xa2 ) @ R3 )
=> ~ ( member_o @ Y2 @ Q3 ) ) ) )
=> ( wf_o @ R3 ) ) ).
% wfI_min
thf(fact_2905_wfI__min,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ! [X3: product_unit,Q3: set_Product_unit] :
( ( member_Product_unit @ X3 @ Q3 )
=> ? [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ Q3 )
& ! [Y2: product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ Xa2 ) @ R3 )
=> ~ ( member_Product_unit @ Y2 @ Q3 ) ) ) )
=> ( wf_Product_unit @ R3 ) ) ).
% wfI_min
thf(fact_2906_wfI__min,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Q3: set_nat] :
( ( member_nat @ X3 @ Q3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ Q3 )
& ! [Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Xa2 ) @ R3 )
=> ~ ( member_nat @ Y2 @ Q3 ) ) ) )
=> ( wf_nat @ R3 ) ) ).
% wfI_min
thf(fact_2907_wfI__min,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ! [X3: int,Q3: set_int] :
( ( member_int @ X3 @ Q3 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ Q3 )
& ! [Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Xa2 ) @ R3 )
=> ~ ( member_int @ Y2 @ Q3 ) ) ) )
=> ( wf_int @ R3 ) ) ).
% wfI_min
thf(fact_2908_wfE__min,axiom,
! [R3: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Q2: set_Pr4532377907799695533_nat_o] :
( ( wf_Pro8787509627421085110_nat_o @ R3 )
=> ( ( member6576561426505652726_nat_o @ X @ Q2 )
=> ~ ! [Z3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Z3 @ Q2 )
=> ~ ! [Y5: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ Z3 ) @ R3 )
=> ~ ( member6576561426505652726_nat_o @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_2909_wfE__min,axiom,
! [R3: set_Product_prod_o_o,X: $o,Q2: set_o] :
( ( wf_o @ R3 )
=> ( ( member_o @ X @ Q2 )
=> ~ ! [Z3: $o] :
( ( member_o @ Z3 @ Q2 )
=> ~ ! [Y5: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_o @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_2910_wfE__min,axiom,
! [R3: set_Pr5094982260447487303t_unit,X: product_unit,Q2: set_Product_unit] :
( ( wf_Product_unit @ R3 )
=> ( ( member_Product_unit @ X @ Q2 )
=> ~ ! [Z3: product_unit] :
( ( member_Product_unit @ Z3 @ Q2 )
=> ~ ! [Y5: product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_Product_unit @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_2911_wfE__min,axiom,
! [R3: set_Pr1261947904930325089at_nat,X: nat,Q2: set_nat] :
( ( wf_nat @ R3 )
=> ( ( member_nat @ X @ Q2 )
=> ~ ! [Z3: nat] :
( ( member_nat @ Z3 @ Q2 )
=> ~ ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_2912_wfE__min,axiom,
! [R3: set_Pr958786334691620121nt_int,X: int,Q2: set_int] :
( ( wf_int @ R3 )
=> ( ( member_int @ X @ Q2 )
=> ~ ! [Z3: int] :
( ( member_int @ Z3 @ Q2 )
=> ~ ! [Y5: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_int @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_2913_wf__def,axiom,
( wf_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [P2: nat > $o] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
=> ( P2 @ Y3 ) )
=> ( P2 @ X2 ) )
=> ! [X7: nat] : ( P2 @ X7 ) ) ) ) ).
% wf_def
thf(fact_2914_wf__def,axiom,
( wf_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [P2: int > $o] :
( ! [X2: int] :
( ! [Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
=> ( P2 @ Y3 ) )
=> ( P2 @ X2 ) )
=> ! [X7: int] : ( P2 @ X7 ) ) ) ) ).
% wf_def
thf(fact_2915_well__order__on__domain,axiom,
! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( order_2118566763991480939_nat_o @ A @ R2 )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
& ( member6576561426505652726_nat_o @ B @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_2916_well__order__on__domain,axiom,
! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( order_2615775362358406454r_on_o @ A @ R2 )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( ( member_o @ A2 @ A )
& ( member_o @ B @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_2917_well__order__on__domain,axiom,
! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( order_3164716736300062379t_unit @ A @ R2 )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
=> ( ( member_Product_unit @ A2 @ A )
& ( member_Product_unit @ B @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_2918_well__order__on__domain,axiom,
! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( order_2888998067076097458on_nat @ A @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( member_nat @ A2 @ A )
& ( member_nat @ B @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_2919_well__order__on__domain,axiom,
! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( order_2886507596567047182on_int @ A @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( member_int @ A2 @ A )
& ( member_int @ B @ A ) ) ) ) ).
% well_order_on_domain
thf(fact_2920_underS__E,axiom,
! [I2: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,J: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) )
=> ( ( I2 != J )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_2921_underS__E,axiom,
! [I2: $o,R3: set_Product_prod_o_o,J: $o] :
( ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) )
=> ( ( I2 != J )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_2922_underS__E,axiom,
! [I2: product_unit,R3: set_Pr5094982260447487303t_unit,J: product_unit] :
( ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) )
=> ( ( I2 != J )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_2923_underS__E,axiom,
! [I2: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) )
=> ( ( I2 != J )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_2924_underS__E,axiom,
! [I2: int,R3: set_Pr958786334691620121nt_int,J: int] :
( ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) )
=> ( ( I2 != J )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_2925_underS__I,axiom,
! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
( ( I2 != J )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
=> ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_2926_underS__I,axiom,
! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
( ( I2 != J )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
=> ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_2927_underS__I,axiom,
! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
( ( I2 != J )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
=> ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_2928_underS__I,axiom,
! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
( ( I2 != J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
=> ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_2929_underS__I,axiom,
! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
( ( I2 != J )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
=> ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_2930_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,J: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) )
=> ( member6576561426505652726_nat_o @ I2 @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_2931_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) )
=> ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_2932_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: int,R3: set_Pr958786334691620121nt_int,J: int] :
( ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) )
=> ( member_int @ I2 @ ( field_int @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_2933_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: $o,R3: set_Product_prod_o_o,J: $o] :
( ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) )
=> ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_2934_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I2: product_unit,R3: set_Pr5094982260447487303t_unit,J: product_unit] :
( ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) )
=> ( member_Product_unit @ I2 @ ( field_Product_unit @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_2935_wo__rel_OTOTALS,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ! [X6: nat] :
( ( member_nat @ X6 @ ( field_nat @ R2 ) )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( field_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa2 ) @ R2 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_2936_wo__rel_OTOTALS,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ! [X6: int] :
( ( member_int @ X6 @ ( field_int @ R2 ) )
=> ! [Xa2: int] :
( ( member_int @ Xa2 @ ( field_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Xa2 ) @ R2 )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_2937_wfE__min_H,axiom,
! [R3: set_Pr5094982260447487303t_unit,Q2: set_Product_unit] :
( ( wf_Product_unit @ R3 )
=> ( ( Q2 != bot_bo3957492148770167129t_unit )
=> ~ ! [Z3: product_unit] :
( ( member_Product_unit @ Z3 @ Q2 )
=> ~ ! [Y5: product_unit] :
( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_Product_unit @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_2938_wfE__min_H,axiom,
! [R3: set_Product_prod_o_o,Q2: set_o] :
( ( wf_o @ R3 )
=> ( ( Q2 != bot_bot_set_o )
=> ~ ! [Z3: $o] :
( ( member_o @ Z3 @ Q2 )
=> ~ ! [Y5: $o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_o @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_2939_wfE__min_H,axiom,
! [R3: set_Pr1261947904930325089at_nat,Q2: set_nat] :
( ( wf_nat @ R3 )
=> ( ( Q2 != bot_bot_set_nat )
=> ~ ! [Z3: nat] :
( ( member_nat @ Z3 @ Q2 )
=> ~ ! [Y5: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_2940_wfE__min_H,axiom,
! [R3: set_Pr958786334691620121nt_int,Q2: set_int] :
( ( wf_int @ R3 )
=> ( ( Q2 != bot_bot_set_int )
=> ~ ! [Z3: int] :
( ( member_int @ Z3 @ Q2 )
=> ~ ! [Y5: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ Z3 ) @ R3 )
=> ~ ( member_int @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_2941_wfE__min_H,axiom,
! [R3: set_Pr2161125870931222855_nat_o,Q2: set_Pr4532377907799695533_nat_o] :
( ( wf_Pro8787509627421085110_nat_o @ R3 )
=> ( ( Q2 != bot_bo7824918357723582233_nat_o )
=> ~ ! [Z3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Z3 @ Q2 )
=> ~ ! [Y5: produc3658429121746597890et_nat > $o] :
( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ Z3 ) @ R3 )
=> ~ ( member6576561426505652726_nat_o @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_2942_well__order__on__empty,axiom,
order_3164716736300062379t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% well_order_on_empty
thf(fact_2943_well__order__on__empty,axiom,
order_2615775362358406454r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% well_order_on_empty
thf(fact_2944_well__order__on__empty,axiom,
order_2888998067076097458on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% well_order_on_empty
thf(fact_2945_well__order__on__empty,axiom,
order_2886507596567047182on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% well_order_on_empty
thf(fact_2946_well__order__on__empty,axiom,
order_2118566763991480939_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% well_order_on_empty
thf(fact_2947_well__order__induct__imp,axiom,
! [R2: set_Pr2161125870931222855_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,A2: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ! [Y5: produc3658429121746597890et_nat > $o] :
( ( ( Y5 != X3 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ X3 ) @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ Y5 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( P @ Y5 ) ) )
=> ( ( member6576561426505652726_nat_o @ X3 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2948_well__order__induct__imp,axiom,
! [R2: set_Product_prod_o_o,P: $o > $o,A2: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ! [X3: $o] :
( ! [Y5: $o] :
( ( ( Y5 != X3 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X3 ) @ R2 ) )
=> ( ( member_o @ Y5 @ ( field_o @ R2 ) )
=> ( P @ Y5 ) ) )
=> ( ( member_o @ X3 @ ( field_o @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2949_well__order__induct__imp,axiom,
! [R2: set_Pr5094982260447487303t_unit,P: product_unit > $o,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ! [X3: product_unit] :
( ! [Y5: product_unit] :
( ( ( Y5 != X3 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ X3 ) @ R2 ) )
=> ( ( member_Product_unit @ Y5 @ ( field_Product_unit @ R2 ) )
=> ( P @ Y5 ) ) )
=> ( ( member_Product_unit @ X3 @ ( field_Product_unit @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2950_well__order__induct__imp,axiom,
! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ! [X3: nat] :
( ! [Y5: nat] :
( ( ( Y5 != X3 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 ) )
=> ( ( member_nat @ Y5 @ ( field_nat @ R2 ) )
=> ( P @ Y5 ) ) )
=> ( ( member_nat @ X3 @ ( field_nat @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2951_well__order__induct__imp,axiom,
! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ! [X3: int] :
( ! [Y5: int] :
( ( ( Y5 != X3 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 ) )
=> ( ( member_int @ Y5 @ ( field_int @ R2 ) )
=> ( P @ Y5 ) ) )
=> ( ( member_int @ X3 @ ( field_int @ R2 ) )
=> ( P @ X3 ) ) )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( P @ A2 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2952_underS__empty,axiom,
! [A2: product_unit,R2: set_Pr5094982260447487303t_unit] :
( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( order_5809128134054296019t_unit @ R2 @ A2 )
= bot_bo3957492148770167129t_unit ) ) ).
% underS_empty
thf(fact_2953_underS__empty,axiom,
! [A2: $o,R2: set_Product_prod_o_o] :
( ~ ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( order_underS_o @ R2 @ A2 )
= bot_bot_set_o ) ) ).
% underS_empty
thf(fact_2954_underS__empty,axiom,
! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
( ~ ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( order_underS_nat @ R2 @ A2 )
= bot_bot_set_nat ) ) ).
% underS_empty
thf(fact_2955_underS__empty,axiom,
! [A2: int,R2: set_Pr958786334691620121nt_int] :
( ~ ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( order_underS_int @ R2 @ A2 )
= bot_bot_set_int ) ) ).
% underS_empty
thf(fact_2956_underS__empty,axiom,
! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
( ~ ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( order_7262550146012401043_nat_o @ R2 @ A2 )
= bot_bo7824918357723582233_nat_o ) ) ).
% underS_empty
thf(fact_2957_wfI__pf,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ! [A9: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A9 @ ( image_8431784415008483091t_unit @ R3 @ A9 ) )
=> ( A9 = bot_bo3957492148770167129t_unit ) )
=> ( wf_Product_unit @ R3 ) ) ).
% wfI_pf
thf(fact_2958_wfI__pf,axiom,
! [R3: set_Product_prod_o_o] :
( ! [A9: set_o] :
( ( ord_less_eq_set_o @ A9 @ ( image_o_o @ R3 @ A9 ) )
=> ( A9 = bot_bot_set_o ) )
=> ( wf_o @ R3 ) ) ).
% wfI_pf
thf(fact_2959_wfI__pf,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ! [A9: set_int] :
( ( ord_less_eq_set_int @ A9 @ ( image_int_int @ R3 @ A9 ) )
=> ( A9 = bot_bot_set_int ) )
=> ( wf_int @ R3 ) ) ).
% wfI_pf
thf(fact_2960_wfI__pf,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ! [A9: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A9 @ ( image_2232984160439939859_nat_o @ R3 @ A9 ) )
=> ( A9 = bot_bo7824918357723582233_nat_o ) )
=> ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% wfI_pf
thf(fact_2961_wfI__pf,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ! [A9: set_nat] :
( ( ord_less_eq_set_nat @ A9 @ ( image_nat_nat @ R3 @ A9 ) )
=> ( A9 = bot_bot_set_nat ) )
=> ( wf_nat @ R3 ) ) ).
% wfI_pf
thf(fact_2962_wfE__pf,axiom,
! [R3: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
( ( wf_Product_unit @ R3 )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_8431784415008483091t_unit @ R3 @ A ) )
=> ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% wfE_pf
thf(fact_2963_wfE__pf,axiom,
! [R3: set_Product_prod_o_o,A: set_o] :
( ( wf_o @ R3 )
=> ( ( ord_less_eq_set_o @ A @ ( image_o_o @ R3 @ A ) )
=> ( A = bot_bot_set_o ) ) ) ).
% wfE_pf
thf(fact_2964_wfE__pf,axiom,
! [R3: set_Pr958786334691620121nt_int,A: set_int] :
( ( wf_int @ R3 )
=> ( ( ord_less_eq_set_int @ A @ ( image_int_int @ R3 @ A ) )
=> ( A = bot_bot_set_int ) ) ) ).
% wfE_pf
thf(fact_2965_wfE__pf,axiom,
! [R3: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( wf_Pro8787509627421085110_nat_o @ R3 )
=> ( ( ord_le2965882846123202637_nat_o @ A @ ( image_2232984160439939859_nat_o @ R3 @ A ) )
=> ( A = bot_bo7824918357723582233_nat_o ) ) ) ).
% wfE_pf
thf(fact_2966_wfE__pf,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat] :
( ( wf_nat @ R3 )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ R3 @ A ) )
=> ( A = bot_bot_set_nat ) ) ) ).
% wfE_pf
thf(fact_2967_wf__eq__minimal2,axiom,
( wf_Product_unit
= ( ^ [R4: set_Pr5094982260447487303t_unit] :
! [A3: set_Product_unit] :
( ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R4 ) )
& ( A3 != bot_bo3957492148770167129t_unit ) )
=> ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A3 )
& ! [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ A3 )
=> ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2968_wf__eq__minimal2,axiom,
( wf_o
= ( ^ [R4: set_Product_prod_o_o] :
! [A3: set_o] :
( ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R4 ) )
& ( A3 != bot_bot_set_o ) )
=> ? [X2: $o] :
( ( member_o @ X2 @ A3 )
& ! [Y3: $o] :
( ( member_o @ Y3 @ A3 )
=> ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2969_wf__eq__minimal2,axiom,
( wf_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [A3: set_int] :
( ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R4 ) )
& ( A3 != bot_bot_set_int ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ! [Y3: int] :
( ( member_int @ Y3 @ A3 )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2970_wf__eq__minimal2,axiom,
( wf_Pro8787509627421085110_nat_o
= ( ^ [R4: set_Pr2161125870931222855_nat_o] :
! [A3: set_Pr4532377907799695533_nat_o] :
( ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R4 ) )
& ( A3 != bot_bo7824918357723582233_nat_o ) )
=> ? [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A3 )
& ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ A3 )
=> ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2971_wf__eq__minimal2,axiom,
( wf_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [A3: set_nat] :
( ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R4 ) )
& ( A3 != bot_bot_set_nat ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2972_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) )
=> ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2973_wo__rel_Ocases__Total,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: $o > $o > $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
=> ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2974_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: int > int > $o] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) )
=> ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2975_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,Phi: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2976_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: nat > nat > $o] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 )
=> ( Phi @ A2 @ B ) )
=> ( Phi @ A2 @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2977_wf__no__path,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) )
= bot_bo3957492148770167129t_unit )
=> ( wf_Product_unit @ R3 ) ) ).
% wf_no_path
thf(fact_2978_wf__no__path,axiom,
! [R3: set_Product_prod_o_o] :
( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) )
= bot_bot_set_o )
=> ( wf_o @ R3 ) ) ).
% wf_no_path
thf(fact_2979_wf__no__path,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) )
= bot_bot_set_nat )
=> ( wf_nat @ R3 ) ) ).
% wf_no_path
thf(fact_2980_wf__no__path,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( ( inf_inf_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) )
= bot_bot_set_int )
=> ( wf_int @ R3 ) ) ).
% wf_no_path
thf(fact_2981_wf__no__path,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) )
= bot_bo7824918357723582233_nat_o )
=> ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% wf_no_path
thf(fact_2982_wf__min,axiom,
! [R3: set_Pr2161125870931222855_nat_o] :
( ( wf_Pro8787509627421085110_nat_o @ R3 )
=> ( ( R3 != bot_bo3657187498359500723_nat_o )
=> ~ ! [M2: produc3658429121746597890et_nat > $o] :
~ ( member6576561426505652726_nat_o @ M2 @ ( minus_1801376950450012436_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) ) ) ) ) ).
% wf_min
thf(fact_2983_wf__min,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( wf_int @ R3 )
=> ( ( R3 != bot_bo1796632182523588997nt_int )
=> ~ ! [M2: int] :
~ ( member_int @ M2 @ ( minus_minus_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) ) ) ) ) ).
% wf_min
thf(fact_2984_wf__min,axiom,
! [R3: set_Product_prod_o_o] :
( ( wf_o @ R3 )
=> ( ( R3 != bot_bo7073875226086086771od_o_o )
=> ~ ! [M2: $o] :
~ ( member_o @ M2 @ ( minus_minus_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) ) ) ) ) ).
% wf_min
thf(fact_2985_wf__min,axiom,
! [R3: set_Pr5094982260447487303t_unit] :
( ( wf_Product_unit @ R3 )
=> ( ( R3 != bot_bo1654022762305692595t_unit )
=> ~ ! [M2: product_unit] :
~ ( member_Product_unit @ M2 @ ( minus_6452836326544984404t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) ) ) ) ) ).
% wf_min
thf(fact_2986_wf__min,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( wf_nat @ R3 )
=> ( ( R3 != bot_bo2099793752762293965at_nat )
=> ~ ! [M2: nat] :
~ ( member_nat @ M2 @ ( minus_minus_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) ) ) ) ) ).
% wf_min
thf(fact_2987_underS__incl__iff,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ord_le2965882846123202637_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( order_7262550146012401043_nat_o @ R2 @ B ) )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_2988_underS__incl__iff,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( ord_less_eq_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( order_underS_o @ R2 @ B ) )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_2989_underS__incl__iff,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( order_5809128134054296019t_unit @ R2 @ B ) )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_2990_underS__incl__iff,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( ord_less_eq_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( order_underS_int @ R2 @ B ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_2991_underS__incl__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( order_underS_nat @ R2 @ B ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_2992_wo__rel_OWell__order__isMinim__exists,axiom,
! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ? [X_1: product_unit] : ( bNF_We909621271159000003t_unit @ R2 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_2993_wo__rel_OWell__order__isMinim__exists,axiom,
! [R2: set_Product_prod_o_o,B4: set_o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
=> ( ( B4 != bot_bot_set_o )
=> ? [X_1: $o] : ( bNF_We4647864369647134286inim_o @ R2 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_2994_wo__rel_OWell__order__isMinim__exists,axiom,
! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( B4 != bot_bot_set_int )
=> ? [X_1: int] : ( bNF_We6511469253196982518im_int @ R2 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_2995_wo__rel_OWell__order__isMinim__exists,axiom,
! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ? [X_1: produc3658429121746597890et_nat > $o] : ( bNF_We5608535359669600131_nat_o @ R2 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_2996_wo__rel_OWell__order__isMinim__exists,axiom,
! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( B4 != bot_bot_set_nat )
=> ? [X_1: nat] : ( bNF_We6513959723706032794im_nat @ R2 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_2997_Refl__under__underS,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( refl_on_Product_unit @ ( field_Product_unit @ R2 ) @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( order_6949322460238004850t_unit @ R2 @ A2 )
= ( sup_su793286257634532545t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Refl_under_underS
thf(fact_2998_Refl__under__underS,axiom,
! [R2: set_Product_prod_o_o,A2: $o] :
( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( order_under_o @ R2 @ A2 )
= ( sup_sup_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ).
% Refl_under_underS
thf(fact_2999_Refl__under__underS,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( order_under_nat @ R2 @ A2 )
= ( sup_sup_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ).
% Refl_under_underS
thf(fact_3000_Refl__under__underS,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int] :
( ( refl_on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( order_under_int @ R2 @ A2 )
= ( sup_sup_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ).
% Refl_under_underS
thf(fact_3001_Refl__under__underS,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( refl_o1075864025857821388_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( order_2885205318754725938_nat_o @ R2 @ A2 )
= ( sup_su5209123915105501825_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% Refl_under_underS
thf(fact_3002_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ ( field_Product_unit @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_3003_wo__rel_Ominim__inField,axiom,
! [R2: set_Product_prod_o_o,B4: set_o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
=> ( ( B4 != bot_bot_set_o )
=> ( member_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ ( field_o @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_3004_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( B4 != bot_bot_set_int )
=> ( member_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ ( field_int @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_3005_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( member6576561426505652726_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_3006_wo__rel_Ominim__inField,axiom,
! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( member_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ ( field_nat @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_3007_wo__rel_Ominim__in,axiom,
! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_3008_wo__rel_Ominim__in,axiom,
! [R2: set_Product_prod_o_o,B4: set_o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
=> ( ( B4 != bot_bot_set_o )
=> ( member_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_3009_wo__rel_Ominim__in,axiom,
! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( B4 != bot_bot_set_int )
=> ( member_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_3010_wo__rel_Ominim__in,axiom,
! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( member6576561426505652726_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_3011_wo__rel_Ominim__in,axiom,
! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( member_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_3012_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ A2 @ B4 )
=> ( ! [B2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B2 @ B4 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 ) )
=> ( A2
= ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_3013_wo__rel_Oequals__minim,axiom,
! [R2: set_Product_prod_o_o,B4: set_o,A2: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
=> ( ( member_o @ A2 @ B4 )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ B4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R2 ) )
=> ( A2
= ( bNF_We537455116209922916inim_o @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_3014_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,A2: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ A2 @ B4 )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ B4 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B2 ) @ R2 ) )
=> ( A2
= ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_3015_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr958786334691620121nt_int,B4: set_int,A2: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( member_int @ A2 @ B4 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ B4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) )
=> ( A2
= ( bNF_We3272633801743244640im_int @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_3016_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,A2: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ A2 @ B4 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ B4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) )
=> ( A2
= ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_3017_wo__rel_Ominim__least,axiom,
! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ B4 )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_3018_wo__rel_Ominim__least,axiom,
! [R2: set_Product_prod_o_o,B4: set_o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ B4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_3019_wo__rel_Ominim__least,axiom,
! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ B4 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_3020_wo__rel_Ominim__least,axiom,
! [R2: set_Pr958786334691620121nt_int,B4: set_int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ B4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_3021_wo__rel_Ominim__least,axiom,
! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ B4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_3022_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
& ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_3023_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
& ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_3024_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
& ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_3025_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
& ( member_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_3026_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
& ( member6576561426505652726_nat_o @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_3027_wo__rel_Omax2__def,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
= B ) )
& ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
= A2 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_3028_wo__rel_Omax2__def,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
= B ) )
& ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
= A2 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_3029_under__incr,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( trans_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_int @ ( order_under_int @ R2 @ A2 ) @ ( order_under_int @ R2 @ B ) ) ) ) ).
% under_incr
thf(fact_3030_under__incr,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( trans_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
=> ( ord_less_eq_set_nat @ ( order_under_nat @ R2 @ A2 ) @ ( order_under_nat @ R2 @ B ) ) ) ) ).
% under_incr
thf(fact_3031_wo__rel_Ominim__isMinim,axiom,
! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( bNF_We909621271159000003t_unit @ R2 @ B4 @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_3032_wo__rel_Ominim__isMinim,axiom,
! [R2: set_Product_prod_o_o,B4: set_o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
=> ( ( B4 != bot_bot_set_o )
=> ( bNF_We4647864369647134286inim_o @ R2 @ B4 @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_3033_wo__rel_Ominim__isMinim,axiom,
! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
=> ( ( B4 != bot_bot_set_int )
=> ( bNF_We6511469253196982518im_int @ R2 @ B4 @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_3034_wo__rel_Ominim__isMinim,axiom,
! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( bNF_We5608535359669600131_nat_o @ R2 @ B4 @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_3035_wo__rel_Ominim__isMinim,axiom,
! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( bNF_We6513959723706032794im_nat @ R2 @ B4 @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_3036_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( bNF_We5608535359669600131_nat_o @ R2 @ A @ B )
= ( ( member6576561426505652726_nat_o @ B @ A )
& ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ A )
=> ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_3037_wo__rel_OisMinim__def,axiom,
! [R2: set_Product_prod_o_o,A: set_o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( bNF_We4647864369647134286inim_o @ R2 @ A @ B )
= ( ( member_o @ B @ A )
& ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_3038_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr5094982260447487303t_unit,A: set_Product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( bNF_We909621271159000003t_unit @ R2 @ A @ B )
= ( ( member_Product_unit @ B @ A )
& ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_3039_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( bNF_We6513959723706032794im_nat @ R2 @ A @ B )
= ( ( member_nat @ B @ A )
& ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_3040_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr958786334691620121nt_int,A: set_int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( bNF_We6511469253196982518im_int @ R2 @ A @ B )
= ( ( member_int @ B @ A )
& ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_3041_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
& ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_3042_wo__rel_Omax2__greater,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
& ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_3043_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
& ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_3044_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_3045_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_3046_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B )
= B )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_3047_wo__rel_Omax2__equals2,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B )
= B )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_3048_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B )
= B )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_3049_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
= B )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_3050_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
= B )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_3051_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B )
= A2 )
= ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_3052_wo__rel_Omax2__equals1,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B )
= A2 )
= ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_3053_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( ( ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B )
= A2 )
= ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_3054_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
= A2 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_3055_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
= A2 )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_3056_wo__rel_Omax2__among,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
( ( bNF_We2091578341578594809t_unit @ R2 )
=> ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
=> ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
=> ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_3057_wo__rel_Omax2__among,axiom,
! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
( ( bNF_We6437023980930612612_rel_o @ R2 )
=> ( ( member_o @ A2 @ ( field_o @ R2 ) )
=> ( ( member_o @ B @ ( field_o @ R2 ) )
=> ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_3058_wo__rel_Omax2__among,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
( ( bNF_We3818239936649020644el_nat @ R2 )
=> ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
=> ( ( member_nat @ B @ ( field_nat @ R2 ) )
=> ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_3059_wo__rel_Omax2__among,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
( ( bNF_We3815749466139970368el_int @ R2 )
=> ( ( member_int @ A2 @ ( field_int @ R2 ) )
=> ( ( member_int @ B @ ( field_int @ R2 ) )
=> ( member_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_3060_wo__rel_Omax2__among,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
( ( bNF_We740377018556551609_nat_o @ R2 )
=> ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
=> ( member6576561426505652726_nat_o @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_3061_chains__extend,axiom,
! [C: set_set_nat,S: set_set_nat,Z: set_nat] :
( ( member_set_set_nat @ C @ ( chains_nat2 @ S ) )
=> ( ( member_set_nat @ Z @ S )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z @ bot_bot_set_set_nat ) @ C ) @ ( chains_nat2 @ S ) ) ) ) ) ).
% chains_extend
thf(fact_3062_Zorns__po__lemma,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( order_5251275573222108571on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ! [C6: set_nat] :
( ( member_set_nat @ C6 @ ( chains_nat @ R2 ) )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( field_nat @ R2 ) )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ C6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X6 ) @ R2 ) ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( field_nat @ R2 ) )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( field_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_3063_Zorns__po__lemma,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( order_5248785102713058295on_int @ ( field_int @ R2 ) @ R2 )
=> ( ! [C6: set_int] :
( ( member_set_int @ C6 @ ( chains_int @ R2 ) )
=> ? [X6: int] :
( ( member_int @ X6 @ ( field_int @ R2 ) )
& ! [Xa3: int] :
( ( member_int @ Xa3 @ C6 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa3 @ X6 ) @ R2 ) ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ ( field_int @ R2 ) )
& ! [Xa2: int] :
( ( member_int @ Xa2 @ ( field_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Xa2 ) @ R2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_3064_bsqr__max2,axiom,
! [R2: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% bsqr_max2
thf(fact_3065_bsqr__max2,axiom,
! [R2: set_Pr958786334691620121nt_int,A1: int,A22: int,B1: int,B22: int] :
( ( order_2886507596567047182on_int @ ( field_int @ R2 ) @ R2 )
=> ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( product_Pair_int_int @ B1 @ B22 ) ) @ ( bNF_We1730399552001240015qr_int @ R2 ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A1 @ A22 ) @ ( bNF_We2378448595015613790x2_int @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% bsqr_max2
thf(fact_3066_wf__Union,axiom,
! [R3: set_se8994758365886191399t_unit] :
( ! [X3: set_Pr5094982260447487303t_unit] :
( ( member1248447801810324848t_unit @ X3 @ R3 )
=> ( wf_Product_unit @ X3 ) )
=> ( ! [X3: set_Pr5094982260447487303t_unit] :
( ( member1248447801810324848t_unit @ X3 @ R3 )
=> ! [Xa3: set_Pr5094982260447487303t_unit] :
( ( member1248447801810324848t_unit @ Xa3 @ R3 )
=> ( ( X3 != Xa3 )
=> ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ X3 ) @ ( range_6617724647562260917t_unit @ Xa3 ) )
= bot_bo3957492148770167129t_unit ) ) ) )
=> ( wf_Product_unit @ ( comple2798140142350441396t_unit @ R3 ) ) ) ) ).
% wf_Union
thf(fact_3067_wf__Union,axiom,
! [R3: set_se491199571785101031od_o_o] :
( ! [X3: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X3 @ R3 )
=> ( wf_o @ X3 ) )
=> ( ! [X3: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ X3 @ R3 )
=> ! [Xa3: set_Product_prod_o_o] :
( ( member3157015407008874800od_o_o @ Xa3 @ R3 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_o @ ( domain_o_o @ X3 ) @ ( range_o_o @ Xa3 ) )
= bot_bot_set_o ) ) ) )
=> ( wf_o @ ( comple2567785453016644468od_o_o @ R3 ) ) ) ) ).
% wf_Union
thf(fact_3068_wf__Union,axiom,
! [R3: set_se7855581050983116737at_nat] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ R3 )
=> ( wf_nat @ X3 ) )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ R3 )
=> ! [Xa3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Xa3 @ R3 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_nat @ ( domain_nat_nat @ X3 ) @ ( range_nat_nat @ Xa3 ) )
= bot_bot_set_nat ) ) ) )
=> ( wf_nat @ ( comple5685304695842803022at_nat @ R3 ) ) ) ) ).
% wf_Union
thf(fact_3069_wf__Union,axiom,
! [R3: set_se6260736226359567993nt_int] :
( ! [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ R3 )
=> ( wf_int @ X3 ) )
=> ( ! [X3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ X3 @ R3 )
=> ! [Xa3: set_Pr958786334691620121nt_int] :
( ( member2340774599025711042nt_int @ Xa3 @ R3 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_int @ ( domain_int_int @ X3 ) @ ( range_int_int @ Xa3 ) )
= bot_bot_set_int ) ) ) )
=> ( wf_int @ ( comple5382143125604098054nt_int @ R3 ) ) ) ) ).
% wf_Union
thf(fact_3070_wf__Union,axiom,
! [R3: set_se7663487083892625191_nat_o] :
( ! [X3: set_Pr2161125870931222855_nat_o] :
( ( member5475208728337331568_nat_o @ X3 @ R3 )
=> ( wf_Pro8787509627421085110_nat_o @ X3 ) )
=> ( ! [X3: set_Pr2161125870931222855_nat_o] :
( ( member5475208728337331568_nat_o @ X3 @ R3 )
=> ! [Xa3: set_Pr2161125870931222855_nat_o] :
( ( member5475208728337331568_nat_o @ Xa3 @ R3 )
=> ( ( X3 != Xa3 )
=> ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ X3 ) @ ( range_5289318983786108341_nat_o @ Xa3 ) )
= bot_bo7824918357723582233_nat_o ) ) ) )
=> ( wf_Pro8787509627421085110_nat_o @ ( comple8164645552537203636_nat_o @ R3 ) ) ) ) ).
% wf_Union
thf(fact_3071_Pow__singleton__iff,axiom,
! [X4: set_Product_unit,Y6: set_Product_unit] :
( ( ( pow_Product_unit @ X4 )
= ( insert8986028188087627965t_unit @ Y6 @ bot_bo6284674557190688313t_unit ) )
= ( ( X4 = bot_bo3957492148770167129t_unit )
& ( Y6 = bot_bo3957492148770167129t_unit ) ) ) ).
% Pow_singleton_iff
thf(fact_3072_Pow__singleton__iff,axiom,
! [X4: set_o,Y6: set_o] :
( ( ( pow_o @ X4 )
= ( insert_set_o @ Y6 @ bot_bot_set_set_o ) )
= ( ( X4 = bot_bot_set_o )
& ( Y6 = bot_bot_set_o ) ) ) ).
% Pow_singleton_iff
thf(fact_3073_Pow__singleton__iff,axiom,
! [X4: set_nat,Y6: set_nat] :
( ( ( pow_nat @ X4 )
= ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y6 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_3074_Pow__singleton__iff,axiom,
! [X4: set_int,Y6: set_int] :
( ( ( pow_int @ X4 )
= ( insert_set_int @ Y6 @ bot_bot_set_set_int ) )
= ( ( X4 = bot_bot_set_int )
& ( Y6 = bot_bot_set_int ) ) ) ).
% Pow_singleton_iff
thf(fact_3075_Pow__singleton__iff,axiom,
! [X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
( ( ( pow_Pr7829143134171777386_nat_o @ X4 )
= ( insert2507855216428400253_nat_o @ Y6 @ bot_bo3736544561982865913_nat_o ) )
= ( ( X4 = bot_bo7824918357723582233_nat_o )
& ( Y6 = bot_bo7824918357723582233_nat_o ) ) ) ).
% Pow_singleton_iff
thf(fact_3076_Pow__empty,axiom,
( ( pow_Product_unit @ bot_bo3957492148770167129t_unit )
= ( insert8986028188087627965t_unit @ bot_bo3957492148770167129t_unit @ bot_bo6284674557190688313t_unit ) ) ).
% Pow_empty
thf(fact_3077_Pow__empty,axiom,
( ( pow_o @ bot_bot_set_o )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% Pow_empty
thf(fact_3078_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_3079_Pow__empty,axiom,
( ( pow_int @ bot_bot_set_int )
= ( insert_set_int @ bot_bot_set_int @ bot_bot_set_set_int ) ) ).
% Pow_empty
thf(fact_3080_Pow__empty,axiom,
( ( pow_Pr7829143134171777386_nat_o @ bot_bo7824918357723582233_nat_o )
= ( insert2507855216428400253_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3736544561982865913_nat_o ) ) ).
% Pow_empty
thf(fact_3081_rtrancl__mapI,axiom,
! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2905341329935302413cl_nat @ ( image_5168914502847457605at_nat @ ( pairself_nat_nat @ F ) @ E ) ) ) ) ).
% rtrancl_mapI
thf(fact_3082_rtrancl__mapI,axiom,
! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: nat > int] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ E ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2902850859426252137cl_int @ ( image_1990417195239315453nt_int @ ( pairself_nat_int @ F ) @ E ) ) ) ) ).
% rtrancl_mapI
thf(fact_3083_rtrancl__mapI,axiom,
! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ E ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2905341329935302413cl_nat @ ( image_5831868185956570253at_nat @ ( pairself_int_nat @ F ) @ E ) ) ) ) ).
% rtrancl_mapI
thf(fact_3084_rtrancl__mapI,axiom,
! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ E ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2902850859426252137cl_int @ ( image_2653370878348428101nt_int @ ( pairself_int_int @ F ) @ E ) ) ) ) ).
% rtrancl_mapI
thf(fact_3085_disjoint__image__subset,axiom,
! [A11: set_set_nat,F: set_nat > set_nat] :
( ( pairwise_set_nat @ disjnt_nat @ A11 )
=> ( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A11 )
=> ( ord_less_eq_set_nat @ ( F @ X8 ) @ X8 ) )
=> ( pairwise_set_nat @ disjnt_nat @ ( image_7916887816326733075et_nat @ F @ A11 ) ) ) ) ).
% disjoint_image_subset
thf(fact_3086_pairself_Opelims,axiom,
! [X: nat > nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( pairself_nat_nat @ X @ Xa )
= Y )
=> ( ( accp_P4147487641329543982at_nat @ pairself_rel_nat_nat @ ( produc4237282728380940073at_nat @ X @ Xa ) )
=> ~ ! [A4: nat,B2: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A4 @ B2 ) )
=> ( ( Y
= ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) )
=> ~ ( accp_P4147487641329543982at_nat @ pairself_rel_nat_nat @ ( produc4237282728380940073at_nat @ X @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_3087_pairself_Opelims,axiom,
! [X: nat > int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
( ( ( pairself_nat_int @ X @ Xa )
= Y )
=> ( ( accp_P3050404660685545426at_nat @ pairself_rel_nat_int @ ( produc3140199747736941517at_nat @ X @ Xa ) )
=> ~ ! [A4: nat,B2: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A4 @ B2 ) )
=> ( ( Y
= ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) )
=> ~ ( accp_P3050404660685545426at_nat @ pairself_rel_nat_int @ ( produc3140199747736941517at_nat @ X @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_3088_pairself_Opelims,axiom,
! [X: int > nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
( ( ( pairself_int_nat @ X @ Xa )
= Y )
=> ( ( accp_P5604362734843950218nt_int @ pairself_rel_int_nat @ ( produc2818822084525909125nt_int @ X @ Xa ) )
=> ~ ! [A4: int,B2: int] :
( ( Xa
= ( product_Pair_int_int @ A4 @ B2 ) )
=> ( ( Y
= ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) )
=> ~ ( accp_P5604362734843950218nt_int @ pairself_rel_int_nat @ ( produc2818822084525909125nt_int @ X @ ( product_Pair_int_int @ A4 @ B2 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_3089_pairself_Opelims,axiom,
! [X: int > int,Xa: product_prod_int_int,Y: product_prod_int_int] :
( ( ( pairself_int_int @ X @ Xa )
= Y )
=> ( ( accp_P4507279754199951662nt_int @ pairself_rel_int_int @ ( produc1721739103881910569nt_int @ X @ Xa ) )
=> ~ ! [A4: int,B2: int] :
( ( Xa
= ( product_Pair_int_int @ A4 @ B2 ) )
=> ( ( Y
= ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) )
=> ~ ( accp_P4507279754199951662nt_int @ pairself_rel_int_int @ ( produc1721739103881910569nt_int @ X @ ( product_Pair_int_int @ A4 @ B2 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_3090_Pow__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% Pow_iff
thf(fact_3091_PowI,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( member_set_nat @ A @ ( pow_nat @ B4 ) ) ) ).
% PowI
thf(fact_3092_Pow__Int__eq,axiom,
! [A: set_nat,B4: set_nat] :
( ( pow_nat @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_inf_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% Pow_Int_eq
thf(fact_3093_pairwiseI,axiom,
! [S: set_Pr4532377907799695533_nat_o,R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ S )
=> ( ( member6576561426505652726_nat_o @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwi2432143822833250616_nat_o @ R3 @ S ) ) ).
% pairwiseI
thf(fact_3094_pairwiseI,axiom,
! [S: set_nat,R3: nat > nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ S )
=> ( ( member_nat @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwise_nat @ R3 @ S ) ) ).
% pairwiseI
thf(fact_3095_pairwiseI,axiom,
! [S: set_int,R3: int > int > $o] :
( ! [X3: int,Y2: int] :
( ( member_int @ X3 @ S )
=> ( ( member_int @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwise_int @ R3 @ S ) ) ).
% pairwiseI
thf(fact_3096_pairwiseI,axiom,
! [S: set_o,R3: $o > $o > $o] :
( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ S )
=> ( ( member_o @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwise_o @ R3 @ S ) ) ).
% pairwiseI
thf(fact_3097_pairwiseI,axiom,
! [S: set_Product_unit,R3: product_unit > product_unit > $o] :
( ! [X3: product_unit,Y2: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( ( member_Product_unit @ Y2 @ S )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwi7919127943033184120t_unit @ R3 @ S ) ) ).
% pairwiseI
thf(fact_3098_pairwiseD,axiom,
! [R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( pairwi2432143822833250616_nat_o @ R3 @ S )
=> ( ( member6576561426505652726_nat_o @ X @ S )
=> ( ( member6576561426505652726_nat_o @ Y @ S )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_3099_pairwiseD,axiom,
! [R3: nat > nat > $o,S: set_nat,X: nat,Y: nat] :
( ( pairwise_nat @ R3 @ S )
=> ( ( member_nat @ X @ S )
=> ( ( member_nat @ Y @ S )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_3100_pairwiseD,axiom,
! [R3: int > int > $o,S: set_int,X: int,Y: int] :
( ( pairwise_int @ R3 @ S )
=> ( ( member_int @ X @ S )
=> ( ( member_int @ Y @ S )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_3101_pairwiseD,axiom,
! [R3: $o > $o > $o,S: set_o,X: $o,Y: $o] :
( ( pairwise_o @ R3 @ S )
=> ( ( member_o @ X @ S )
=> ( ( member_o @ Y @ S )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_3102_pairwiseD,axiom,
! [R3: product_unit > product_unit > $o,S: set_Product_unit,X: product_unit,Y: product_unit] :
( ( pairwi7919127943033184120t_unit @ R3 @ S )
=> ( ( member_Product_unit @ X @ S )
=> ( ( member_Product_unit @ Y @ S )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_3103_refl__on__UNION,axiom,
! [S: set_nat,A: nat > set_nat,R2: nat > set_Pr1261947904930325089at_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( refl_on_nat @ ( A @ X3 ) @ ( R2 @ X3 ) ) )
=> ( refl_on_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ S ) ) @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ R2 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_3104_Pow__bottom,axiom,
! [B4: set_Product_unit] : ( member5877623283571906838t_unit @ bot_bo3957492148770167129t_unit @ ( pow_Product_unit @ B4 ) ) ).
% Pow_bottom
thf(fact_3105_Pow__bottom,axiom,
! [B4: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B4 ) ) ).
% Pow_bottom
thf(fact_3106_Pow__bottom,axiom,
! [B4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B4 ) ) ).
% Pow_bottom
thf(fact_3107_Pow__bottom,axiom,
! [B4: set_int] : ( member_set_int @ bot_bot_set_int @ ( pow_int @ B4 ) ) ).
% Pow_bottom
thf(fact_3108_Pow__bottom,axiom,
! [B4: set_Pr4532377907799695533_nat_o] : ( member1452989533377600726_nat_o @ bot_bo7824918357723582233_nat_o @ ( pow_Pr7829143134171777386_nat_o @ B4 ) ) ).
% Pow_bottom
thf(fact_3109_Pow__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% Pow_mono
thf(fact_3110_PowD,axiom,
! [A: set_nat,B4: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% PowD
thf(fact_3111_Un__Pow__subset,axiom,
! [A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_3112_pairwise__imageI,axiom,
! [A: set_Code_integer,F: code_integer > code_integer,P: code_integer > code_integer > $o] :
( ! [X3: code_integer,Y2: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ( member_Code_integer @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( F @ X3 )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwi4741737654671186738nteger @ P @ ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% pairwise_imageI
thf(fact_3113_pairwise__imageI,axiom,
! [A: set_nat,F: nat > set_nat,P: set_nat > set_nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( F @ X3 )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_set_nat @ P @ ( image_nat_set_nat @ F @ A ) ) ) ).
% pairwise_imageI
thf(fact_3114_pairwise__imageI,axiom,
! [A: set_nat,F: nat > nat,P: nat > nat > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( F @ X3 )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise_nat @ P @ ( image_nat_nat2 @ F @ A ) ) ) ).
% pairwise_imageI
thf(fact_3115_pairwise__imageI,axiom,
! [A: set_nat,F: nat > code_integer,P: code_integer > code_integer > $o] :
( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( F @ X3 )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwi4741737654671186738nteger @ P @ ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% pairwise_imageI
thf(fact_3116_pairwise__empty,axiom,
! [P: product_unit > product_unit > $o] : ( pairwi7919127943033184120t_unit @ P @ bot_bo3957492148770167129t_unit ) ).
% pairwise_empty
thf(fact_3117_pairwise__empty,axiom,
! [P: $o > $o > $o] : ( pairwise_o @ P @ bot_bot_set_o ) ).
% pairwise_empty
thf(fact_3118_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_3119_pairwise__empty,axiom,
! [P: int > int > $o] : ( pairwise_int @ P @ bot_bot_set_int ) ).
% pairwise_empty
thf(fact_3120_pairwise__empty,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] : ( pairwi2432143822833250616_nat_o @ P @ bot_bo7824918357723582233_nat_o ) ).
% pairwise_empty
thf(fact_3121_pairwise__subset,axiom,
! [P: nat > nat > $o,S: set_nat,T2: set_nat] :
( ( pairwise_nat @ P @ S )
=> ( ( ord_less_eq_set_nat @ T2 @ S )
=> ( pairwise_nat @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_3122_pairwise__mono,axiom,
! [P: nat > nat > $o,A: set_nat,Q2: nat > nat > $o,B4: set_nat] :
( ( pairwise_nat @ P @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( Q2 @ X3 @ Y2 ) )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( pairwise_nat @ Q2 @ B4 ) ) ) ) ).
% pairwise_mono
thf(fact_3123_pairwise__insert,axiom,
! [R2: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,S2: set_Pr4532377907799695533_nat_o] :
( ( pairwi2432143822833250616_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ S2 ) )
= ( ! [Y3: produc3658429121746597890et_nat > $o] :
( ( ( member6576561426505652726_nat_o @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwi2432143822833250616_nat_o @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_3124_pairwise__insert,axiom,
! [R2: nat > nat > $o,X: nat,S2: set_nat] :
( ( pairwise_nat @ R2 @ ( insert_nat @ X @ S2 ) )
= ( ! [Y3: nat] :
( ( ( member_nat @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise_nat @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_3125_pairwise__insert,axiom,
! [R2: int > int > $o,X: int,S2: set_int] :
( ( pairwise_int @ R2 @ ( insert_int @ X @ S2 ) )
= ( ! [Y3: int] :
( ( ( member_int @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise_int @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_3126_pairwise__insert,axiom,
! [R2: $o > $o > $o,X: $o,S2: set_o] :
( ( pairwise_o @ R2 @ ( insert_o @ X @ S2 ) )
= ( ! [Y3: $o] :
( ( ( member_o @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwise_o @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_3127_pairwise__insert,axiom,
! [R2: product_unit > product_unit > $o,X: product_unit,S2: set_Product_unit] :
( ( pairwi7919127943033184120t_unit @ R2 @ ( insert_Product_unit @ X @ S2 ) )
= ( ! [Y3: product_unit] :
( ( ( member_Product_unit @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R2 @ X @ Y3 )
& ( R2 @ Y3 @ X ) ) )
& ( pairwi7919127943033184120t_unit @ R2 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_3128_finite__UNION__then__finite,axiom,
! [B4: ( produc3658429121746597890et_nat > $o ) > set_nat,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ B4 @ A ) ) )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3129_finite__UNION__then__finite,axiom,
! [B4: nat > set_nat,A: set_nat,A2: nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3130_finite__UNION__then__finite,axiom,
! [B4: int > set_nat,A: set_int,A2: int] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ A ) ) )
=> ( ( member_int @ A2 @ A )
=> ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3131_finite__UNION__then__finite,axiom,
! [B4: $o > set_nat,A: set_o,A2: $o] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3132_finite__UNION__then__finite,axiom,
! [B4: product_unit > set_nat,A: set_Product_unit,A2: product_unit] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B4 @ A ) ) )
=> ( ( member_Product_unit @ A2 @ A )
=> ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3133_finite__UNION__then__finite,axiom,
! [B4: ( produc3658429121746597890et_nat > $o ) > set_Code_integer,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_8657708916238040603nteger @ B4 @ A ) ) )
=> ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3134_finite__UNION__then__finite,axiom,
! [B4: nat > set_Code_integer,A: set_nat,A2: nat] :
( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_2385391725447763818nteger @ B4 @ A ) ) )
=> ( ( member_nat @ A2 @ A )
=> ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3135_finite__UNION__then__finite,axiom,
! [B4: int > set_Code_integer,A: set_int,A2: int] :
( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_75573793354812870nteger @ B4 @ A ) ) )
=> ( ( member_int @ A2 @ A )
=> ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3136_finite__UNION__then__finite,axiom,
! [B4: $o > set_Code_integer,A: set_o,A2: $o] :
( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_6491638074533506790nteger @ B4 @ A ) ) )
=> ( ( member_o @ A2 @ A )
=> ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3137_finite__UNION__then__finite,axiom,
! [B4: product_unit > set_Code_integer,A: set_Product_unit,A2: product_unit] :
( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_4571654851662262875nteger @ B4 @ A ) ) )
=> ( ( member_Product_unit @ A2 @ A )
=> ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3138_image__Pow__surj,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
( ( ( image_nat_set_nat @ F @ A )
= B4 )
=> ( ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_set_nat @ B4 ) ) ) ).
% image_Pow_surj
thf(fact_3139_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ( image_nat_nat2 @ F @ A )
= B4 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B4 ) ) ) ).
% image_Pow_surj
thf(fact_3140_image__Pow__surj,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
( ( ( image_4470545334726330049nteger @ F @ A )
= B4 )
=> ( ( image_6270750266955655469nteger @ ( image_4470545334726330049nteger @ F ) @ ( pow_Code_integer @ A ) )
= ( pow_Code_integer @ B4 ) ) ) ).
% image_Pow_surj
thf(fact_3141_image__Pow__surj,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
( ( ( image_1215581382706833972nteger @ F @ A )
= B4 )
=> ( ( image_6916300684244607776nteger @ ( image_1215581382706833972nteger @ F ) @ ( pow_nat @ A ) )
= ( pow_Code_integer @ B4 ) ) ) ).
% image_Pow_surj
thf(fact_3142_Pow__insert,axiom,
! [A2: nat,A: set_nat] :
( ( pow_nat @ ( insert_nat @ A2 @ A ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A2 ) @ ( pow_nat @ A ) ) ) ) ).
% Pow_insert
thf(fact_3143_Pow__insert,axiom,
! [A2: int,A: set_int] :
( ( pow_int @ ( insert_int @ A2 @ A ) )
= ( sup_sup_set_set_int @ ( pow_int @ A ) @ ( image_524474410958335435et_int @ ( insert_int @ A2 ) @ ( pow_int @ A ) ) ) ) ).
% Pow_insert
thf(fact_3144_Pow__insert,axiom,
! [A2: $o,A: set_o] :
( ( pow_o @ ( insert_o @ A2 @ A ) )
= ( sup_sup_set_set_o @ ( pow_o @ A ) @ ( image_set_o_set_o @ ( insert_o @ A2 ) @ ( pow_o @ A ) ) ) ) ).
% Pow_insert
thf(fact_3145_Pow__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( pow_Product_unit @ ( insert_Product_unit @ A2 @ A ) )
= ( sup_su5975557051335651745t_unit @ ( pow_Product_unit @ A ) @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ A2 ) @ ( pow_Product_unit @ A ) ) ) ) ).
% Pow_insert
thf(fact_3146_pairself_Osimps,axiom,
! [F: nat > nat,A2: nat,B: nat] :
( ( pairself_nat_nat @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
= ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% pairself.simps
thf(fact_3147_pairself_Osimps,axiom,
! [F: nat > int,A2: nat,B: nat] :
( ( pairself_nat_int @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
= ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% pairself.simps
thf(fact_3148_pairself_Osimps,axiom,
! [F: int > nat,A2: int,B: int] :
( ( pairself_int_nat @ F @ ( product_Pair_int_int @ A2 @ B ) )
= ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% pairself.simps
thf(fact_3149_pairself_Osimps,axiom,
! [F: int > int,A2: int,B: int] :
( ( pairself_int_int @ F @ ( product_Pair_int_int @ A2 @ B ) )
= ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% pairself.simps
thf(fact_3150_pairself_Oelims,axiom,
! [X: nat > nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ( pairself_nat_nat @ X @ Xa )
= Y )
=> ~ ! [A4: nat,B2: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A4 @ B2 ) )
=> ( Y
!= ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% pairself.elims
thf(fact_3151_pairself_Oelims,axiom,
! [X: nat > int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
( ( ( pairself_nat_int @ X @ Xa )
= Y )
=> ~ ! [A4: nat,B2: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A4 @ B2 ) )
=> ( Y
!= ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% pairself.elims
thf(fact_3152_pairself_Oelims,axiom,
! [X: int > nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
( ( ( pairself_int_nat @ X @ Xa )
= Y )
=> ~ ! [A4: int,B2: int] :
( ( Xa
= ( product_Pair_int_int @ A4 @ B2 ) )
=> ( Y
!= ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% pairself.elims
thf(fact_3153_pairself_Oelims,axiom,
! [X: int > int,Xa: product_prod_int_int,Y: product_prod_int_int] :
( ( ( pairself_int_int @ X @ Xa )
= Y )
=> ~ ! [A4: int,B2: int] :
( ( Xa
= ( product_Pair_int_int @ A4 @ B2 ) )
=> ( Y
!= ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% pairself.elims
thf(fact_3154_pairwise__singleton,axiom,
! [P: product_unit > product_unit > $o,A: product_unit] : ( pairwi7919127943033184120t_unit @ P @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% pairwise_singleton
thf(fact_3155_pairwise__singleton,axiom,
! [P: $o > $o > $o,A: $o] : ( pairwise_o @ P @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% pairwise_singleton
thf(fact_3156_pairwise__singleton,axiom,
! [P: nat > nat > $o,A: nat] : ( pairwise_nat @ P @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_3157_pairwise__singleton,axiom,
! [P: int > int > $o,A: int] : ( pairwise_int @ P @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% pairwise_singleton
thf(fact_3158_pairwise__singleton,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,A: produc3658429121746597890et_nat > $o] : ( pairwi2432143822833250616_nat_o @ P @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% pairwise_singleton
thf(fact_3159_finite__Sup__in,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( member_set_nat @ Y2 @ A )
=> ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ A ) ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_3160_finite__Sup__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( member_o @ ( sup_sup_o @ X3 @ Y2 ) @ A ) ) )
=> ( member_o @ ( complete_Sup_Sup_o @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_3161_finite__Sup__in,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) @ A ) ) )
=> ( member6576561426505652726_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_3162_finite__Sup__in,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y2: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( member_Product_unit @ Y2 @ A )
=> ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ A ) ) )
=> ( member_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_3163_Sup__fin__Sup,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ A )
= ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_fin_Sup
thf(fact_3164_Sup__fin__Sup,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( lattic7320199455484906628_nat_o @ A )
= ( comple4225350640907012154_nat_o @ A ) ) ) ) ).
% Sup_fin_Sup
thf(fact_3165_Sup__fin__Sup,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A )
= ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% Sup_fin_Sup
thf(fact_3166_insert__partition,axiom,
! [X: set_Product_unit,F2: set_set_Product_unit] :
( ~ ( member5877623283571906838t_unit @ X @ F2 )
=> ( ! [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ ( insert8986028188087627965t_unit @ X @ F2 ) )
=> ! [Xa3: set_Product_unit] :
( ( member5877623283571906838t_unit @ Xa3 @ ( insert8986028188087627965t_unit @ X @ F2 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_in4660618365625256667t_unit @ X3 @ Xa3 )
= bot_bo3957492148770167129t_unit ) ) ) )
=> ( ( inf_in4660618365625256667t_unit @ X @ ( comple3349607820504768346t_unit @ F2 ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% insert_partition
thf(fact_3167_insert__partition,axiom,
! [X: set_o,F2: set_set_o] :
( ~ ( member_set_o @ X @ F2 )
=> ( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( insert_set_o @ X @ F2 ) )
=> ! [Xa3: set_o] :
( ( member_set_o @ Xa3 @ ( insert_set_o @ X @ F2 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_o @ X3 @ Xa3 )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F2 ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_3168_insert__partition,axiom,
! [X: set_nat,F2: set_set_nat] :
( ~ ( member_set_nat @ X @ F2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( insert_set_nat @ X @ F2 ) )
=> ! [Xa3: set_nat] :
( ( member_set_nat @ Xa3 @ ( insert_set_nat @ X @ F2 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_nat @ X3 @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F2 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_3169_insert__partition,axiom,
! [X: set_int,F2: set_set_int] :
( ~ ( member_set_int @ X @ F2 )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ ( insert_set_int @ X @ F2 ) )
=> ! [Xa3: set_int] :
( ( member_set_int @ Xa3 @ ( insert_set_int @ X @ F2 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_int @ X3 @ Xa3 )
= bot_bot_set_int ) ) ) )
=> ( ( inf_inf_set_int @ X @ ( comple3221217463730067765et_int @ F2 ) )
= bot_bot_set_int ) ) ) ).
% insert_partition
thf(fact_3170_insert__partition,axiom,
! [X: set_Pr4532377907799695533_nat_o,F2: set_se1353816476878259085_nat_o] :
( ~ ( member1452989533377600726_nat_o @ X @ F2 )
=> ( ! [X3: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X3 @ ( insert2507855216428400253_nat_o @ X @ F2 ) )
=> ! [Xa3: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ Xa3 @ ( insert2507855216428400253_nat_o @ X @ F2 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_in1906310914598751387_nat_o @ X3 @ Xa3 )
= bot_bo7824918357723582233_nat_o ) ) ) )
=> ( ( inf_in1906310914598751387_nat_o @ X @ ( comple3382087578465172250_nat_o @ F2 ) )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% insert_partition
thf(fact_3171_image__Pow__mono,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_set_nat @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_3172_image__Pow__mono,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
=> ( ord_le1914454125413604393nteger @ ( image_6270750266955655469nteger @ ( image_4470545334726330049nteger @ F ) @ ( pow_Code_integer @ A ) ) @ ( pow_Code_integer @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_3173_image__Pow__mono,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
=> ( ord_le1914454125413604393nteger @ ( image_6916300684244607776nteger @ ( image_1215581382706833972nteger @ F ) @ ( pow_nat @ A ) ) @ ( pow_Code_integer @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_3174_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_3175_pairwise__alt,axiom,
( pairwi7919127943033184120t_unit
= ( ^ [R: product_unit > product_unit > $o,S6: set_Product_unit] :
! [X2: product_unit] :
( ( member_Product_unit @ X2 @ S6 )
=> ! [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ ( minus_6452836326544984404t_unit @ S6 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_3176_pairwise__alt,axiom,
( pairwise_o
= ( ^ [R: $o > $o > $o,S6: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ S6 )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ ( minus_minus_set_o @ S6 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_3177_pairwise__alt,axiom,
( pairwise_int
= ( ^ [R: int > int > $o,S6: set_int] :
! [X2: int] :
( ( member_int @ X2 @ S6 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ ( minus_minus_set_int @ S6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_3178_pairwise__alt,axiom,
( pairwi2432143822833250616_nat_o
= ( ^ [R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S6: set_Pr4532377907799695533_nat_o] :
! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ S6 )
=> ! [Y3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Y3 @ ( minus_1801376950450012436_nat_o @ S6 @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_3179_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ S6 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( minus_minus_set_nat @ S6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( R @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_3180_Sup__insert,axiom,
! [A2: $o,A: set_o] :
( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ A ) )
= ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_insert
thf(fact_3181_Sup__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ A ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_insert
thf(fact_3182_Sup__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ A ) )
= ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ A ) ) ) ).
% Sup_insert
thf(fact_3183_ccpo__Sup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_3184_ccpo__Sup__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o] :
( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_3185_ccpo__Sup__singleton,axiom,
! [X: product_unit] :
( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_3186_cSup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cSup_singleton
thf(fact_3187_cSup__singleton,axiom,
! [X: int] :
( ( complete_Sup_Sup_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% cSup_singleton
thf(fact_3188_cSup__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o] :
( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X ) ).
% cSup_singleton
thf(fact_3189_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_3190_cSup__singleton,axiom,
! [X: product_unit] :
( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% cSup_singleton
thf(fact_3191_Sup__empty,axiom,
( ( comple3349607820504768346t_unit @ bot_bo6284674557190688313t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Sup_empty
thf(fact_3192_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_3193_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_3194_Sup__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Sup_empty
thf(fact_3195_Sup__empty,axiom,
( ( comple3382087578465172250_nat_o @ bot_bo3736544561982865913_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Sup_empty
thf(fact_3196_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_3197_Sup__empty,axiom,
( ( comple4225350640907012154_nat_o @ bot_bo7824918357723582233_nat_o )
= bot_bo6993851861578111033_nat_o ) ).
% Sup_empty
thf(fact_3198_Sup__empty,axiom,
( ( comple4687483117567863418t_unit @ bot_bo3957492148770167129t_unit )
= bot_bot_Product_unit ) ).
% Sup_empty
thf(fact_3199_Sup__bot__conv_I1_J,axiom,
! [A: set_set_Product_unit] :
( ( ( comple3349607820504768346t_unit @ A )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: set_Product_unit] :
( ( member5877623283571906838t_unit @ X2 @ A )
=> ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3200_Sup__bot__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3201_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3202_Sup__bot__conv_I1_J,axiom,
! [A: set_set_int] :
( ( ( comple3221217463730067765et_int @ A )
= bot_bot_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( X2 = bot_bot_set_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3203_Sup__bot__conv_I1_J,axiom,
! [A: set_se1353816476878259085_nat_o] :
( ( ( comple3382087578465172250_nat_o @ A )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X2 @ A )
=> ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3204_Sup__bot__conv_I1_J,axiom,
! [A: set_Product_unit] :
( ( ( comple4687483117567863418t_unit @ A )
= bot_bot_Product_unit )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( X2 = bot_bot_Product_unit ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3205_Sup__bot__conv_I2_J,axiom,
! [A: set_set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( comple3349607820504768346t_unit @ A ) )
= ( ! [X2: set_Product_unit] :
( ( member5877623283571906838t_unit @ X2 @ A )
=> ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3206_Sup__bot__conv_I2_J,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3207_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3208_Sup__bot__conv_I2_J,axiom,
! [A: set_set_int] :
( ( bot_bot_set_int
= ( comple3221217463730067765et_int @ A ) )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( X2 = bot_bot_set_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3209_Sup__bot__conv_I2_J,axiom,
! [A: set_se1353816476878259085_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( comple3382087578465172250_nat_o @ A ) )
= ( ! [X2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X2 @ A )
=> ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3210_Sup__bot__conv_I2_J,axiom,
! [A: set_Product_unit] :
( ( bot_bot_Product_unit
= ( comple4687483117567863418t_unit @ A ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( X2 = bot_bot_Product_unit ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3211_cSup__eq__Sup__fin,axiom,
! [X4: set_o] :
( ( finite_finite_o @ X4 )
=> ( ( X4 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o @ X4 )
= ( lattic1508158080041050831_fin_o @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_3212_cSup__eq__Sup__fin,axiom,
! [X4: set_int] :
( ( finite_finite_int @ X4 )
=> ( ( X4 != bot_bot_set_int )
=> ( ( complete_Sup_Sup_int @ X4 )
= ( lattic1091506334969745077in_int @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_3213_cSup__eq__Sup__fin,axiom,
! [X4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ X4 )
=> ( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ( comple4225350640907012154_nat_o @ X4 )
= ( lattic7320199455484906628_nat_o @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_3214_cSup__eq__Sup__fin,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X4 )
= ( lattic1093996805478795353in_nat @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_3215_cSup__eq__Sup__fin,axiom,
! [X4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit @ X4 )
= ( lattic5294303975357428420t_unit @ X4 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_3216_Union__image__insert,axiom,
! [F: nat > set_nat,A2: nat,B4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A2 @ B4 ) ) )
= ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ).
% Union_image_insert
thf(fact_3217_Union__image__insert,axiom,
! [F: int > set_nat,A2: int,B4: set_int] :
( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( insert_int @ A2 @ B4 ) ) )
= ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ B4 ) ) ) ) ).
% Union_image_insert
thf(fact_3218_Union__image__insert,axiom,
! [F: $o > set_nat,A2: $o,B4: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ B4 ) ) )
= ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ).
% Union_image_insert
thf(fact_3219_Union__image__insert,axiom,
! [F: product_unit > set_nat,A2: product_unit,B4: set_Product_unit] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ B4 ) ) )
= ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ).
% Union_image_insert
thf(fact_3220_Union__image__empty,axiom,
! [A: set_nat,F: product_unit > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ bot_bo3957492148770167129t_unit ) ) )
= A ) ).
% Union_image_empty
thf(fact_3221_Union__image__empty,axiom,
! [A: set_nat,F: $o > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
= A ) ).
% Union_image_empty
thf(fact_3222_Union__image__empty,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_3223_Union__image__empty,axiom,
! [A: set_nat,F: int > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ bot_bot_set_int ) ) )
= A ) ).
% Union_image_empty
thf(fact_3224_Union__image__empty,axiom,
! [A: set_nat,F: ( produc3658429121746597890et_nat > $o ) > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ F @ bot_bo7824918357723582233_nat_o ) ) )
= A ) ).
% Union_image_empty
thf(fact_3225_Sup__SUP__eq,axiom,
( comple3590721164184202819at_o_o
= ( ^ [S6: set_Pr6409761941060870310at_o_o,X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ ( comple3382087578465172250_nat_o @ ( image_8639947063239330806_nat_o @ collec939566748876313656_nat_o @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3226_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S6: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3227_Sup__SUP__eq,axiom,
( comple6496622788309502864_int_o
= ( ^ [S6: set_int_o,X2: int] : ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_o_set_int @ collect_int @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3228_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S6: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3229_Sup__SUP__eq,axiom,
( comple4773641153059738115unit_o
= ( ^ [S6: set_Product_unit_o,X2: product_unit] : ( member_Product_unit @ X2 @ ( comple3349607820504768346t_unit @ ( image_5138372075374782070t_unit @ collect_Product_unit @ S6 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3230_empty__Union__conv,axiom,
! [A: set_set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( comple3349607820504768346t_unit @ A ) )
= ( ! [X2: set_Product_unit] :
( ( member5877623283571906838t_unit @ X2 @ A )
=> ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% empty_Union_conv
thf(fact_3231_empty__Union__conv,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_3232_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_3233_empty__Union__conv,axiom,
! [A: set_set_int] :
( ( bot_bot_set_int
= ( comple3221217463730067765et_int @ A ) )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( X2 = bot_bot_set_int ) ) ) ) ).
% empty_Union_conv
thf(fact_3234_empty__Union__conv,axiom,
! [A: set_se1353816476878259085_nat_o] :
( ( bot_bo7824918357723582233_nat_o
= ( comple3382087578465172250_nat_o @ A ) )
= ( ! [X2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X2 @ A )
=> ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% empty_Union_conv
thf(fact_3235_Union__empty__conv,axiom,
! [A: set_set_Product_unit] :
( ( ( comple3349607820504768346t_unit @ A )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: set_Product_unit] :
( ( member5877623283571906838t_unit @ X2 @ A )
=> ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% Union_empty_conv
thf(fact_3236_Union__empty__conv,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ A )
=> ( X2 = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_3237_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_3238_Union__empty__conv,axiom,
! [A: set_set_int] :
( ( ( comple3221217463730067765et_int @ A )
= bot_bot_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( X2 = bot_bot_set_int ) ) ) ) ).
% Union_empty_conv
thf(fact_3239_Union__empty__conv,axiom,
! [A: set_se1353816476878259085_nat_o] :
( ( ( comple3382087578465172250_nat_o @ A )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X2 @ A )
=> ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% Union_empty_conv
thf(fact_3240_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V3: $o] :
( ( member_o @ V3 @ A )
=> ( ord_less_eq_o @ U @ V3 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_3241_less__eq__Sup,axiom,
! [A: set_Pr4532377907799695533_nat_o,U: produc3658429121746597890et_nat > $o] :
( ! [V3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ V3 @ A )
=> ( ord_le729326519192465773_nat_o @ U @ V3 ) )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ord_le729326519192465773_nat_o @ U @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_3242_less__eq__Sup,axiom,
! [A: set_filter_nat,U: filter_nat] :
( ! [V3: filter_nat] :
( ( member_filter_nat @ V3 @ A )
=> ( ord_le2510731241096832064er_nat @ U @ V3 ) )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ord_le2510731241096832064er_nat @ U @ ( comple2454491259644266675er_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_3243_less__eq__Sup,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V3: set_nat] :
( ( member_set_nat @ V3 @ A )
=> ( ord_less_eq_set_nat @ U @ V3 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_3244_less__eq__Sup,axiom,
! [A: set_Product_unit,U: product_unit] :
( ! [V3: product_unit] :
( ( member_Product_unit @ V3 @ A )
=> ( ord_le3221252021190050221t_unit @ U @ V3 ) )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_3245_cSup__least,axiom,
! [X4: set_o,Z: $o] :
( ( X4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X4 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3246_cSup__least,axiom,
! [X4: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X4 )
=> ( ord_le729326519192465773_nat_o @ X3 @ Z ) )
=> ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3247_cSup__least,axiom,
! [X4: set_filter_nat,Z: filter_nat] :
( ( X4 != bot_bo498966703094740906er_nat )
=> ( ! [X3: filter_nat] :
( ( member_filter_nat @ X3 @ X4 )
=> ( ord_le2510731241096832064er_nat @ X3 @ Z ) )
=> ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3248_cSup__least,axiom,
! [X4: set_set_nat,Z: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X4 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3249_cSup__least,axiom,
! [X4: set_int,Z: int] :
( ( X4 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ( ord_less_eq_int @ X3 @ Z ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3250_cSup__least,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3251_cSup__least,axiom,
! [X4: set_Product_unit,Z: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ X3 @ Z ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ X4 ) @ Z ) ) ) ).
% cSup_least
thf(fact_3252_cSup__eq__non__empty,axiom,
! [X4: set_o,A2: $o] :
( ( X4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X4 )
=> ( ord_less_eq_o @ X3 @ A2 ) )
=> ( ! [Y2: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X4 )
=> ( ord_less_eq_o @ X6 @ Y2 ) )
=> ( ord_less_eq_o @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3253_cSup__eq__non__empty,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X4 )
=> ( ord_le729326519192465773_nat_o @ X3 @ A2 ) )
=> ( ! [Y2: produc3658429121746597890et_nat > $o] :
( ! [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ X4 )
=> ( ord_le729326519192465773_nat_o @ X6 @ Y2 ) )
=> ( ord_le729326519192465773_nat_o @ A2 @ Y2 ) )
=> ( ( comple4225350640907012154_nat_o @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3254_cSup__eq__non__empty,axiom,
! [X4: set_filter_nat,A2: filter_nat] :
( ( X4 != bot_bo498966703094740906er_nat )
=> ( ! [X3: filter_nat] :
( ( member_filter_nat @ X3 @ X4 )
=> ( ord_le2510731241096832064er_nat @ X3 @ A2 ) )
=> ( ! [Y2: filter_nat] :
( ! [X6: filter_nat] :
( ( member_filter_nat @ X6 @ X4 )
=> ( ord_le2510731241096832064er_nat @ X6 @ Y2 ) )
=> ( ord_le2510731241096832064er_nat @ A2 @ Y2 ) )
=> ( ( comple2454491259644266675er_nat @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3255_cSup__eq__non__empty,axiom,
! [X4: set_set_nat,A2: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X4 )
=> ( ord_less_eq_set_nat @ X3 @ A2 ) )
=> ( ! [Y2: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ X4 )
=> ( ord_less_eq_set_nat @ X6 @ Y2 ) )
=> ( ord_less_eq_set_nat @ A2 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3256_cSup__eq__non__empty,axiom,
! [X4: set_int,A2: int] :
( ( X4 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ( ord_less_eq_int @ X3 @ A2 ) )
=> ( ! [Y2: int] :
( ! [X6: int] :
( ( member_int @ X6 @ X4 )
=> ( ord_less_eq_int @ X6 @ Y2 ) )
=> ( ord_less_eq_int @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_int @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3257_cSup__eq__non__empty,axiom,
! [X4: set_nat,A2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ord_less_eq_nat @ X3 @ A2 ) )
=> ( ! [Y2: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X4 )
=> ( ord_less_eq_nat @ X6 @ Y2 ) )
=> ( ord_less_eq_nat @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3258_cSup__eq__non__empty,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ X3 @ A2 ) )
=> ( ! [Y2: product_unit] :
( ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ X4 )
=> ( ord_le3221252021190050221t_unit @ X6 @ Y2 ) )
=> ( ord_le3221252021190050221t_unit @ A2 @ Y2 ) )
=> ( ( comple4687483117567863418t_unit @ X4 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3259_SUP__eq__const,axiom,
! [I: set_nat,F: nat > set_nat,X: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_3260_SUP__eq__const,axiom,
! [I: set_Product_unit,F: product_unit > product_unit,X: product_unit] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_3261_SUP__eq__const,axiom,
! [I: set_o,F: $o > product_unit,X: product_unit] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_3262_SUP__eq__const,axiom,
! [I: set_nat,F: nat > product_unit,X: product_unit] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_3263_SUP__eq__const,axiom,
! [I: set_int,F: int > product_unit,X: product_unit] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_3264_SUP__eq__const,axiom,
! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > product_unit,X: product_unit] :
( ( I != bot_bo7824918357723582233_nat_o )
=> ( ! [I3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8451000719036923781t_unit @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_3265_Union__disjoint,axiom,
! [C5: set_set_Product_unit,A: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( comple3349607820504768346t_unit @ C5 ) @ A )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: set_Product_unit] :
( ( member5877623283571906838t_unit @ X2 @ C5 )
=> ( ( inf_in4660618365625256667t_unit @ X2 @ A )
= bot_bo3957492148770167129t_unit ) ) ) ) ).
% Union_disjoint
thf(fact_3266_Union__disjoint,axiom,
! [C5: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C5 ) @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C5 )
=> ( ( inf_inf_set_o @ X2 @ A )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_3267_Union__disjoint,axiom,
! [C5: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C5 ) @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C5 )
=> ( ( inf_inf_set_nat @ X2 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_3268_Union__disjoint,axiom,
! [C5: set_set_int,A: set_int] :
( ( ( inf_inf_set_int @ ( comple3221217463730067765et_int @ C5 ) @ A )
= bot_bot_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ C5 )
=> ( ( inf_inf_set_int @ X2 @ A )
= bot_bot_set_int ) ) ) ) ).
% Union_disjoint
thf(fact_3269_Union__disjoint,axiom,
! [C5: set_se1353816476878259085_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) @ A )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X2 @ C5 )
=> ( ( inf_in1906310914598751387_nat_o @ X2 @ A )
= bot_bo7824918357723582233_nat_o ) ) ) ) ).
% Union_disjoint
thf(fact_3270_Union__empty,axiom,
( ( comple3349607820504768346t_unit @ bot_bo6284674557190688313t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Union_empty
thf(fact_3271_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_3272_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_3273_Union__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Union_empty
thf(fact_3274_Union__empty,axiom,
( ( comple3382087578465172250_nat_o @ bot_bo3736544561982865913_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Union_empty
thf(fact_3275_SUP__eq__iff,axiom,
! [I: set_Product_unit,C: product_unit,F: product_unit > product_unit] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
= C )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3276_SUP__eq__iff,axiom,
! [I: set_o,C: product_unit,F: $o > product_unit] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3277_SUP__eq__iff,axiom,
! [I: set_nat,C: product_unit,F: nat > product_unit] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3278_SUP__eq__iff,axiom,
! [I: set_int,C: product_unit,F: int > product_unit] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
= C )
= ( ! [X2: int] :
( ( member_int @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3279_SUP__eq__iff,axiom,
! [I: set_Product_unit,C: filter_nat,F: product_unit > filter_nat] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple2454491259644266675er_nat @ ( image_1853478764106424488er_nat @ F @ I ) )
= C )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3280_SUP__eq__iff,axiom,
! [I: set_o,C: filter_nat,F: $o > filter_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple2454491259644266675er_nat @ ( image_o_filter_nat @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3281_SUP__eq__iff,axiom,
! [I: set_nat,C: filter_nat,F: nat > filter_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3282_SUP__eq__iff,axiom,
! [I: set_int,C: filter_nat,F: int > filter_nat] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple2454491259644266675er_nat @ ( image_int_filter_nat @ F @ I ) )
= C )
= ( ! [X2: int] :
( ( member_int @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3283_SUP__eq__iff,axiom,
! [I: set_Product_unit,C: set_nat,F: product_unit > set_nat] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
= C )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3284_SUP__eq__iff,axiom,
! [I: set_o,C: set_nat,F: $o > set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3285_cSUP__least,axiom,
! [A: set_Product_unit,F: product_unit > int,M: int] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3286_cSUP__least,axiom,
! [A: set_o,F: $o > int,M: int] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3287_cSUP__least,axiom,
! [A: set_nat,F: nat > int,M: int] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3288_cSUP__least,axiom,
! [A: set_int,F: int > int,M: int] :
( ( A != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3289_cSUP__least,axiom,
! [A: set_Product_unit,F: product_unit > nat,M: nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3290_cSUP__least,axiom,
! [A: set_o,F: $o > nat,M: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3291_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3292_cSUP__least,axiom,
! [A: set_int,F: int > nat,M: nat] :
( ( A != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3293_cSUP__least,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,M: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ M ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3294_cSUP__least,axiom,
! [A: set_o,F: $o > product_unit,M: product_unit] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ M ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_3295_Sup__inf__eq__bot__iff,axiom,
! [B4: set_set_Product_unit,A2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( comple3349607820504768346t_unit @ B4 ) @ A2 )
= bot_bo3957492148770167129t_unit )
= ( ! [X2: set_Product_unit] :
( ( member5877623283571906838t_unit @ X2 @ B4 )
=> ( ( inf_in4660618365625256667t_unit @ X2 @ A2 )
= bot_bo3957492148770167129t_unit ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3296_Sup__inf__eq__bot__iff,axiom,
! [B4: set_set_o,A2: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B4 ) @ A2 )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ B4 )
=> ( ( inf_inf_set_o @ X2 @ A2 )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3297_Sup__inf__eq__bot__iff,axiom,
! [B4: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B4 )
=> ( ( inf_inf_set_nat @ X2 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3298_Sup__inf__eq__bot__iff,axiom,
! [B4: set_set_int,A2: set_int] :
( ( ( inf_inf_set_int @ ( comple3221217463730067765et_int @ B4 ) @ A2 )
= bot_bot_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B4 )
=> ( ( inf_inf_set_int @ X2 @ A2 )
= bot_bot_set_int ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3299_Sup__inf__eq__bot__iff,axiom,
! [B4: set_se1353816476878259085_nat_o,A2: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ ( comple3382087578465172250_nat_o @ B4 ) @ A2 )
= bot_bo7824918357723582233_nat_o )
= ( ! [X2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X2 @ B4 )
=> ( ( inf_in1906310914598751387_nat_o @ X2 @ A2 )
= bot_bo7824918357723582233_nat_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3300_Sup__inf__eq__bot__iff,axiom,
! [B4: set_Product_unit,A2: product_unit] :
( ( ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ B4 ) @ A2 )
= bot_bot_Product_unit )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ B4 )
=> ( ( inf_inf_Product_unit @ X2 @ A2 )
= bot_bot_Product_unit ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3301_UNION__fun__upd,axiom,
! [A: product_unit > set_Product_unit,I2: product_unit,B4: set_Product_unit,J2: set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ ( fun_up488859312337516701t_unit @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_Product_unit @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3302_UNION__fun__upd,axiom,
! [A: product_unit > set_o,I2: product_unit,B4: set_o,J2: set_Product_unit] :
( ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ ( fun_up1762901246852280360_set_o @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_o @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3303_UNION__fun__upd,axiom,
! [A: product_unit > set_nat,I2: product_unit,B4: set_nat,J2: set_Product_unit] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ ( fun_up4569834200710200790et_nat @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_nat @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_3304_UNION__fun__upd,axiom,
! [A: product_unit > set_int,I2: product_unit,B4: set_int,J2: set_Product_unit] :
( ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ ( fun_up391983181201004082et_int @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_int @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_int ) ) ) ).
% UNION_fun_upd
thf(fact_3305_UNION__fun__upd,axiom,
! [A: $o > set_Product_unit,I2: $o,B4: set_Product_unit,J2: set_o] :
( ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ ( fun_up4718931970374059986t_unit @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_Product_unit @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3306_UNION__fun__upd,axiom,
! [A: $o > set_o,I2: $o,B4: set_o,J2: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3307_UNION__fun__upd,axiom,
! [A: $o > set_nat,I2: $o,B4: set_nat,J2: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( fun_upd_o_set_nat @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_nat @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_3308_UNION__fun__upd,axiom,
! [A: $o > set_int,I2: $o,B4: set_int,J2: set_o] :
( ( comple3221217463730067765et_int @ ( image_o_set_int @ ( fun_upd_o_set_int @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_o_set_int @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_int @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_int ) ) ) ).
% UNION_fun_upd
thf(fact_3309_UNION__fun__upd,axiom,
! [A: int > set_Product_unit,I2: int,B4: set_Product_unit,J2: set_int] :
( ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ ( fun_up5054257004045450754t_unit @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ A @ ( minus_minus_set_int @ J2 @ ( insert_int @ I2 @ bot_bot_set_int ) ) ) ) @ ( if_set_Product_unit @ ( member_int @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3310_UNION__fun__upd,axiom,
! [A: int > set_o,I2: int,B4: set_o,J2: set_int] :
( ( comple90263536869209701_set_o @ ( image_int_set_o @ ( fun_upd_int_set_o @ A @ I2 @ B4 ) @ J2 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_int_set_o @ A @ ( minus_minus_set_int @ J2 @ ( insert_int @ I2 @ bot_bot_set_int ) ) ) ) @ ( if_set_o @ ( member_int @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3311_cSUP__union,axiom,
! [A: set_Product_unit,F: product_unit > nat,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ B4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3312_cSUP__union,axiom,
! [A: set_o,F: $o > nat,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ B4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3313_cSUP__union,axiom,
! [A: set_nat,F: nat > nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ B4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3314_cSUP__union,axiom,
! [A: set_int,F: int > nat,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ B4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3315_cSUP__union,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3316_cSUP__union,axiom,
! [A: set_o,F: $o > product_unit,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ B4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3317_cSUP__union,axiom,
! [A: set_nat,F: nat > product_unit,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3318_cSUP__union,axiom,
! [A: set_int,F: int > product_unit,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
=> ( ( B4 != bot_bot_set_int )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3319_cSUP__union,axiom,
! [A: set_Product_unit,F: product_unit > set_nat,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3320_cSUP__union,axiom,
! [A: set_o,F: $o > set_nat,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A ) )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ B4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_3321_pair__in__swap__image,axiom,
! [Y: set_nat,X: heap_e7401611519738050253t_unit,A: set_Pr3948176798113811640et_nat] :
( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ Y @ X ) @ ( image_1893317993577548229t_unit @ produc6300697153377358172et_nat @ A ) )
= ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3322_pair__in__swap__image,axiom,
! [Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr3286484037609594932et_nat] :
( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ Y @ X ) @ ( image_6229134628169569989_nat_o @ produc3251101204053596376et_nat @ A ) )
= ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3323_pair__in__swap__image,axiom,
! [Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr8536935166611901872et_nat] :
( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ Y @ X ) @ ( image_4071802983491761093_nat_o @ produc1151471490888544852et_nat @ A ) )
= ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3324_pair__in__swap__image,axiom,
! [Y: heap_e7401611519738050253t_unit,X: set_nat,A: set_Pr4071100604818519410t_unit] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y @ X ) @ ( image_5838825082199883205et_nat @ produc1264744871843189710t_unit @ A ) )
= ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3325_pair__in__swap__image,axiom,
! [Y: nat,X: nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3326_pair__in__swap__image,axiom,
! [Y: int,X: int,A: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( image_2653370878348428101nt_int @ product_swap_int_int @ A ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3327_pair__in__swap__image,axiom,
! [Y: produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat,A: set_Pr4208978187861016310_nat_o] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y @ X ) @ ( image_4501728140019976901et_nat @ produc1078497009909708626_nat_o @ A ) )
= ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3328_pair__in__swap__image,axiom,
! [Y: produc3658429121746597890et_nat > $o,X: produc3925858234332021118et_nat,A: set_Pr7705059561495379578_nat_o] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y @ X ) @ ( image_7255662321850427333et_nat @ produc2580581461503396054_nat_o @ A ) )
= ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_3329_cSUP__insert,axiom,
! [A: set_Product_unit,F: product_unit > nat,A2: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
= ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3330_cSUP__insert,axiom,
! [A: set_o,F: $o > nat,A2: $o] :
( ( A != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ ( insert_o @ A2 @ A ) ) )
= ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3331_cSUP__insert,axiom,
! [A: set_nat,F: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ A ) ) )
= ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3332_cSUP__insert,axiom,
! [A: set_int,F: int > nat,A2: int] :
( ( A != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
=> ( ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ ( insert_int @ A2 @ A ) ) )
= ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3333_cSUP__insert,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,A2: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A2 @ A ) ) )
= ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3334_cSUP__insert,axiom,
! [A: set_o,F: $o > product_unit,A2: $o] :
( ( A != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ ( insert_o @ A2 @ A ) ) )
= ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3335_cSUP__insert,axiom,
! [A: set_nat,F: nat > product_unit,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ A ) ) )
= ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3336_cSUP__insert,axiom,
! [A: set_int,F: int > product_unit,A2: int] :
( ( A != bot_bot_set_int )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ A ) ) )
= ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3337_cSUP__insert,axiom,
! [A: set_Product_unit,F: product_unit > set_nat,A2: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
=> ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
= ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3338_cSUP__insert,axiom,
! [A: set_o,F: $o > set_nat,A2: $o] :
( ( A != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ A ) ) )
= ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3339_cSup__inter__less__eq,axiom,
! [A: set_o,B4: set_o] :
( ( condit5488710616941104124bove_o @ A )
=> ( ( condit5488710616941104124bove_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A @ B4 ) ) @ ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3340_cSup__inter__less__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( condit4352657709669162033_nat_o @ A )
=> ( ( condit4352657709669162033_nat_o @ B4 )
=> ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
!= bot_bo7824918357723582233_nat_o )
=> ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) @ ( sup_su5453871518329203617_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3341_cSup__inter__less__eq,axiom,
! [A: set_filter_nat,B4: set_filter_nat] :
( ( condit4029664723990530684er_nat @ A )
=> ( ( condit4029664723990530684er_nat @ B4 )
=> ( ( ( inf_in7395780696554941096er_nat @ A @ B4 )
!= bot_bo498966703094740906er_nat )
=> ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( inf_in7395780696554941096er_nat @ A @ B4 ) ) @ ( sup_sup_filter_nat @ ( comple2454491259644266675er_nat @ A ) @ ( comple2454491259644266675er_nat @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3342_cSup__inter__less__eq,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( condit5477540289124974626et_nat @ A )
=> ( ( condit5477540289124974626et_nat @ B4 )
=> ( ( ( inf_inf_set_set_nat @ A @ B4 )
!= bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) @ ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3343_cSup__inter__less__eq,axiom,
! [A: set_int,B4: set_int] :
( ( condit2212336002400062152ve_int @ A )
=> ( ( condit2212336002400062152ve_int @ B4 )
=> ( ( ( inf_inf_set_int @ A @ B4 )
!= bot_bot_set_int )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( inf_inf_set_int @ A @ B4 ) ) @ ( sup_sup_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3344_cSup__inter__less__eq,axiom,
! [A: set_nat,B4: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ( ( condit2214826472909112428ve_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3345_cSup__inter__less__eq,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( condit2080475194861589617t_unit @ A )
=> ( ( condit2080475194861589617t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
!= bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) @ ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_3346_cSUP__subset__mono,axiom,
! [A: set_Product_unit,G: product_unit > int,B4: set_Product_unit,F: product_unit > int] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ G @ B4 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3347_cSUP__subset__mono,axiom,
! [A: set_o,G: $o > int,B4: set_o,F: $o > int] :
( ( A != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_o_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3348_cSUP__subset__mono,axiom,
! [A: set_int,G: int > int,B4: set_int,F: int > int] :
( ( A != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_int_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3349_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > int,B4: set_nat,F: nat > int] :
( ( A != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3350_cSUP__subset__mono,axiom,
! [A: set_Product_unit,G: product_unit > nat,B4: set_Product_unit,F: product_unit > nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ G @ B4 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3351_cSUP__subset__mono,axiom,
! [A: set_o,G: $o > nat,B4: set_o,F: $o > nat] :
( ( A != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3352_cSUP__subset__mono,axiom,
! [A: set_int,G: int > nat,B4: set_int,F: int > nat] :
( ( A != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3353_cSUP__subset__mono,axiom,
! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3354_cSUP__subset__mono,axiom,
! [A: set_Product_unit,G: product_unit > product_unit,B4: set_Product_unit,F: product_unit > product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3355_cSUP__subset__mono,axiom,
! [A: set_o,G: $o > product_unit,B4: set_o,F: $o > product_unit] :
( ( A != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3356_Image__Int__eq,axiom,
! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
( ( single5507084439971879973at_nat @ ( converse_nat_nat @ R3 ) )
=> ( ( image_nat_nat @ R3 @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ) ).
% Image_Int_eq
thf(fact_3357_image__update,axiom,
! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer,N2: code_integer] :
( ~ ( member_Code_integer @ X @ A )
=> ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ N2 ) @ A )
= ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% image_update
thf(fact_3358_image__update,axiom,
! [X: nat,A: set_nat,F: nat > set_nat,N2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ X @ N2 ) @ A )
= ( image_nat_set_nat @ F @ A ) ) ) ).
% image_update
thf(fact_3359_image__update,axiom,
! [X: nat,A: set_nat,F: nat > nat,N2: nat] :
( ~ ( member_nat @ X @ A )
=> ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X @ N2 ) @ A )
= ( image_nat_nat2 @ F @ A ) ) ) ).
% image_update
thf(fact_3360_image__update,axiom,
! [X: nat,A: set_nat,F: nat > code_integer,N2: code_integer] :
( ~ ( member_nat @ X @ A )
=> ( ( image_1215581382706833972nteger @ ( fun_up3904115110581602988nteger @ F @ X @ N2 ) @ A )
= ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% image_update
thf(fact_3361_bdd__above__empty,axiom,
condit2080475194861589617t_unit @ bot_bo3957492148770167129t_unit ).
% bdd_above_empty
thf(fact_3362_bdd__above__empty,axiom,
condit5488710616941104124bove_o @ bot_bot_set_o ).
% bdd_above_empty
thf(fact_3363_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_3364_bdd__above__empty,axiom,
condit2212336002400062152ve_int @ bot_bot_set_int ).
% bdd_above_empty
thf(fact_3365_bdd__above__empty,axiom,
condit4352657709669162033_nat_o @ bot_bo7824918357723582233_nat_o ).
% bdd_above_empty
thf(fact_3366_bdd__above__insert,axiom,
! [A2: nat,A: set_nat] :
( ( condit2214826472909112428ve_nat @ ( insert_nat @ A2 @ A ) )
= ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above_insert
thf(fact_3367_bdd__above__insert,axiom,
! [A2: int,A: set_int] :
( ( condit2212336002400062152ve_int @ ( insert_int @ A2 @ A ) )
= ( condit2212336002400062152ve_int @ A ) ) ).
% bdd_above_insert
thf(fact_3368_bdd__above__insert,axiom,
! [A2: $o,A: set_o] :
( ( condit5488710616941104124bove_o @ ( insert_o @ A2 @ A ) )
= ( condit5488710616941104124bove_o @ A ) ) ).
% bdd_above_insert
thf(fact_3369_bdd__above__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( condit2080475194861589617t_unit @ ( insert_Product_unit @ A2 @ A ) )
= ( condit2080475194861589617t_unit @ A ) ) ).
% bdd_above_insert
thf(fact_3370_swap__simp,axiom,
! [X: set_nat,Y: heap_e7401611519738050253t_unit] :
( ( produc1264744871843189710t_unit @ ( produc2471974422597015918t_unit @ X @ Y ) )
= ( produc7507926704131184380et_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_3371_swap__simp,axiom,
! [X: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat > $o] :
( ( produc1078497009909708626_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) )
= ( produc5001842942810119800et_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_3372_swap__simp,axiom,
! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o] :
( ( produc2580581461503396054_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) )
= ( produc2245416461498447860et_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_3373_swap__simp,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( produc6300697153377358172et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
= ( produc2471974422597015918t_unit @ Y @ X ) ) ).
% swap_simp
thf(fact_3374_swap__simp,axiom,
! [X: nat,Y: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= ( product_Pair_nat_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_3375_swap__simp,axiom,
! [X: int,Y: int] :
( ( product_swap_int_int @ ( product_Pair_int_int @ X @ Y ) )
= ( product_Pair_int_int @ Y @ X ) ) ).
% swap_simp
thf(fact_3376_swap__simp,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( produc3251101204053596376et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( produc2829238748666232050_nat_o @ Y @ X ) ) ).
% swap_simp
thf(fact_3377_swap__simp,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( produc1151471490888544852et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( produc3674526432113299062_nat_o @ Y @ X ) ) ).
% swap_simp
thf(fact_3378_single__valuedD,axiom,
! [R2: set_Pr3948176798113811640et_nat,X: heap_e7401611519738050253t_unit,Y: set_nat,Z: set_nat] :
( ( single6620584844202787856et_nat @ R2 )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ R2 )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Z ) @ R2 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_3379_single__valuedD,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
( ( single5507084439971879973at_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_3380_single__valuedD,axiom,
! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
( ( single6503947681363884253nt_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_3381_single__valuedD,axiom,
! [R2: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Z: produc3658429121746597890et_nat] :
( ( single7715245794981374860et_nat @ R2 )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ R2 )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Z ) @ R2 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_3382_single__valuedD,axiom,
! [R2: set_Pr8536935166611901872et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Z: produc3925858234332021118et_nat] :
( ( single261872189793494280et_nat @ R2 )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ R2 )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Z ) @ R2 )
=> ( Y = Z ) ) ) ) ).
% single_valuedD
thf(fact_3383_single__valuedI,axiom,
! [R2: set_Pr3948176798113811640et_nat] :
( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single6620584844202787856et_nat @ R2 ) ) ).
% single_valuedI
thf(fact_3384_single__valuedI,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ! [X3: nat,Y2: nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single5507084439971879973at_nat @ R2 ) ) ).
% single_valuedI
thf(fact_3385_single__valuedI,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y2: int,Z3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single6503947681363884253nt_int @ R2 ) ) ).
% single_valuedI
thf(fact_3386_single__valuedI,axiom,
! [R2: set_Pr3286484037609594932et_nat] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single7715245794981374860et_nat @ R2 ) ) ).
% single_valuedI
thf(fact_3387_single__valuedI,axiom,
! [R2: set_Pr8536935166611901872et_nat] :
( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single261872189793494280et_nat @ R2 ) ) ).
% single_valuedI
thf(fact_3388_single__valued__def,axiom,
( single6620584844202787856et_nat
= ( ^ [R4: set_Pr3948176798113811640et_nat] :
! [X2: heap_e7401611519738050253t_unit,Y3: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y3 ) @ R4 )
=> ! [Z2: set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R4 )
=> ( Y3 = Z2 ) ) ) ) ) ).
% single_valued_def
thf(fact_3389_single__valued__def,axiom,
( single5507084439971879973at_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
! [X2: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
=> ! [Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R4 )
=> ( Y3 = Z2 ) ) ) ) ) ).
% single_valued_def
thf(fact_3390_single__valued__def,axiom,
( single6503947681363884253nt_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
! [X2: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
=> ! [Z2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R4 )
=> ( Y3 = Z2 ) ) ) ) ) ).
% single_valued_def
thf(fact_3391_single__valued__def,axiom,
( single7715245794981374860et_nat
= ( ^ [R4: set_Pr3286484037609594932et_nat] :
! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y3 ) @ R4 )
=> ! [Z2: produc3658429121746597890et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R4 )
=> ( Y3 = Z2 ) ) ) ) ) ).
% single_valued_def
thf(fact_3392_single__valued__def,axiom,
( single261872189793494280et_nat
= ( ^ [R4: set_Pr8536935166611901872et_nat] :
! [X2: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y3 ) @ R4 )
=> ! [Z2: produc3925858234332021118et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R4 )
=> ( Y3 = Z2 ) ) ) ) ) ).
% single_valued_def
thf(fact_3393_single__valued__confluent,axiom,
! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
( ( single5507084439971879973at_nat @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_3394_single__valued__confluent,axiom,
! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
( ( single6503947681363884253nt_int @ R2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_3395_cSup__le__iff,axiom,
! [S: set_o,A2: $o] :
( ( S != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ S )
=> ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S ) @ A2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ S )
=> ( ord_less_eq_o @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3396_cSup__le__iff,axiom,
! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( S != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ S )
=> ( ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ S ) @ A2 )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ S )
=> ( ord_le729326519192465773_nat_o @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3397_cSup__le__iff,axiom,
! [S: set_filter_nat,A2: filter_nat] :
( ( S != bot_bo498966703094740906er_nat )
=> ( ( condit4029664723990530684er_nat @ S )
=> ( ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ S ) @ A2 )
= ( ! [X2: filter_nat] :
( ( member_filter_nat @ X2 @ S )
=> ( ord_le2510731241096832064er_nat @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3398_cSup__le__iff,axiom,
! [S: set_set_nat,A2: set_nat] :
( ( S != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ S )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ S ) @ A2 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ord_less_eq_set_nat @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3399_cSup__le__iff,axiom,
! [S: set_int,A2: int] :
( ( S != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ S )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ S ) @ A2 )
= ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ord_less_eq_int @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3400_cSup__le__iff,axiom,
! [S: set_nat,A2: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ord_less_eq_nat @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3401_cSup__le__iff,axiom,
! [S: set_Product_unit,A2: product_unit] :
( ( S != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ S )
=> ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ S ) @ A2 )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ S )
=> ( ord_le3221252021190050221t_unit @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_3402_cSup__mono,axiom,
! [B4: set_o,A: set_o] :
( ( B4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ B4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ B2 @ X6 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B4 ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3403_cSup__mono,axiom,
! [B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ A )
=> ( ! [B2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B2 @ B4 )
=> ? [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ A )
& ( ord_le729326519192465773_nat_o @ B2 @ X6 ) ) )
=> ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ B4 ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3404_cSup__mono,axiom,
! [B4: set_filter_nat,A: set_filter_nat] :
( ( B4 != bot_bo498966703094740906er_nat )
=> ( ( condit4029664723990530684er_nat @ A )
=> ( ! [B2: filter_nat] :
( ( member_filter_nat @ B2 @ B4 )
=> ? [X6: filter_nat] :
( ( member_filter_nat @ X6 @ A )
& ( ord_le2510731241096832064er_nat @ B2 @ X6 ) ) )
=> ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ B4 ) @ ( comple2454491259644266675er_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3405_cSup__mono,axiom,
! [B4: set_set_nat,A: set_set_nat] :
( ( B4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ B4 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ B2 @ X6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3406_cSup__mono,axiom,
! [B4: set_int,A: set_int] :
( ( B4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ A )
=> ( ! [B2: int] :
( ( member_int @ B2 @ B4 )
=> ? [X6: int] :
( ( member_int @ X6 @ A )
& ( ord_less_eq_int @ B2 @ X6 ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ B4 ) @ ( complete_Sup_Sup_int @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3407_cSup__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ B2 @ X6 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B4 ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3408_cSup__mono,axiom,
! [B4: set_Product_unit,A: set_Product_unit] :
( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ A )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ B4 )
=> ? [X6: product_unit] :
( ( member_Product_unit @ X6 @ A )
& ( ord_le3221252021190050221t_unit @ B2 @ X6 ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ B4 ) @ ( comple4687483117567863418t_unit @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_3409_cSUP__le__iff,axiom,
! [A: set_Product_unit,F: product_unit > int,U: int] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ U )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3410_cSUP__le__iff,axiom,
! [A: set_o,F: $o > int,U: int] :
( ( A != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int2 @ F @ A ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ U )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3411_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > int,U: int] :
( ( A != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int2 @ F @ A ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ U )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3412_cSUP__le__iff,axiom,
! [A: set_int,F: int > int,U: int] :
( ( A != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int2 @ F @ A ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ U )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3413_cSUP__le__iff,axiom,
! [A: set_Product_unit,F: product_unit > nat,U: nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ U )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3414_cSUP__le__iff,axiom,
! [A: set_o,F: $o > nat,U: nat] :
( ( A != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ U )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3415_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ U )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3416_cSUP__le__iff,axiom,
! [A: set_int,F: int > nat,U: nat] :
( ( A != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ U )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3417_cSUP__le__iff,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ U )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3418_cSUP__le__iff,axiom,
! [A: set_o,F: $o > product_unit,U: product_unit] :
( ( A != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
=> ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ U )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3419_cSUP__mono,axiom,
! [A: set_Product_unit,G: nat > set_nat,B4: set_nat,F: product_unit > set_nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
=> ( ! [N3: product_unit] :
( ( member_Product_unit @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3420_cSUP__mono,axiom,
! [A: set_o,G: nat > set_nat,B4: set_nat,F: $o > set_nat] :
( ( A != bot_bot_set_o )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
=> ( ! [N3: $o] :
( ( member_o @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3421_cSUP__mono,axiom,
! [A: set_nat,G: nat > set_nat,B4: set_nat,F: nat > set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3422_cSUP__mono,axiom,
! [A: set_int,G: nat > set_nat,B4: set_nat,F: int > set_nat] :
( ( A != bot_bot_set_int )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
=> ( ! [N3: int] :
( ( member_int @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3423_cSUP__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,G: nat > set_nat,B4: set_nat,F: ( produc3658429121746597890et_nat > $o ) > set_nat] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
=> ( ! [N3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3424_cSUP__mono,axiom,
! [A: set_Product_unit,G: nat > nat,B4: set_nat,F: product_unit > nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ! [N3: product_unit] :
( ( member_Product_unit @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3425_cSUP__mono,axiom,
! [A: set_o,G: nat > nat,B4: set_nat,F: $o > nat] :
( ( A != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ! [N3: $o] :
( ( member_o @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3426_cSUP__mono,axiom,
! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3427_cSUP__mono,axiom,
! [A: set_int,G: nat > nat,B4: set_nat,F: int > nat] :
( ( A != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ! [N3: int] :
( ( member_int @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3428_cSUP__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,G: nat > nat,B4: set_nat,F: ( produc3658429121746597890et_nat > $o ) > nat] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ! [N3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ N3 @ A )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B4 )
& ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2174381888560928856_o_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3429_cSup__subset__mono,axiom,
! [A: set_o,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B4 )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3430_cSup__subset__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ B4 )
=> ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3431_cSup__subset__mono,axiom,
! [A: set_filter_nat,B4: set_filter_nat] :
( ( A != bot_bo498966703094740906er_nat )
=> ( ( condit4029664723990530684er_nat @ B4 )
=> ( ( ord_le2426478655948331894er_nat @ A @ B4 )
=> ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ A ) @ ( comple2454491259644266675er_nat @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3432_cSup__subset__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3433_cSup__subset__mono,axiom,
! [A: set_int,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3434_cSup__subset__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3435_cSup__subset__mono,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ B4 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_3436_cSup__insert__If,axiom,
! [X4: set_set_nat,A2: set_nat] :
( ( condit5477540289124974626et_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_3437_cSup__insert__If,axiom,
! [X4: set_o,A2: $o] :
( ( condit5488710616941104124bove_o @ X4 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ X4 ) )
= ( ( ( X4 = bot_bot_set_o )
=> A2 )
& ( ( X4 != bot_bot_set_o )
=> ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_3438_cSup__insert__If,axiom,
! [X4: set_int,A2: int] :
( ( condit2212336002400062152ve_int @ X4 )
=> ( ( ( X4 = bot_bot_set_int )
=> ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bot_set_int )
=> ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
= ( sup_sup_int @ A2 @ ( complete_Sup_Sup_int @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_3439_cSup__insert__If,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( condit4352657709669162033_nat_o @ X4 )
=> ( ( ( X4 = bot_bo7824918357723582233_nat_o )
=> ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= ( sup_su5453871518329203617_nat_o @ A2 @ ( comple4225350640907012154_nat_o @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_3440_cSup__insert__If,axiom,
! [X4: set_nat,A2: nat] :
( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
= ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_3441_cSup__insert__If,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( condit2080475194861589617t_unit @ X4 )
=> ( ( ( X4 = bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
= ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_3442_cSup__insert,axiom,
! [X4: set_set_nat,A2: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_3443_cSup__insert,axiom,
! [X4: set_o,A2: $o] :
( ( X4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ X4 )
=> ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ X4 ) )
= ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_3444_cSup__insert,axiom,
! [X4: set_int,A2: int] :
( ( X4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ X4 )
=> ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
= ( sup_sup_int @ A2 @ ( complete_Sup_Sup_int @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_3445_cSup__insert,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ X4 )
=> ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= ( sup_su5453871518329203617_nat_o @ A2 @ ( comple4225350640907012154_nat_o @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_3446_cSup__insert,axiom,
! [X4: set_nat,A2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
= ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_3447_cSup__insert,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ X4 )
=> ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
= ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_3448_cSup__union__distrib,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A )
=> ( ( B4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B4 )
=> ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_3449_cSup__union__distrib,axiom,
! [A: set_o,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ B4 )
=> ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B4 ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_3450_cSup__union__distrib,axiom,
! [A: set_int,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ A )
=> ( ( B4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ B4 )
=> ( ( complete_Sup_Sup_int @ ( sup_sup_set_int @ A @ B4 ) )
= ( sup_sup_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_3451_cSup__union__distrib,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ A )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ B4 )
=> ( ( comple4225350640907012154_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( sup_su5453871518329203617_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_3452_cSup__union__distrib,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B4 )
=> ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_3453_cSup__union__distrib,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ A )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ B4 )
=> ( ( comple4687483117567863418t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_3454_fun__upd__image,axiom,
! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer,Y: code_integer] :
( ( ( member_Code_integer @ X @ A )
=> ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ Y ) @ A )
= ( insert_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) )
& ( ~ ( member_Code_integer @ X @ A )
=> ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ Y ) @ A )
= ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3455_fun__upd__image,axiom,
! [X: product_unit,A: set_Product_unit,F: product_unit > nat,Y: nat] :
( ( ( member_Product_unit @ X @ A )
=> ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X @ Y ) @ A )
= ( insert_nat @ Y @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X @ Y ) @ A )
= ( image_875570014554754200it_nat @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3456_fun__upd__image,axiom,
! [X: product_unit,A: set_Product_unit,F: product_unit > int,Y: int] :
( ( ( member_Product_unit @ X @ A )
=> ( ( image_873079544045703924it_int @ ( fun_up6470560416331323516it_int @ F @ X @ Y ) @ A )
= ( insert_int @ Y @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( image_873079544045703924it_int @ ( fun_up6470560416331323516it_int @ F @ X @ Y ) @ A )
= ( image_873079544045703924it_int @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3457_fun__upd__image,axiom,
! [X: product_unit,A: set_Product_unit,F: product_unit > $o,Y: $o] :
( ( ( member_Product_unit @ X @ A )
=> ( ( image_Product_unit_o2 @ ( fun_up5432024876597553480unit_o @ F @ X @ Y ) @ A )
= ( insert_o @ Y @ ( image_Product_unit_o2 @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( image_Product_unit_o2 @ ( fun_up5432024876597553480unit_o @ F @ X @ Y ) @ A )
= ( image_Product_unit_o2 @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3458_fun__upd__image,axiom,
! [X: product_unit,A: set_Product_unit,F: product_unit > product_unit,Y: product_unit] :
( ( ( member_Product_unit @ X @ A )
=> ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X @ Y ) @ A )
= ( insert_Product_unit @ Y @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X @ Y ) @ A )
= ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3459_fun__upd__image,axiom,
! [X: $o,A: set_o,F: $o > nat,Y: nat] :
( ( ( member_o @ X @ A )
=> ( ( image_o_nat2 @ ( fun_upd_o_nat @ F @ X @ Y ) @ A )
= ( insert_nat @ Y @ ( image_o_nat2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( image_o_nat2 @ ( fun_upd_o_nat @ F @ X @ Y ) @ A )
= ( image_o_nat2 @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3460_fun__upd__image,axiom,
! [X: $o,A: set_o,F: $o > int,Y: int] :
( ( ( member_o @ X @ A )
=> ( ( image_o_int2 @ ( fun_upd_o_int @ F @ X @ Y ) @ A )
= ( insert_int @ Y @ ( image_o_int2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( image_o_int2 @ ( fun_upd_o_int @ F @ X @ Y ) @ A )
= ( image_o_int2 @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3461_fun__upd__image,axiom,
! [X: $o,A: set_o,F: $o > $o,Y: $o] :
( ( ( member_o @ X @ A )
=> ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X @ Y ) @ A )
= ( insert_o @ Y @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X @ Y ) @ A )
= ( image_o_o2 @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3462_fun__upd__image,axiom,
! [X: $o,A: set_o,F: $o > product_unit,Y: product_unit] :
( ( ( member_o @ X @ A )
=> ( ( image_o_Product_unit2 @ ( fun_up846054379331416818t_unit @ F @ X @ Y ) @ A )
= ( insert_Product_unit @ Y @ ( image_o_Product_unit2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( image_o_Product_unit2 @ ( fun_up846054379331416818t_unit @ F @ X @ Y ) @ A )
= ( image_o_Product_unit2 @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3463_fun__upd__image,axiom,
! [X: int,A: set_int,F: int > nat,Y: nat] :
( ( ( member_int @ X @ A )
=> ( ( image_int_nat2 @ ( fun_upd_int_nat @ F @ X @ Y ) @ A )
= ( insert_nat @ Y @ ( image_int_nat2 @ F @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( image_int_nat2 @ ( fun_upd_int_nat @ F @ X @ Y ) @ A )
= ( image_int_nat2 @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_3464_Sup__fin_Oeq__fold,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
= ( finite_fold_int_int @ sup_sup_int @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3465_Sup__fin_Oeq__fold,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( finite_fold_o_o @ sup_sup_o @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3466_Sup__fin_Oeq__fold,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3467_Sup__fin_Oeq__fold,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
= ( finite677925301803182934et_nat @ sup_sup_set_nat @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3468_Sup__fin_Oeq__fold,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
= ( finite1440822074540989594n_assn @ sup_sup_assn @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3469_Sup__fin_Oeq__fold,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
= ( finite8834431295148555784t_unit @ sup_sup_Product_unit @ X @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_3470_Inf__fin_Oeq__fold,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
= ( finite_fold_int_int @ inf_inf_int @ X @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3471_Inf__fin_Oeq__fold,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( finite_fold_o_o @ inf_inf_o @ X @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3472_Inf__fin_Oeq__fold,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( finite_fold_nat_nat @ inf_inf_nat @ X @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3473_Inf__fin_Oeq__fold,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
= ( finite677925301803182934et_nat @ inf_inf_set_nat @ X @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3474_Inf__fin_Oeq__fold,axiom,
! [A: set_assn,X: assn] :
( ( finite_finite_assn @ A )
=> ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
= ( finite1440822074540989594n_assn @ inf_inf_assn @ X @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3475_Inf__fin_Oeq__fold,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
= ( finite8834431295148555784t_unit @ inf_inf_Product_unit @ X @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3476_wf__bounded__set,axiom,
! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_Code_integer,F: nat > set_Code_integer] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
=> ( ( finite6017078050557962740nteger @ ( Ub @ A4 ) )
& ( ord_le7084787975880047091nteger @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
& ( ord_le7084787975880047091nteger @ ( F @ B2 ) @ ( Ub @ A4 ) )
& ( ord_le1307284697595431911nteger @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
=> ( wf_nat @ R2 ) ) ).
% wf_bounded_set
thf(fact_3477_wf__bounded__set,axiom,
! [R2: set_Pr958786334691620121nt_int,Ub: int > set_Code_integer,F: int > set_Code_integer] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
=> ( ( finite6017078050557962740nteger @ ( Ub @ A4 ) )
& ( ord_le7084787975880047091nteger @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
& ( ord_le7084787975880047091nteger @ ( F @ B2 ) @ ( Ub @ A4 ) )
& ( ord_le1307284697595431911nteger @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
=> ( wf_int @ R2 ) ) ).
% wf_bounded_set
thf(fact_3478_wf__bounded__set,axiom,
! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_nat,F: nat > set_nat] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
=> ( ( finite_finite_nat @ ( Ub @ A4 ) )
& ( ord_less_eq_set_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_eq_set_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_set_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
=> ( wf_nat @ R2 ) ) ).
% wf_bounded_set
thf(fact_3479_wf__bounded__set,axiom,
! [R2: set_Pr958786334691620121nt_int,Ub: int > set_nat,F: int > set_nat] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
=> ( ( finite_finite_nat @ ( Ub @ A4 ) )
& ( ord_less_eq_set_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_eq_set_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_set_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
=> ( wf_int @ R2 ) ) ).
% wf_bounded_set
thf(fact_3480_psubset__insert__iff,axiom,
! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
( ( ord_le8056459307392131481t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
= ( ( ( member_Product_unit @ X @ B4 )
=> ( ord_le8056459307392131481t_unit @ A @ B4 ) )
& ( ~ ( member_Product_unit @ X @ B4 )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3481_psubset__insert__iff,axiom,
! [A: set_o,X: $o,B4: set_o] :
( ( ord_less_set_o @ A @ ( insert_o @ X @ B4 ) )
= ( ( ( member_o @ X @ B4 )
=> ( ord_less_set_o @ A @ B4 ) )
& ( ~ ( member_o @ X @ B4 )
=> ( ( ( member_o @ X @ A )
=> ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
& ( ~ ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3482_psubset__insert__iff,axiom,
! [A: set_int,X: int,B4: set_int] :
( ( ord_less_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( ( ( member_int @ X @ B4 )
=> ( ord_less_set_int @ A @ B4 ) )
& ( ~ ( member_int @ X @ B4 )
=> ( ( ( member_int @ X @ A )
=> ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
& ( ~ ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3483_psubset__insert__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2453136405763929_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
= ( ( ( member6576561426505652726_nat_o @ X @ B4 )
=> ( ord_le2453136405763929_nat_o @ A @ B4 ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
=> ( ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3484_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ B4 )
=> ( ord_less_set_nat @ A @ B4 ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( ( member_nat @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3485_finite__induct__select,axiom,
! [S: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ S )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [T3: set_Code_integer] :
( ( ord_le1307284697595431911nteger @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: code_integer] :
( ( member_Code_integer @ X6 @ ( minus_2355218937544613996nteger @ S @ T3 ) )
& ( P @ ( insert_Code_integer @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3486_finite__induct__select,axiom,
! [S: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [T3: set_Product_unit] :
( ( ord_le8056459307392131481t_unit @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: product_unit] :
( ( member_Product_unit @ X6 @ ( minus_6452836326544984404t_unit @ S @ T3 ) )
& ( P @ ( insert_Product_unit @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3487_finite__induct__select,axiom,
! [S: set_o,P: set_o > $o] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [T3: set_o] :
( ( ord_less_set_o @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: $o] :
( ( member_o @ X6 @ ( minus_minus_set_o @ S @ T3 ) )
& ( P @ ( insert_o @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3488_finite__induct__select,axiom,
! [S: set_int,P: set_int > $o] :
( ( finite_finite_int @ S )
=> ( ( P @ bot_bot_set_int )
=> ( ! [T3: set_int] :
( ( ord_less_set_int @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: int] :
( ( member_int @ X6 @ ( minus_minus_set_int @ S @ T3 ) )
& ( P @ ( insert_int @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3489_finite__induct__select,axiom,
! [S: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
( ( finite3252695134891459830_nat_o @ S )
=> ( ( P @ bot_bo7824918357723582233_nat_o )
=> ( ! [T3: set_Pr4532377907799695533_nat_o] :
( ( ord_le2453136405763929_nat_o @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ ( minus_1801376950450012436_nat_o @ S @ T3 ) )
& ( P @ ( insert5175938949040314269_nat_o @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3490_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3491_Max_Osubset__imp,axiom,
! [A: set_Code_integer,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ A @ B4 )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3492_Max_Osubset__imp,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ ( lattic1210392890313158908t_unit @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3493_Max_Osubset__imp,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3494_Max_Osubset__imp,axiom,
! [A: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A @ B4 )
=> ( ( A != bot_bot_set_rat )
=> ( ( finite_finite_rat @ B4 )
=> ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3495_Max_Osubset__imp,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3496_Max_Osubset__imp,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3497_Max__mono,axiom,
! [M: set_Code_integer,N: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ M @ N )
=> ( ( M != bot_bo3990330152332043303nteger )
=> ( ( finite6017078050557962740nteger @ N )
=> ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ M ) @ ( lattic4901227151466704046nteger @ N ) ) ) ) ) ).
% Max_mono
thf(fact_3498_Max__mono,axiom,
! [M: set_Product_unit,N: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ M @ N )
=> ( ( M != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ N )
=> ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ M ) @ ( lattic1210392890313158908t_unit @ N ) ) ) ) ) ).
% Max_mono
thf(fact_3499_Max__mono,axiom,
! [M: set_o,N: set_o] :
( ( ord_less_eq_set_o @ M @ N )
=> ( ( M != bot_bot_set_o )
=> ( ( finite_finite_o @ N )
=> ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ M ) @ ( lattic1921953407002678535_Max_o @ N ) ) ) ) ) ).
% Max_mono
thf(fact_3500_Max__mono,axiom,
! [M: set_rat,N: set_rat] :
( ( ord_less_eq_set_rat @ M @ N )
=> ( ( M != bot_bot_set_rat )
=> ( ( finite_finite_rat @ N )
=> ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ M ) @ ( lattic7630753665789217321ax_rat @ N ) ) ) ) ) ).
% Max_mono
thf(fact_3501_Max__mono,axiom,
! [M: set_int,N: set_int] :
( ( ord_less_eq_set_int @ M @ N )
=> ( ( M != bot_bot_set_int )
=> ( ( finite_finite_int @ N )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M ) @ ( lattic8263393255366662781ax_int @ N ) ) ) ) ) ).
% Max_mono
thf(fact_3502_Max__mono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).
% Max_mono
thf(fact_3503_image__vimage__eq,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( image_4470545334726330049nteger @ F @ ( vimage1318793397690208119nteger @ F @ A ) )
= ( inf_in1364745209274528805nteger @ A @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ) ).
% image_vimage_eq
thf(fact_3504_image__vimage__eq,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( image_875570014554754200it_nat @ F @ ( vimage6253328473476588386it_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% image_vimage_eq
thf(fact_3505_image__vimage__eq,axiom,
! [F: $o > nat,A: set_nat] :
( ( image_o_nat2 @ F @ ( vimage_o_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).
% image_vimage_eq
thf(fact_3506_image__vimage__eq,axiom,
! [F: nat > set_nat,A: set_set_nat] :
( ( image_nat_set_nat @ F @ ( vimage_nat_set_nat @ F @ A ) )
= ( inf_inf_set_set_nat @ A @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3507_image__vimage__eq,axiom,
! [F: nat > code_integer,A: set_Code_integer] :
( ( image_1215581382706833972nteger @ F @ ( vimage6104394758324456170nteger @ F @ A ) )
= ( inf_in1364745209274528805nteger @ A @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3508_image__vimage__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3509_image__vimage__eq,axiom,
! [F: int > nat,A: set_nat] :
( ( image_int_nat2 @ F @ ( vimage_int_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_int_nat2 @ F @ top_top_set_int ) ) ) ).
% image_vimage_eq
thf(fact_3510_UNIV__I,axiom,
! [X: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ top_to4869887016220417533_nat_o ) ).
% UNIV_I
thf(fact_3511_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_3512_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_3513_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_3514_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_3515_Int__UNIV,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B4 = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_3516_Int__UNIV,axiom,
! [A: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A @ B4 )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B4 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_3517_Int__UNIV,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B4 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_3518_Int__UNIV,axiom,
! [A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ A @ B4 )
= top_top_set_int )
= ( ( A = top_top_set_int )
& ( B4 = top_top_set_int ) ) ) ).
% Int_UNIV
thf(fact_3519_psubsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != B4 )
=> ( ord_less_set_nat @ A @ B4 ) ) ) ).
% psubsetI
thf(fact_3520_vimage__UNIV,axiom,
! [F: product_unit > product_unit] :
( ( vimage7995052115951654139t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_3521_vimage__UNIV,axiom,
! [F: $o > product_unit] :
( ( vimage4222377557387472176t_unit @ F @ top_to1996260823553986621t_unit )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_3522_vimage__UNIV,axiom,
! [F: nat > product_unit] :
( ( vimage4884490618288580032t_unit @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_3523_vimage__UNIV,axiom,
! [F: int > product_unit] :
( ( vimage2000912565490760292t_unit @ F @ top_to1996260823553986621t_unit )
= top_top_set_int ) ).
% vimage_UNIV
thf(fact_3524_vimage__UNIV,axiom,
! [F: product_unit > $o] :
( ( vimage8808348054653608838unit_o @ F @ top_top_set_o )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_3525_vimage__UNIV,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ top_top_set_o )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_3526_vimage__UNIV,axiom,
! [F: nat > $o] :
( ( vimage_nat_o @ F @ top_top_set_o )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_3527_vimage__UNIV,axiom,
! [F: int > $o] :
( ( vimage_int_o @ F @ top_top_set_o )
= top_top_set_int ) ).
% vimage_UNIV
thf(fact_3528_vimage__UNIV,axiom,
! [F: product_unit > nat] :
( ( vimage6253328473476588386it_nat @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_3529_vimage__UNIV,axiom,
! [F: $o > nat] :
( ( vimage_o_nat @ F @ top_top_set_nat )
= top_top_set_o ) ).
% vimage_UNIV
thf(fact_3530_Pow__UNIV,axiom,
( ( pow_Product_unit @ top_to1996260823553986621t_unit )
= top_to1767297665138865437t_unit ) ).
% Pow_UNIV
thf(fact_3531_Pow__UNIV,axiom,
( ( pow_o @ top_top_set_o )
= top_top_set_set_o ) ).
% Pow_UNIV
thf(fact_3532_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_3533_Pow__UNIV,axiom,
( ( pow_int @ top_top_set_int )
= top_top_set_set_int ) ).
% Pow_UNIV
thf(fact_3534_Diff__UNIV,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( minus_1801376950450012436_nat_o @ A @ top_to4869887016220417533_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Diff_UNIV
thf(fact_3535_Diff__UNIV,axiom,
! [A: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_3536_Diff__UNIV,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_3537_Diff__UNIV,axiom,
! [A: set_int] :
( ( minus_minus_set_int @ A @ top_top_set_int )
= bot_bot_set_int ) ).
% Diff_UNIV
thf(fact_3538_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_3539_Max__singleton,axiom,
! [X: product_unit] :
( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% Max_singleton
thf(fact_3540_Max__singleton,axiom,
! [X: $o] :
( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% Max_singleton
thf(fact_3541_Max__singleton,axiom,
! [X: int] :
( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% Max_singleton
thf(fact_3542_Max__singleton,axiom,
! [X: nat] :
( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Max_singleton
thf(fact_3543_range__snd,axiom,
( ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ top_to2524957550908328808et_nat )
= top_top_set_set_nat ) ).
% range_snd
thf(fact_3544_range__snd,axiom,
( ( image_5042161079198086560nt_int @ product_snd_int_int @ top_to4366644338036079209nt_int )
= top_top_set_int ) ).
% range_snd
thf(fact_3545_Domain__Id,axiom,
( ( domain4292024440186670924t_unit @ id_Product_unit )
= top_to1996260823553986621t_unit ) ).
% Domain_Id
thf(fact_3546_Domain__Id,axiom,
( ( domain_o_o @ id_o2 )
= top_top_set_o ) ).
% Domain_Id
thf(fact_3547_Domain__Id,axiom,
( ( domain_nat_nat @ id_nat )
= top_top_set_nat ) ).
% Domain_Id
thf(fact_3548_Domain__Id,axiom,
( ( domain_int_int @ id_int )
= top_top_set_int ) ).
% Domain_Id
thf(fact_3549_Range__Id,axiom,
( ( range_6617724647562260917t_unit @ id_Product_unit )
= top_to1996260823553986621t_unit ) ).
% Range_Id
thf(fact_3550_Range__Id,axiom,
( ( range_o_o @ id_o2 )
= top_top_set_o ) ).
% Range_Id
thf(fact_3551_Range__Id,axiom,
( ( range_nat_nat @ id_nat )
= top_top_set_nat ) ).
% Range_Id
thf(fact_3552_Range__Id,axiom,
( ( range_int_int @ id_int )
= top_top_set_int ) ).
% Range_Id
thf(fact_3553_Max_Obounded__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
= ( ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le3102999989581377725nteger @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3554_Max_Obounded__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3555_Max_Obounded__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3556_Max_Obounded__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
= ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3557_Max_Obounded__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3558_Max_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3559_Max__less__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le361264281704409273t_unit @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3560_Max__less__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_o @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3561_Max__less__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3562_Max__less__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le6747313008572928689nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
= ( ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le6747313008572928689nteger @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3563_Max__less__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
= ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_rat @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3564_Max__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3565_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_3566_lt__ex,axiom,
! [X: rat] :
? [Y2: rat] : ( ord_less_rat @ Y2 @ X ) ).
% lt_ex
thf(fact_3567_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_3568_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_3569_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_3570_dense,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ? [Z3: rat] :
( ( ord_less_rat @ X @ Z3 )
& ( ord_less_rat @ Z3 @ Y ) ) ) ).
% dense
thf(fact_3571_less__imp__neq,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_3572_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_3573_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_3574_less__imp__neq,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_3575_less__imp__neq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_3576_order_Oasym,axiom,
! [A2: assn,B: assn] :
( ( ord_less_assn @ A2 @ B )
=> ~ ( ord_less_assn @ B @ A2 ) ) ).
% order.asym
thf(fact_3577_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_3578_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_3579_order_Oasym,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ~ ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% order.asym
thf(fact_3580_order_Oasym,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ~ ( ord_less_rat @ B @ A2 ) ) ).
% order.asym
thf(fact_3581_ord__eq__less__trans,axiom,
! [A2: assn,B: assn,C: assn] :
( ( A2 = B )
=> ( ( ord_less_assn @ B @ C )
=> ( ord_less_assn @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3582_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3583_ord__eq__less__trans,axiom,
! [A2: int,B: int,C: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3584_ord__eq__less__trans,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( A2 = B )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3585_ord__eq__less__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( A2 = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_3586_ord__less__eq__trans,axiom,
! [A2: assn,B: assn,C: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_assn @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3587_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3588_ord__less__eq__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3589_ord__less__eq__trans,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( B = C )
=> ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3590_ord__less__eq__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_3591_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_3592_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_3593_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_3594_antisym__conv3,axiom,
! [Y: code_integer,X: code_integer] :
( ~ ( ord_le6747313008572928689nteger @ Y @ X )
=> ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_3595_antisym__conv3,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_rat @ Y @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_3596_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_3597_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_3598_linorder__cases,axiom,
! [X: code_integer,Y: code_integer] :
( ~ ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ( X != Y )
=> ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_3599_linorder__cases,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_3600_dual__order_Oasym,axiom,
! [B: assn,A2: assn] :
( ( ord_less_assn @ B @ A2 )
=> ~ ( ord_less_assn @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_3601_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_3602_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_3603_dual__order_Oasym,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ~ ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_3604_dual__order_Oasym,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ~ ( ord_less_rat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_3605_dual__order_Oirrefl,axiom,
! [A2: assn] :
~ ( ord_less_assn @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_3606_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_3607_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_3608_dual__order_Oirrefl,axiom,
! [A2: code_integer] :
~ ( ord_le6747313008572928689nteger @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_3609_dual__order_Oirrefl,axiom,
! [A2: rat] :
~ ( ord_less_rat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_3610_exists__least__iff,axiom,
( ( ^ [P6: nat > $o] :
? [X9: nat] : ( P6 @ X9 ) )
= ( ^ [P2: nat > $o] :
? [N4: nat] :
( ( P2 @ N4 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ~ ( P2 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_3611_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B2: nat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3612_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B2: int] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3613_linorder__less__wlog,axiom,
! [P: code_integer > code_integer > $o,A2: code_integer,B: code_integer] :
( ! [A4: code_integer,B2: code_integer] :
( ( ord_le6747313008572928689nteger @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: code_integer] : ( P @ A4 @ A4 )
=> ( ! [A4: code_integer,B2: code_integer] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3614_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A2: rat,B: rat] :
( ! [A4: rat,B2: rat] :
( ( ord_less_rat @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: rat] : ( P @ A4 @ A4 )
=> ( ! [A4: rat,B2: rat] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_3615_order_Ostrict__trans,axiom,
! [A2: assn,B: assn,C: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_assn @ B @ C )
=> ( ord_less_assn @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_3616_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_3617_order_Ostrict__trans,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_3618_order_Ostrict__trans,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_3619_order_Ostrict__trans,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_3620_top_Oextremum__strict,axiom,
! [A2: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A2 ) ).
% top.extremum_strict
thf(fact_3621_top_Oextremum__strict,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ top_top_set_o @ A2 ) ).
% top.extremum_strict
thf(fact_3622_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_3623_top_Oextremum__strict,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ top_top_set_int @ A2 ) ).
% top.extremum_strict
thf(fact_3624_top_Oextremum__strict,axiom,
! [A2: assn] :
~ ( ord_less_assn @ top_top_assn @ A2 ) ).
% top.extremum_strict
thf(fact_3625_top_Onot__eq__extremum,axiom,
! [A2: set_Product_unit] :
( ( A2 != top_to1996260823553986621t_unit )
= ( ord_le8056459307392131481t_unit @ A2 @ top_to1996260823553986621t_unit ) ) ).
% top.not_eq_extremum
thf(fact_3626_top_Onot__eq__extremum,axiom,
! [A2: set_o] :
( ( A2 != top_top_set_o )
= ( ord_less_set_o @ A2 @ top_top_set_o ) ) ).
% top.not_eq_extremum
thf(fact_3627_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_3628_top_Onot__eq__extremum,axiom,
! [A2: set_int] :
( ( A2 != top_top_set_int )
= ( ord_less_set_int @ A2 @ top_top_set_int ) ) ).
% top.not_eq_extremum
thf(fact_3629_top_Onot__eq__extremum,axiom,
! [A2: assn] :
( ( A2 != top_top_assn )
= ( ord_less_assn @ A2 @ top_top_assn ) ) ).
% top.not_eq_extremum
thf(fact_3630_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3631_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3632_not__less__iff__gr__or__eq,axiom,
! [X: code_integer,Y: code_integer] :
( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
= ( ( ord_le6747313008572928689nteger @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3633_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ( ord_less_rat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_3634_dual__order_Ostrict__trans,axiom,
! [B: assn,A2: assn,C: assn] :
( ( ord_less_assn @ B @ A2 )
=> ( ( ord_less_assn @ C @ B )
=> ( ord_less_assn @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_3635_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_3636_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_3637_dual__order_Ostrict__trans,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( ( ord_le6747313008572928689nteger @ C @ B )
=> ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_3638_dual__order_Ostrict__trans,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_3639_order_Ostrict__implies__not__eq,axiom,
! [A2: assn,B: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3640_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3641_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3642_order_Ostrict__implies__not__eq,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3643_order_Ostrict__implies__not__eq,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_3644_dual__order_Ostrict__implies__not__eq,axiom,
! [B: assn,A2: assn] :
( ( ord_less_assn @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3645_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3646_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3647_dual__order_Ostrict__implies__not__eq,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3648_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_3649_psubsetD,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
( ( ord_le2453136405763929_nat_o @ A @ B4 )
=> ( ( member6576561426505652726_nat_o @ C @ A )
=> ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_3650_psubsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_3651_psubsetD,axiom,
! [A: set_int,B4: set_int,C: int] :
( ( ord_less_set_int @ A @ B4 )
=> ( ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_3652_psubsetD,axiom,
! [A: set_o,B4: set_o,C: $o] :
( ( ord_less_set_o @ A @ B4 )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_3653_psubsetD,axiom,
! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
( ( ord_le8056459307392131481t_unit @ A @ B4 )
=> ( ( member_Product_unit @ C @ A )
=> ( member_Product_unit @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_3654_UNIV__eq__I,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ! [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A )
=> ( top_to4869887016220417533_nat_o = A ) ) ).
% UNIV_eq_I
thf(fact_3655_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_3656_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_3657_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_3658_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X3: int] : ( member_int @ X3 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_3659_UNIV__witness,axiom,
? [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ top_to4869887016220417533_nat_o ) ).
% UNIV_witness
thf(fact_3660_UNIV__witness,axiom,
? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_3661_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_3662_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_3663_UNIV__witness,axiom,
? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_3664_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_3665_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_3666_linorder__neqE,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
=> ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_3667_linorder__neqE,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_3668_order__less__asym,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ~ ( ord_less_assn @ Y @ X ) ) ).
% order_less_asym
thf(fact_3669_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_3670_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_3671_order__less__asym,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% order_less_asym
thf(fact_3672_order__less__asym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_asym
thf(fact_3673_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3674_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3675_linorder__neq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
= ( ( ord_le6747313008572928689nteger @ X @ Y )
| ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3676_linorder__neq__iff,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
= ( ( ord_less_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_3677_order__less__asym_H,axiom,
! [A2: assn,B: assn] :
( ( ord_less_assn @ A2 @ B )
=> ~ ( ord_less_assn @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_3678_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_3679_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_3680_order__less__asym_H,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ~ ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_3681_order__less__asym_H,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ~ ( ord_less_rat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_3682_order__less__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( ( ord_less_assn @ X @ Y )
=> ( ( ord_less_assn @ Y @ Z )
=> ( ord_less_assn @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3683_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3684_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3685_order__less__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ( ord_le6747313008572928689nteger @ Y @ Z )
=> ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3686_order__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_3687_ord__eq__less__subst,axiom,
! [A2: assn,F: assn > assn,B: assn,C: assn] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3688_ord__eq__less__subst,axiom,
! [A2: nat,F: assn > nat,B: assn,C: assn] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3689_ord__eq__less__subst,axiom,
! [A2: int,F: assn > int,B: assn,C: assn] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3690_ord__eq__less__subst,axiom,
! [A2: code_integer,F: assn > code_integer,B: assn,C: assn] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3691_ord__eq__less__subst,axiom,
! [A2: rat,F: assn > rat,B: assn,C: assn] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3692_ord__eq__less__subst,axiom,
! [A2: assn,F: nat > assn,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3693_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3694_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3695_ord__eq__less__subst,axiom,
! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3696_ord__eq__less__subst,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_3697_ord__less__eq__subst,axiom,
! [A2: assn,B: assn,F: assn > assn,C: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3698_ord__less__eq__subst,axiom,
! [A2: assn,B: assn,F: assn > nat,C: nat] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3699_ord__less__eq__subst,axiom,
! [A2: assn,B: assn,F: assn > int,C: int] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3700_ord__less__eq__subst,axiom,
! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3701_ord__less__eq__subst,axiom,
! [A2: assn,B: assn,F: assn > rat,C: rat] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3702_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > assn,C: assn] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3703_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3704_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3705_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3706_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_3707_order__less__irrefl,axiom,
! [X: assn] :
~ ( ord_less_assn @ X @ X ) ).
% order_less_irrefl
thf(fact_3708_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_3709_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_3710_order__less__irrefl,axiom,
! [X: code_integer] :
~ ( ord_le6747313008572928689nteger @ X @ X ) ).
% order_less_irrefl
thf(fact_3711_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_3712_order__less__subst1,axiom,
! [A2: assn,F: assn > assn,B: assn,C: assn] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3713_order__less__subst1,axiom,
! [A2: assn,F: nat > assn,B: nat,C: nat] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3714_order__less__subst1,axiom,
! [A2: assn,F: int > assn,B: int,C: int] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3715_order__less__subst1,axiom,
! [A2: assn,F: code_integer > assn,B: code_integer,C: code_integer] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3716_order__less__subst1,axiom,
! [A2: assn,F: rat > assn,B: rat,C: rat] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3717_order__less__subst1,axiom,
! [A2: nat,F: assn > nat,B: assn,C: assn] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3718_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3719_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3720_order__less__subst1,axiom,
! [A2: nat,F: code_integer > nat,B: code_integer,C: code_integer] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3721_order__less__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_3722_order__less__subst2,axiom,
! [A2: assn,B: assn,F: assn > assn,C: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_assn @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3723_order__less__subst2,axiom,
! [A2: assn,B: assn,F: assn > nat,C: nat] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3724_order__less__subst2,axiom,
! [A2: assn,B: assn,F: assn > int,C: int] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3725_order__less__subst2,axiom,
! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3726_order__less__subst2,axiom,
! [A2: assn,B: assn,F: assn > rat,C: rat] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3727_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > assn,C: assn] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_assn @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3728_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3729_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3730_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3731_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_3732_order__less__not__sym,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ~ ( ord_less_assn @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_3733_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_3734_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_3735_order__less__not__sym,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_3736_order__less__not__sym,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_3737_order__less__imp__triv,axiom,
! [X: assn,Y: assn,P: $o] :
( ( ord_less_assn @ X @ Y )
=> ( ( ord_less_assn @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3738_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3739_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3740_order__less__imp__triv,axiom,
! [X: code_integer,Y: code_integer,P: $o] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ( ord_le6747313008572928689nteger @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3741_order__less__imp__triv,axiom,
! [X: rat,Y: rat,P: $o] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_3742_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_3743_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_3744_linorder__less__linear,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
| ( X = Y )
| ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_3745_linorder__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
| ( X = Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_3746_order__less__imp__not__eq,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_3747_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_3748_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_3749_order__less__imp__not__eq,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_3750_order__less__imp__not__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_3751_order__less__imp__not__eq2,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_3752_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_3753_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_3754_order__less__imp__not__eq2,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_3755_order__less__imp__not__eq2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_3756_order__less__imp__not__less,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ~ ( ord_less_assn @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_3757_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_3758_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_3759_order__less__imp__not__less,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_3760_order__less__imp__not__less,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ~ ( ord_less_rat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_3761_Max__gr__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ X @ ( lattic1210392890313158908t_unit @ A ) )
= ( ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
& ( ord_le361264281704409273t_unit @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3762_Max__gr__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_o @ X @ ( lattic1921953407002678535_Max_o @ A ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3763_Max__gr__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3764_Max__gr__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le6747313008572928689nteger @ X @ ( lattic4901227151466704046nteger @ A ) )
= ( ? [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
& ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3765_Max__gr__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_rat @ X @ ( lattic7630753665789217321ax_rat @ A ) )
= ( ? [X2: rat] :
( ( member_rat @ X2 @ A )
& ( ord_less_rat @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3766_Max__gr__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3767_top__greatest,axiom,
! [A2: assn] : ( ord_less_eq_assn @ A2 @ top_top_assn ) ).
% top_greatest
thf(fact_3768_top__greatest,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_3769_top__greatest,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% top_greatest
thf(fact_3770_top__greatest,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% top_greatest
thf(fact_3771_top__greatest,axiom,
! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ top_top_filter_nat ) ).
% top_greatest
thf(fact_3772_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_3773_top_Oextremum__unique,axiom,
! [A2: assn] :
( ( ord_less_eq_assn @ top_top_assn @ A2 )
= ( A2 = top_top_assn ) ) ).
% top.extremum_unique
thf(fact_3774_top_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
= ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_3775_top_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
= ( A2 = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_3776_top_Oextremum__unique,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
= ( A2 = top_top_set_int ) ) ).
% top.extremum_unique
thf(fact_3777_top_Oextremum__unique,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
= ( A2 = top_top_filter_nat ) ) ).
% top.extremum_unique
thf(fact_3778_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_3779_top_Oextremum__uniqueI,axiom,
! [A2: assn] :
( ( ord_less_eq_assn @ top_top_assn @ A2 )
=> ( A2 = top_top_assn ) ) ).
% top.extremum_uniqueI
thf(fact_3780_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
=> ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_3781_top_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
=> ( A2 = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_3782_top_Oextremum__uniqueI,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
=> ( A2 = top_top_set_int ) ) ).
% top.extremum_uniqueI
thf(fact_3783_top_Oextremum__uniqueI,axiom,
! [A2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
=> ( A2 = top_top_filter_nat ) ) ).
% top.extremum_uniqueI
thf(fact_3784_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_3785_leD,axiom,
! [Y: assn,X: assn] :
( ( ord_less_eq_assn @ Y @ X )
=> ~ ( ord_less_assn @ X @ Y ) ) ).
% leD
thf(fact_3786_leD,axiom,
! [Y: code_integer,X: code_integer] :
( ( ord_le3102999989581377725nteger @ Y @ X )
=> ~ ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% leD
thf(fact_3787_leD,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ~ ( ord_less_filter_nat @ X @ Y ) ) ).
% leD
thf(fact_3788_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_3789_leD,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ~ ( ord_less_rat @ X @ Y ) ) ).
% leD
thf(fact_3790_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_3791_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_3792_leI,axiom,
! [X: code_integer,Y: code_integer] :
( ~ ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ord_le3102999989581377725nteger @ Y @ X ) ) ).
% leI
thf(fact_3793_leI,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ Y @ X ) ) ).
% leI
thf(fact_3794_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_3795_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_3796_nless__le,axiom,
! [A2: assn,B: assn] :
( ( ~ ( ord_less_assn @ A2 @ B ) )
= ( ~ ( ord_less_eq_assn @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3797_nless__le,axiom,
! [A2: code_integer,B: code_integer] :
( ( ~ ( ord_le6747313008572928689nteger @ A2 @ B ) )
= ( ~ ( ord_le3102999989581377725nteger @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3798_nless__le,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ~ ( ord_less_filter_nat @ A2 @ B ) )
= ( ~ ( ord_le2510731241096832064er_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3799_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3800_nless__le,axiom,
! [A2: rat,B: rat] :
( ( ~ ( ord_less_rat @ A2 @ B ) )
= ( ~ ( ord_less_eq_rat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3801_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3802_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_3803_antisym__conv1,axiom,
! [X: assn,Y: assn] :
( ~ ( ord_less_assn @ X @ Y )
=> ( ( ord_less_eq_assn @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3804_antisym__conv1,axiom,
! [X: code_integer,Y: code_integer] :
( ~ ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ( ord_le3102999989581377725nteger @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3805_antisym__conv1,axiom,
! [X: filter_nat,Y: filter_nat] :
( ~ ( ord_less_filter_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3806_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3807_antisym__conv1,axiom,
! [X: rat,Y: rat] :
( ~ ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3808_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3809_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_3810_antisym__conv2,axiom,
! [X: assn,Y: assn] :
( ( ord_less_eq_assn @ X @ Y )
=> ( ( ~ ( ord_less_assn @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3811_antisym__conv2,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ Y )
=> ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3812_antisym__conv2,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ~ ( ord_less_filter_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3813_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3814_antisym__conv2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ~ ( ord_less_rat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3815_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3816_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_3817_dense__ge,axiom,
! [Z: rat,Y: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ Z @ X3 )
=> ( ord_less_eq_rat @ Y @ X3 ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_ge
thf(fact_3818_dense__le,axiom,
! [Y: rat,Z: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ X3 @ Y )
=> ( ord_less_eq_rat @ X3 @ Z ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ).
% dense_le
thf(fact_3819_less__le__not__le,axiom,
( ord_less_assn
= ( ^ [X2: assn,Y3: assn] :
( ( ord_less_eq_assn @ X2 @ Y3 )
& ~ ( ord_less_eq_assn @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3820_less__le__not__le,axiom,
( ord_le6747313008572928689nteger
= ( ^ [X2: code_integer,Y3: code_integer] :
( ( ord_le3102999989581377725nteger @ X2 @ Y3 )
& ~ ( ord_le3102999989581377725nteger @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3821_less__le__not__le,axiom,
( ord_less_filter_nat
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
& ~ ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3822_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3823_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
& ~ ( ord_less_eq_rat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3824_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3825_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_3826_not__le__imp__less,axiom,
! [Y: code_integer,X: code_integer] :
( ~ ( ord_le3102999989581377725nteger @ Y @ X )
=> ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_3827_not__le__imp__less,axiom,
! [Y: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y @ X )
=> ( ord_less_rat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_3828_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_3829_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_3830_order_Oorder__iff__strict,axiom,
( ord_less_eq_assn
= ( ^ [A6: assn,B6: assn] :
( ( ord_less_assn @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3831_order_Oorder__iff__strict,axiom,
( ord_le3102999989581377725nteger
= ( ^ [A6: code_integer,B6: code_integer] :
( ( ord_le6747313008572928689nteger @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3832_order_Oorder__iff__strict,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [A6: filter_nat,B6: filter_nat] :
( ( ord_less_filter_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3833_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3834_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A6: rat,B6: rat] :
( ( ord_less_rat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3835_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3836_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A6: int,B6: int] :
( ( ord_less_int @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_3837_order_Ostrict__iff__order,axiom,
( ord_less_assn
= ( ^ [A6: assn,B6: assn] :
( ( ord_less_eq_assn @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3838_order_Ostrict__iff__order,axiom,
( ord_le6747313008572928689nteger
= ( ^ [A6: code_integer,B6: code_integer] :
( ( ord_le3102999989581377725nteger @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3839_order_Ostrict__iff__order,axiom,
( ord_less_filter_nat
= ( ^ [A6: filter_nat,B6: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3840_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3841_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A6: rat,B6: rat] :
( ( ord_less_eq_rat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3842_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3843_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_3844_order_Ostrict__trans1,axiom,
! [A2: assn,B: assn,C: assn] :
( ( ord_less_eq_assn @ A2 @ B )
=> ( ( ord_less_assn @ B @ C )
=> ( ord_less_assn @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3845_order_Ostrict__trans1,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3846_order_Ostrict__trans1,axiom,
! [A2: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( ord_less_filter_nat @ B @ C )
=> ( ord_less_filter_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3847_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3848_order_Ostrict__trans1,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3849_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3850_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_3851_order_Ostrict__trans2,axiom,
! [A2: assn,B: assn,C: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_eq_assn @ B @ C )
=> ( ord_less_assn @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3852_order_Ostrict__trans2,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3853_order_Ostrict__trans2,axiom,
! [A2: filter_nat,B: filter_nat,C: filter_nat] :
( ( ord_less_filter_nat @ A2 @ B )
=> ( ( ord_le2510731241096832064er_nat @ B @ C )
=> ( ord_less_filter_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3854_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3855_order_Ostrict__trans2,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3856_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3857_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_3858_order_Ostrict__iff__not,axiom,
( ord_less_assn
= ( ^ [A6: assn,B6: assn] :
( ( ord_less_eq_assn @ A6 @ B6 )
& ~ ( ord_less_eq_assn @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3859_order_Ostrict__iff__not,axiom,
( ord_le6747313008572928689nteger
= ( ^ [A6: code_integer,B6: code_integer] :
( ( ord_le3102999989581377725nteger @ A6 @ B6 )
& ~ ( ord_le3102999989581377725nteger @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3860_order_Ostrict__iff__not,axiom,
( ord_less_filter_nat
= ( ^ [A6: filter_nat,B6: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
& ~ ( ord_le2510731241096832064er_nat @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3861_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3862_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A6: rat,B6: rat] :
( ( ord_less_eq_rat @ A6 @ B6 )
& ~ ( ord_less_eq_rat @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3863_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3864_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A6: int,B6: int] :
( ( ord_less_eq_int @ A6 @ B6 )
& ~ ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_3865_dense__ge__bounded,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ Z @ X )
=> ( ! [W2: rat] :
( ( ord_less_rat @ Z @ W2 )
=> ( ( ord_less_rat @ W2 @ X )
=> ( ord_less_eq_rat @ Y @ W2 ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_3866_dense__le__bounded,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ! [W2: rat] :
( ( ord_less_rat @ X @ W2 )
=> ( ( ord_less_rat @ W2 @ Y )
=> ( ord_less_eq_rat @ W2 @ Z ) ) )
=> ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% dense_le_bounded
thf(fact_3867_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_assn
= ( ^ [B6: assn,A6: assn] :
( ( ord_less_assn @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3868_dual__order_Oorder__iff__strict,axiom,
( ord_le3102999989581377725nteger
= ( ^ [B6: code_integer,A6: code_integer] :
( ( ord_le6747313008572928689nteger @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3869_dual__order_Oorder__iff__strict,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [B6: filter_nat,A6: filter_nat] :
( ( ord_less_filter_nat @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3870_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B6: set_nat,A6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3871_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B6: rat,A6: rat] :
( ( ord_less_rat @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3872_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_nat @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3873_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B6: int,A6: int] :
( ( ord_less_int @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_3874_dual__order_Ostrict__iff__order,axiom,
( ord_less_assn
= ( ^ [B6: assn,A6: assn] :
( ( ord_less_eq_assn @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3875_dual__order_Ostrict__iff__order,axiom,
( ord_le6747313008572928689nteger
= ( ^ [B6: code_integer,A6: code_integer] :
( ( ord_le3102999989581377725nteger @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3876_dual__order_Ostrict__iff__order,axiom,
( ord_less_filter_nat
= ( ^ [B6: filter_nat,A6: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3877_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B6: set_nat,A6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3878_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B6: rat,A6: rat] :
( ( ord_less_eq_rat @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3879_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3880_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B6: int,A6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_3881_dual__order_Ostrict__trans1,axiom,
! [B: assn,A2: assn,C: assn] :
( ( ord_less_eq_assn @ B @ A2 )
=> ( ( ord_less_assn @ C @ B )
=> ( ord_less_assn @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3882_dual__order_Ostrict__trans1,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A2 )
=> ( ( ord_le6747313008572928689nteger @ C @ B )
=> ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3883_dual__order_Ostrict__trans1,axiom,
! [B: filter_nat,A2: filter_nat,C: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B @ A2 )
=> ( ( ord_less_filter_nat @ C @ B )
=> ( ord_less_filter_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3884_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3885_dual__order_Ostrict__trans1,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3886_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3887_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_3888_dual__order_Ostrict__trans2,axiom,
! [B: assn,A2: assn,C: assn] :
( ( ord_less_assn @ B @ A2 )
=> ( ( ord_less_eq_assn @ C @ B )
=> ( ord_less_assn @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3889_dual__order_Ostrict__trans2,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( ( ord_le3102999989581377725nteger @ C @ B )
=> ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3890_dual__order_Ostrict__trans2,axiom,
! [B: filter_nat,A2: filter_nat,C: filter_nat] :
( ( ord_less_filter_nat @ B @ A2 )
=> ( ( ord_le2510731241096832064er_nat @ C @ B )
=> ( ord_less_filter_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3891_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3892_dual__order_Ostrict__trans2,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_rat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3893_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3894_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_3895_dual__order_Ostrict__iff__not,axiom,
( ord_less_assn
= ( ^ [B6: assn,A6: assn] :
( ( ord_less_eq_assn @ B6 @ A6 )
& ~ ( ord_less_eq_assn @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3896_dual__order_Ostrict__iff__not,axiom,
( ord_le6747313008572928689nteger
= ( ^ [B6: code_integer,A6: code_integer] :
( ( ord_le3102999989581377725nteger @ B6 @ A6 )
& ~ ( ord_le3102999989581377725nteger @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3897_dual__order_Ostrict__iff__not,axiom,
( ord_less_filter_nat
= ( ^ [B6: filter_nat,A6: filter_nat] :
( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
& ~ ( ord_le2510731241096832064er_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3898_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B6: set_nat,A6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A6 )
& ~ ( ord_less_eq_set_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3899_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B6: rat,A6: rat] :
( ( ord_less_eq_rat @ B6 @ A6 )
& ~ ( ord_less_eq_rat @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3900_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3901_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B6: int,A6: int] :
( ( ord_less_eq_int @ B6 @ A6 )
& ~ ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_3902_order_Ostrict__implies__order,axiom,
! [A2: assn,B: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ord_less_eq_assn @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3903_order_Ostrict__implies__order,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3904_order_Ostrict__implies__order,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ord_less_filter_nat @ A2 @ B )
=> ( ord_le2510731241096832064er_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3905_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3906_order_Ostrict__implies__order,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3907_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3908_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_3909_dual__order_Ostrict__implies__order,axiom,
! [B: assn,A2: assn] :
( ( ord_less_assn @ B @ A2 )
=> ( ord_less_eq_assn @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3910_dual__order_Ostrict__implies__order,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3911_dual__order_Ostrict__implies__order,axiom,
! [B: filter_nat,A2: filter_nat] :
( ( ord_less_filter_nat @ B @ A2 )
=> ( ord_le2510731241096832064er_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3912_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3913_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ord_less_eq_rat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3914_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3915_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_3916_order__le__less,axiom,
( ord_less_eq_assn
= ( ^ [X2: assn,Y3: assn] :
( ( ord_less_assn @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3917_order__le__less,axiom,
( ord_le3102999989581377725nteger
= ( ^ [X2: code_integer,Y3: code_integer] :
( ( ord_le6747313008572928689nteger @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3918_order__le__less,axiom,
( ord_le2510731241096832064er_nat
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_less_filter_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3919_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3920_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X2: rat,Y3: rat] :
( ( ord_less_rat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3921_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3922_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_3923_order__less__le,axiom,
( ord_less_assn
= ( ^ [X2: assn,Y3: assn] :
( ( ord_less_eq_assn @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3924_order__less__le,axiom,
( ord_le6747313008572928689nteger
= ( ^ [X2: code_integer,Y3: code_integer] :
( ( ord_le3102999989581377725nteger @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3925_order__less__le,axiom,
( ord_less_filter_nat
= ( ^ [X2: filter_nat,Y3: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3926_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3927_order__less__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y3: rat] :
( ( ord_less_eq_rat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3928_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3929_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_3930_linorder__not__le,axiom,
! [X: code_integer,Y: code_integer] :
( ( ~ ( ord_le3102999989581377725nteger @ X @ Y ) )
= ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% linorder_not_le
thf(fact_3931_linorder__not__le,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y ) )
= ( ord_less_rat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_3932_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_3933_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_3934_linorder__not__less,axiom,
! [X: code_integer,Y: code_integer] :
( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
= ( ord_le3102999989581377725nteger @ Y @ X ) ) ).
% linorder_not_less
thf(fact_3935_linorder__not__less,axiom,
! [X: rat,Y: rat] :
( ( ~ ( ord_less_rat @ X @ Y ) )
= ( ord_less_eq_rat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_3936_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_3937_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_3938_order__less__imp__le,axiom,
! [X: assn,Y: assn] :
( ( ord_less_assn @ X @ Y )
=> ( ord_less_eq_assn @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3939_order__less__imp__le,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ord_le3102999989581377725nteger @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3940_order__less__imp__le,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_less_filter_nat @ X @ Y )
=> ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3941_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3942_order__less__imp__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3943_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3944_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_3945_order__le__neq__trans,axiom,
! [A2: assn,B: assn] :
( ( ord_less_eq_assn @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_assn @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3946_order__le__neq__trans,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3947_order__le__neq__trans,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3948_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3949_order__le__neq__trans,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3950_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3951_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_3952_order__neq__le__trans,axiom,
! [A2: assn,B: assn] :
( ( A2 != B )
=> ( ( ord_less_eq_assn @ A2 @ B )
=> ( ord_less_assn @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3953_order__neq__le__trans,axiom,
! [A2: code_integer,B: code_integer] :
( ( A2 != B )
=> ( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3954_order__neq__le__trans,axiom,
! [A2: filter_nat,B: filter_nat] :
( ( A2 != B )
=> ( ( ord_le2510731241096832064er_nat @ A2 @ B )
=> ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3955_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3956_order__neq__le__trans,axiom,
! [A2: rat,B: rat] :
( ( A2 != B )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3957_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3958_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_3959_order__le__less__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( ( ord_less_eq_assn @ X @ Y )
=> ( ( ord_less_assn @ Y @ Z )
=> ( ord_less_assn @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3960_order__le__less__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ Y )
=> ( ( ord_le6747313008572928689nteger @ Y @ Z )
=> ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3961_order__le__less__trans,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_less_filter_nat @ Y @ Z )
=> ( ord_less_filter_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3962_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3963_order__le__less__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3964_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3965_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_3966_order__less__le__trans,axiom,
! [X: assn,Y: assn,Z: assn] :
( ( ord_less_assn @ X @ Y )
=> ( ( ord_less_eq_assn @ Y @ Z )
=> ( ord_less_assn @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3967_order__less__le__trans,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ Y )
=> ( ( ord_le3102999989581377725nteger @ Y @ Z )
=> ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3968_order__less__le__trans,axiom,
! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
( ( ord_less_filter_nat @ X @ Y )
=> ( ( ord_le2510731241096832064er_nat @ Y @ Z )
=> ( ord_less_filter_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3969_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3970_order__less__le__trans,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( ord_less_rat @ X @ Y )
=> ( ( ord_less_eq_rat @ Y @ Z )
=> ( ord_less_rat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3971_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3972_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_3973_order__le__less__subst1,axiom,
! [A2: assn,F: assn > assn,B: assn,C: assn] :
( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3974_order__le__less__subst1,axiom,
! [A2: code_integer,F: assn > code_integer,B: assn,C: assn] :
( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
=> ( ( ord_less_assn @ B @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3975_order__le__less__subst1,axiom,
! [A2: assn,F: nat > assn,B: nat,C: nat] :
( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3976_order__le__less__subst1,axiom,
! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3977_order__le__less__subst1,axiom,
! [A2: assn,F: int > assn,B: int,C: int] :
( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3978_order__le__less__subst1,axiom,
! [A2: code_integer,F: int > code_integer,B: int,C: int] :
( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3979_order__le__less__subst1,axiom,
! [A2: assn,F: code_integer > assn,B: code_integer,C: code_integer] :
( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3980_order__le__less__subst1,axiom,
! [A2: code_integer,F: code_integer > code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3981_order__le__less__subst1,axiom,
! [A2: assn,F: rat > assn,B: rat,C: rat] :
( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3982_order__le__less__subst1,axiom,
! [A2: code_integer,F: rat > code_integer,B: rat,C: rat] :
( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_3983_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > assn,C: assn] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_assn @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3984_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > code_integer,C: code_integer] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3985_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3986_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3987_order__le__less__subst2,axiom,
! [A2: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3988_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > assn,C: assn] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_assn @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3989_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3990_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3991_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3992_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_3993_order__less__le__subst1,axiom,
! [A2: assn,F: rat > assn,B: rat,C: rat] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3994_order__less__le__subst1,axiom,
! [A2: code_integer,F: rat > code_integer,B: rat,C: rat] :
( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3995_order__less__le__subst1,axiom,
! [A2: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3996_order__less__le__subst1,axiom,
! [A2: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3997_order__less__le__subst1,axiom,
! [A2: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_eq_rat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3998_order__less__le__subst1,axiom,
! [A2: assn,F: nat > assn,B: nat,C: nat] :
( ( ord_less_assn @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_3999_order__less__le__subst1,axiom,
! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4000_order__less__le__subst1,axiom,
! [A2: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4001_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4002_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_4003_order__less__le__subst2,axiom,
! [A2: assn,B: assn,F: assn > assn,C: assn] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_less_eq_assn @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4004_order__less__le__subst2,axiom,
! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
( ( ord_less_assn @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
=> ( ! [X3: assn,Y2: assn] :
( ( ord_less_assn @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4005_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > assn,C: assn] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_assn @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4006_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4007_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > assn,C: assn] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_assn @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4008_order__less__le__subst2,axiom,
! [A2: int,B: int,F: int > code_integer,C: code_integer] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4009_order__less__le__subst2,axiom,
! [A2: code_integer,B: code_integer,F: code_integer > assn,C: assn] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_less_eq_assn @ ( F @ B ) @ C )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4010_order__less__le__subst2,axiom,
! [A2: code_integer,B: code_integer,F: code_integer > code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4011_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > assn,C: assn] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_assn @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4012_order__less__le__subst2,axiom,
! [A2: rat,B: rat,F: rat > code_integer,C: code_integer] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y2: rat] :
( ( ord_less_rat @ X3 @ Y2 )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_4013_linorder__le__less__linear,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ Y )
| ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_4014_linorder__le__less__linear,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
| ( ord_less_rat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_4015_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_4016_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_4017_order__le__imp__less__or__eq,axiom,
! [X: assn,Y: assn] :
( ( ord_less_eq_assn @ X @ Y )
=> ( ( ord_less_assn @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4018_order__le__imp__less__or__eq,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ Y )
=> ( ( ord_le6747313008572928689nteger @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4019_order__le__imp__less__or__eq,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_less_filter_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4020_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4021_order__le__imp__less__or__eq,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_less_rat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4022_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4023_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_4024_less__numeral__extra_I4_J,axiom,
~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ one_one_Code_natural ) ).
% less_numeral_extra(4)
thf(fact_4025_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_4026_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_4027_less__numeral__extra_I4_J,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% less_numeral_extra(4)
thf(fact_4028_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_4029_diff__strict__mono,axiom,
! [A2: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_4030_diff__strict__mono,axiom,
! [A2: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ D2 @ C )
=> ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_4031_diff__strict__mono,axiom,
! [A2: rat,B: rat,D2: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ D2 @ C )
=> ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_4032_diff__eq__diff__less,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A2 @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_4033_diff__eq__diff__less,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ( minus_8373710615458151222nteger @ A2 @ B )
= ( minus_8373710615458151222nteger @ C @ D2 ) )
=> ( ( ord_le6747313008572928689nteger @ A2 @ B )
= ( ord_le6747313008572928689nteger @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_4034_diff__eq__diff__less,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ( minus_minus_rat @ A2 @ B )
= ( minus_minus_rat @ C @ D2 ) )
=> ( ( ord_less_rat @ A2 @ B )
= ( ord_less_rat @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_4035_diff__strict__left__mono,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4036_diff__strict__left__mono,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A2 ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4037_diff__strict__left__mono,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_4038_diff__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_4039_diff__strict__right__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_4040_diff__strict__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_4041_bot_Oextremum__strict,axiom,
! [A2: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% bot.extremum_strict
thf(fact_4042_bot_Oextremum__strict,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% bot.extremum_strict
thf(fact_4043_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_4044_bot_Oextremum__strict,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_4045_bot_Oextremum__strict,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
~ ( ord_le2453136405763929_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).
% bot.extremum_strict
thf(fact_4046_bot_Oextremum__strict,axiom,
! [A2: assn] :
~ ( ord_less_assn @ A2 @ bot_bot_assn ) ).
% bot.extremum_strict
thf(fact_4047_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_4048_bot_Onot__eq__extremum,axiom,
! [A2: set_Product_unit] :
( ( A2 != bot_bo3957492148770167129t_unit )
= ( ord_le8056459307392131481t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4049_bot_Onot__eq__extremum,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
= ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4050_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4051_bot_Onot__eq__extremum,axiom,
! [A2: set_int] :
( ( A2 != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4052_bot_Onot__eq__extremum,axiom,
! [A2: set_Pr4532377907799695533_nat_o] :
( ( A2 != bot_bo7824918357723582233_nat_o )
= ( ord_le2453136405763929_nat_o @ bot_bo7824918357723582233_nat_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4053_bot_Onot__eq__extremum,axiom,
! [A2: assn] :
( ( A2 != bot_bot_assn )
= ( ord_less_assn @ bot_bot_assn @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4054_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_4055_not__psubset__empty,axiom,
! [A: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% not_psubset_empty
thf(fact_4056_not__psubset__empty,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_4057_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_4058_not__psubset__empty,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_4059_not__psubset__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
~ ( ord_le2453136405763929_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% not_psubset_empty
thf(fact_4060_psubsetE,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% psubsetE
thf(fact_4061_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B7 )
& ( A3 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_4062_psubset__imp__subset,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% psubset_imp_subset
thf(fact_4063_psubset__subset__trans,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C5 )
=> ( ord_less_set_nat @ A @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_4064_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B7 )
& ~ ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_4065_subset__psubset__trans,axiom,
! [A: set_nat,B4: set_nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_set_nat @ B4 @ C5 )
=> ( ord_less_set_nat @ A @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_4066_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B7: set_nat] :
( ( ord_less_set_nat @ A3 @ B7 )
| ( A3 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_4067_rangeI,axiom,
! [F: code_integer > code_integer,X: code_integer] : ( member_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ).
% rangeI
thf(fact_4068_rangeI,axiom,
! [F: product_unit > nat,X: product_unit] : ( member_nat @ ( F @ X ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_4069_rangeI,axiom,
! [F: product_unit > int,X: product_unit] : ( member_int @ ( F @ X ) @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_4070_rangeI,axiom,
! [F: product_unit > $o,X: product_unit] : ( member_o @ ( F @ X ) @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_4071_rangeI,axiom,
! [F: product_unit > product_unit,X: product_unit] : ( member_Product_unit @ ( F @ X ) @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_4072_rangeI,axiom,
! [F: $o > nat,X: $o] : ( member_nat @ ( F @ X ) @ ( image_o_nat2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_4073_rangeI,axiom,
! [F: $o > int,X: $o] : ( member_int @ ( F @ X ) @ ( image_o_int2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_4074_rangeI,axiom,
! [F: $o > $o,X: $o] : ( member_o @ ( F @ X ) @ ( image_o_o2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_4075_rangeI,axiom,
! [F: $o > product_unit,X: $o] : ( member_Product_unit @ ( F @ X ) @ ( image_o_Product_unit2 @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_4076_rangeI,axiom,
! [F: nat > code_integer,X: nat] : ( member_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_4077_range__eqI,axiom,
! [B: code_integer,F: code_integer > code_integer,X: code_integer] :
( ( B
= ( F @ X ) )
=> ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ) ).
% range_eqI
thf(fact_4078_range__eqI,axiom,
! [B: nat,F: product_unit > nat,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_4079_range__eqI,axiom,
! [B: int,F: product_unit > int,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_4080_range__eqI,axiom,
! [B: $o,F: product_unit > $o,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_4081_range__eqI,axiom,
! [B: product_unit,F: product_unit > product_unit,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_Product_unit @ B @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_4082_range__eqI,axiom,
! [B: nat,F: $o > nat,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_4083_range__eqI,axiom,
! [B: int,F: $o > int,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_o_int2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_4084_range__eqI,axiom,
! [B: $o,F: $o > $o,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_4085_range__eqI,axiom,
! [B: product_unit,F: $o > product_unit,X: $o] :
( ( B
= ( F @ X ) )
=> ( member_Product_unit @ B @ ( image_o_Product_unit2 @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_4086_range__eqI,axiom,
! [B: code_integer,F: nat > code_integer,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_4087_empty__not__UNIV,axiom,
bot_bo7824918357723582233_nat_o != top_to4869887016220417533_nat_o ).
% empty_not_UNIV
thf(fact_4088_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_4089_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_4090_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_4091_empty__not__UNIV,axiom,
bot_bot_set_int != top_top_set_int ).
% empty_not_UNIV
thf(fact_4092_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_4093_subset__UNIV,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% subset_UNIV
thf(fact_4094_subset__UNIV,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% subset_UNIV
thf(fact_4095_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_4096_psubset__imp__ex__mem,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2453136405763929_nat_o @ A @ B4 )
=> ? [B2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ B2 @ ( minus_1801376950450012436_nat_o @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4097_psubset__imp__ex__mem,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_set_int @ A @ B4 )
=> ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4098_psubset__imp__ex__mem,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_set_o @ A @ B4 )
=> ? [B2: $o] : ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4099_psubset__imp__ex__mem,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le8056459307392131481t_unit @ A @ B4 )
=> ? [B2: product_unit] : ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4100_psubset__imp__ex__mem,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_4101_insert__UNIV,axiom,
! [X: product_unit] :
( ( insert_Product_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% insert_UNIV
thf(fact_4102_insert__UNIV,axiom,
! [X: $o] :
( ( insert_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_4103_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_4104_insert__UNIV,axiom,
! [X: int] :
( ( insert_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% insert_UNIV
thf(fact_4105_Int__UNIV__left,axiom,
! [B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_4106_Int__UNIV__left,axiom,
! [B4: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_4107_Int__UNIV__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_4108_Int__UNIV__left,axiom,
! [B4: set_int] :
( ( inf_inf_set_int @ top_top_set_int @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_4109_Int__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% Int_UNIV_right
thf(fact_4110_Int__UNIV__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% Int_UNIV_right
thf(fact_4111_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_4112_Int__UNIV__right,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ A @ top_top_set_int )
= A ) ).
% Int_UNIV_right
thf(fact_4113_Un__UNIV__left,axiom,
! [B4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B4 )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_left
thf(fact_4114_Un__UNIV__left,axiom,
! [B4: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B4 )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_4115_Un__UNIV__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_4116_Un__UNIV__left,axiom,
! [B4: set_int] :
( ( sup_sup_set_int @ top_top_set_int @ B4 )
= top_top_set_int ) ).
% Un_UNIV_left
thf(fact_4117_Un__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_right
thf(fact_4118_Un__UNIV__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_4119_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_4120_Un__UNIV__right,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ A @ top_top_set_int )
= top_top_set_int ) ).
% Un_UNIV_right
thf(fact_4121_type__copy__ex__RepI,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,F2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ top_to4869887016220417533_nat_o )
=> ( ( ? [X7: produc3658429121746597890et_nat > $o] : ( F2 @ X7 ) )
= ( ? [B6: assn] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_4122_type__copy__ex__RepI,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,F2: int > $o] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( ? [X7: int] : ( F2 @ X7 ) )
= ( ? [B6: code_integer] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_4123_type__copy__ex__RepI,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,F2: $o > $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( ? [X7: $o] : ( F2 @ X7 ) )
= ( ? [B6: product_unit] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_4124_type__copy__ex__RepI,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,F2: nat > $o] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( ? [X7: nat] : ( F2 @ X7 ) )
= ( ? [B6: code_natural] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_4125_type__copy__obj__one__point__absE,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,S2: assn] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ top_to4869887016220417533_nat_o )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
( S2
!= ( Abs2 @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_4126_type__copy__obj__one__point__absE,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,S2: code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ~ ! [X3: int] :
( S2
!= ( Abs2 @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_4127_type__copy__obj__one__point__absE,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,S2: product_unit] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ~ ! [X3: $o] :
( S2
!= ( Abs2 @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_4128_type__copy__obj__one__point__absE,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,S2: code_natural] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ~ ! [X3: nat] :
( S2
!= ( Abs2 @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_4129_less__1__mult,axiom,
! [M4: code_natural,N2: code_natural] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ M4 )
=> ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ N2 )
=> ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_4130_less__1__mult,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M4 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_4131_less__1__mult,axiom,
! [M4: int,N2: int] :
( ( ord_less_int @ one_one_int @ M4 )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_4132_less__1__mult,axiom,
! [M4: code_integer,N2: code_integer] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M4 )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N2 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_4133_less__1__mult,axiom,
! [M4: rat,N2: rat] :
( ( ord_less_rat @ one_one_rat @ M4 )
=> ( ( ord_less_rat @ one_one_rat @ N2 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M4 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_4134_infinite__growing,axiom,
! [X4: set_Product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ? [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ X4 )
& ( ord_le361264281704409273t_unit @ X3 @ Xa2 ) ) )
=> ~ ( finite4290736615968046902t_unit @ X4 ) ) ) ).
% infinite_growing
thf(fact_4135_infinite__growing,axiom,
! [X4: set_o] :
( ( X4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X4 )
=> ? [Xa2: $o] :
( ( member_o @ Xa2 @ X4 )
& ( ord_less_o @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite_o @ X4 ) ) ) ).
% infinite_growing
thf(fact_4136_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ X4 )
& ( ord_less_nat @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_4137_infinite__growing,axiom,
! [X4: set_int] :
( ( X4 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ? [Xa2: int] :
( ( member_int @ Xa2 @ X4 )
& ( ord_less_int @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite_int @ X4 ) ) ) ).
% infinite_growing
thf(fact_4138_infinite__growing,axiom,
! [X4: set_Code_integer] :
( ( X4 != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ X4 )
=> ? [Xa2: code_integer] :
( ( member_Code_integer @ Xa2 @ X4 )
& ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
=> ~ ( finite6017078050557962740nteger @ X4 ) ) ) ).
% infinite_growing
thf(fact_4139_infinite__growing,axiom,
! [X4: set_rat] :
( ( X4 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ X4 )
=> ? [Xa2: rat] :
( ( member_rat @ Xa2 @ X4 )
& ( ord_less_rat @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite_rat @ X4 ) ) ) ).
% infinite_growing
thf(fact_4140_ex__min__if__finite,axiom,
! [S: set_Product_unit] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( S != bot_bo3957492148770167129t_unit )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
& ~ ? [Xa2: product_unit] :
( ( member_Product_unit @ Xa2 @ S )
& ( ord_le361264281704409273t_unit @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4141_ex__min__if__finite,axiom,
! [S: set_o] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ S )
& ~ ? [Xa2: $o] :
( ( member_o @ Xa2 @ S )
& ( ord_less_o @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4142_ex__min__if__finite,axiom,
! [S: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ S )
=> ( ( S != bot_bo7824918357723582233_nat_o )
=> ? [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ S )
& ~ ? [Xa2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ Xa2 @ S )
& ( ord_le8176553561544679545_nat_o @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4143_ex__min__if__finite,axiom,
! [S: set_assn] :
( ( finite_finite_assn @ S )
=> ( ( S != bot_bot_set_assn )
=> ? [X3: assn] :
( ( member_assn @ X3 @ S )
& ~ ? [Xa2: assn] :
( ( member_assn @ Xa2 @ S )
& ( ord_less_assn @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4144_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa2: nat] :
( ( member_nat @ Xa2 @ S )
& ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4145_ex__min__if__finite,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S )
& ~ ? [Xa2: int] :
( ( member_int @ Xa2 @ S )
& ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4146_ex__min__if__finite,axiom,
! [S: set_Code_integer] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ? [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
& ~ ? [Xa2: code_integer] :
( ( member_Code_integer @ Xa2 @ S )
& ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4147_ex__min__if__finite,axiom,
! [S: set_rat] :
( ( finite_finite_rat @ S )
=> ( ( S != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ S )
& ~ ? [Xa2: rat] :
( ( member_rat @ Xa2 @ S )
& ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4148_less__cSupD,axiom,
! [X4: set_int,Z: int] :
( ( X4 != bot_bot_set_int )
=> ( ( ord_less_int @ Z @ ( complete_Sup_Sup_int @ X4 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ X4 )
& ( ord_less_int @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_4149_less__cSupD,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( ord_less_nat @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_4150_less__cSupD,axiom,
! [X4: set_Product_unit,Z: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ Z @ ( comple4687483117567863418t_unit @ X4 ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
& ( ord_le361264281704409273t_unit @ Z @ X3 ) ) ) ) ).
% less_cSupD
thf(fact_4151_less__cSupE,axiom,
! [Y: int,X4: set_int] :
( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X4 ) )
=> ( ( X4 != bot_bot_set_int )
=> ~ ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ~ ( ord_less_int @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_4152_less__cSupE,axiom,
! [Y: nat,X4: set_nat] :
( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
=> ( ( X4 != bot_bot_set_nat )
=> ~ ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_4153_less__cSupE,axiom,
! [Y: product_unit,X4: set_Product_unit] :
( ( ord_le361264281704409273t_unit @ Y @ ( comple4687483117567863418t_unit @ X4 ) )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ~ ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ~ ( ord_le361264281704409273t_unit @ Y @ X3 ) ) ) ) ).
% less_cSupE
thf(fact_4154_Max__in,axiom,
! [A: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( member_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ A ) ) ) ).
% Max_in
thf(fact_4155_Max__in,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ A ) ) ) ).
% Max_in
thf(fact_4156_Max__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( member_o @ ( lattic1921953407002678535_Max_o @ A ) @ A ) ) ) ).
% Max_in
thf(fact_4157_Max__in,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ).
% Max_in
thf(fact_4158_Max__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).
% Max_in
thf(fact_4159_range__subsetD,axiom,
! [F: code_integer > code_integer,B4: set_Code_integer,I2: code_integer] :
( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) @ B4 )
=> ( member_Code_integer @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4160_range__subsetD,axiom,
! [F: product_unit > int,B4: set_int,I2: product_unit] :
( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) @ B4 )
=> ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4161_range__subsetD,axiom,
! [F: product_unit > $o,B4: set_o,I2: product_unit] :
( ( ord_less_eq_set_o @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) @ B4 )
=> ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4162_range__subsetD,axiom,
! [F: product_unit > product_unit,B4: set_Product_unit,I2: product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) @ B4 )
=> ( member_Product_unit @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4163_range__subsetD,axiom,
! [F: $o > int,B4: set_int,I2: $o] :
( ( ord_less_eq_set_int @ ( image_o_int2 @ F @ top_top_set_o ) @ B4 )
=> ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4164_range__subsetD,axiom,
! [F: $o > $o,B4: set_o,I2: $o] :
( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ top_top_set_o ) @ B4 )
=> ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4165_range__subsetD,axiom,
! [F: $o > product_unit,B4: set_Product_unit,I2: $o] :
( ( ord_le3507040750410214029t_unit @ ( image_o_Product_unit2 @ F @ top_top_set_o ) @ B4 )
=> ( member_Product_unit @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4166_range__subsetD,axiom,
! [F: nat > code_integer,B4: set_Code_integer,I2: nat] :
( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) @ B4 )
=> ( member_Code_integer @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4167_range__subsetD,axiom,
! [F: nat > int,B4: set_int,I2: nat] :
( ( ord_less_eq_set_int @ ( image_nat_int2 @ F @ top_top_set_nat ) @ B4 )
=> ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4168_range__subsetD,axiom,
! [F: nat > $o,B4: set_o,I2: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ top_top_set_nat ) @ B4 )
=> ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_4169_surj__vimage__empty,axiom,
! [F: code_integer > code_integer,A: set_Code_integer] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( vimage1318793397690208119nteger @ F @ A )
= bot_bo3990330152332043303nteger )
= ( A = bot_bo3990330152332043303nteger ) ) ) ).
% surj_vimage_empty
thf(fact_4170_surj__vimage__empty,axiom,
! [F: product_unit > product_unit,A: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage7995052115951654139t_unit @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_4171_surj__vimage__empty,axiom,
! [F: product_unit > $o,A: set_o] :
( ( ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit )
= top_top_set_o )
=> ( ( ( vimage8808348054653608838unit_o @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_4172_surj__vimage__empty,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( vimage6253328473476588386it_nat @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_4173_surj__vimage__empty,axiom,
! [F: product_unit > int,A: set_int] :
( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
= top_top_set_int )
=> ( ( ( vimage6250838002967538110it_int @ F @ A )
= bot_bo3957492148770167129t_unit )
= ( A = bot_bot_set_int ) ) ) ).
% surj_vimage_empty
thf(fact_4174_surj__vimage__empty,axiom,
! [F: $o > product_unit,A: set_Product_unit] :
( ( ( image_o_Product_unit2 @ F @ top_top_set_o )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage4222377557387472176t_unit @ F @ A )
= bot_bot_set_o )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_4175_surj__vimage__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( vimage_o_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_4176_surj__vimage__empty,axiom,
! [F: $o > nat,A: set_nat] :
( ( ( image_o_nat2 @ F @ top_top_set_o )
= top_top_set_nat )
=> ( ( ( vimage_o_nat @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_4177_surj__vimage__empty,axiom,
! [F: $o > int,A: set_int] :
( ( ( image_o_int2 @ F @ top_top_set_o )
= top_top_set_int )
=> ( ( ( vimage_o_int @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_int ) ) ) ).
% surj_vimage_empty
thf(fact_4178_surj__vimage__empty,axiom,
! [F: nat > code_integer,A: set_Code_integer] :
( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
= top_to4645266643341252675nteger )
=> ( ( ( vimage6104394758324456170nteger @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo3990330152332043303nteger ) ) ) ).
% surj_vimage_empty
thf(fact_4179_acyclicI__order,axiom,
! [R2: set_Pr1261947904930325089at_nat,F: nat > assn] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
=> ( ord_less_assn @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% acyclicI_order
thf(fact_4180_acyclicI__order,axiom,
! [R2: set_Pr958786334691620121nt_int,F: int > assn] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
=> ( ord_less_assn @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3136567867383547549ic_int @ R2 ) ) ).
% acyclicI_order
thf(fact_4181_acyclicI__order,axiom,
! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
=> ( ord_less_nat @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% acyclicI_order
thf(fact_4182_acyclicI__order,axiom,
! [R2: set_Pr958786334691620121nt_int,F: int > nat] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
=> ( ord_less_nat @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3136567867383547549ic_int @ R2 ) ) ).
% acyclicI_order
thf(fact_4183_acyclicI__order,axiom,
! [R2: set_Pr1261947904930325089at_nat,F: nat > int] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
=> ( ord_less_int @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% acyclicI_order
thf(fact_4184_acyclicI__order,axiom,
! [R2: set_Pr958786334691620121nt_int,F: int > int] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
=> ( ord_less_int @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3136567867383547549ic_int @ R2 ) ) ).
% acyclicI_order
thf(fact_4185_acyclicI__order,axiom,
! [R2: set_Pr1261947904930325089at_nat,F: nat > code_integer] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
=> ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% acyclicI_order
thf(fact_4186_acyclicI__order,axiom,
! [R2: set_Pr958786334691620121nt_int,F: int > code_integer] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
=> ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3136567867383547549ic_int @ R2 ) ) ).
% acyclicI_order
thf(fact_4187_acyclicI__order,axiom,
! [R2: set_Pr1261947904930325089at_nat,F: nat > rat] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
=> ( ord_less_rat @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3139058337892597825ic_nat @ R2 ) ) ).
% acyclicI_order
thf(fact_4188_acyclicI__order,axiom,
! [R2: set_Pr958786334691620121nt_int,F: int > rat] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
=> ( ord_less_rat @ ( F @ B2 ) @ ( F @ A4 ) ) )
=> ( transi3136567867383547549ic_int @ R2 ) ) ).
% acyclicI_order
thf(fact_4189_type__definition_OAbs__image,axiom,
! [Rep2: set_nat > nat,Abs2: nat > set_nat,A: set_nat] :
( ( type_d202864840109488260at_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_nat_set_nat @ Abs2 @ A )
= top_top_set_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_4190_type__definition_OAbs__image,axiom,
! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,A: set_Code_integer] :
( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ A )
=> ( ( image_4470545334726330049nteger @ Abs2 @ A )
= top_to4645266643341252675nteger ) ) ).
% type_definition.Abs_image
thf(fact_4191_type__definition_OAbs__image,axiom,
! [Rep2: code_integer > nat,Abs2: nat > code_integer,A: set_nat] :
( ( type_d8368584451094728027er_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_1215581382706833972nteger @ Abs2 @ A )
= top_to4645266643341252675nteger ) ) ).
% type_definition.Abs_image
thf(fact_4192_type__definition_OAbs__image,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,A: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_nat_nat2 @ Abs2 @ A )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_4193_type__definition_OAbs__image,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( image_6451031022632655280o_assn @ Abs2 @ A )
= top_top_set_assn ) ) ).
% type_definition.Abs_image
thf(fact_4194_type__definition_OAbs__image,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( image_1587234942943678608nteger @ Abs2 @ A )
= top_to4645266643341252675nteger ) ) ).
% type_definition.Abs_image
thf(fact_4195_type__definition_OAbs__image,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( image_o_Product_unit2 @ Abs2 @ A )
= top_to1996260823553986621t_unit ) ) ).
% type_definition.Abs_image
thf(fact_4196_type__definition_OAbs__image,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_39176534463551487atural @ Abs2 @ A )
= top_to9103716463852441742atural ) ) ).
% type_definition.Abs_image
thf(fact_4197_type__definition_ORep__range,axiom,
! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,A: set_Code_integer] :
( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ A )
=> ( ( image_4470545334726330049nteger @ Rep2 @ top_to4645266643341252675nteger )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4198_type__definition_ORep__range,axiom,
! [Rep2: nat > set_nat,Abs2: set_nat > nat,A: set_set_nat] :
( ( type_d3768499744764857476et_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_nat_set_nat @ Rep2 @ top_top_set_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4199_type__definition_ORep__range,axiom,
! [Rep2: nat > nat,Abs2: nat > nat,A: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_nat_nat2 @ Rep2 @ top_top_set_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4200_type__definition_ORep__range,axiom,
! [Rep2: nat > code_integer,Abs2: code_integer > nat,A: set_Code_integer] :
( ( type_d8633139899873770843nteger @ Rep2 @ Abs2 @ A )
=> ( ( image_1215581382706833972nteger @ Rep2 @ top_top_set_nat )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4201_type__definition_ORep__range,axiom,
! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o] :
( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
=> ( ( image_5465535159827709278_nat_o @ Rep2 @ top_top_set_assn )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4202_type__definition_ORep__range,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
=> ( ( image_948535463418740880er_int @ Rep2 @ top_to4645266643341252675nteger )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4203_type__definition_ORep__range,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
=> ( ( image_Product_unit_o2 @ Rep2 @ top_to1996260823553986621t_unit )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4204_type__definition_ORep__range,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
=> ( ( image_6215854944615398399al_nat @ Rep2 @ top_to9103716463852441742atural )
= A ) ) ).
% type_definition.Rep_range
thf(fact_4205_refl__Id,axiom,
refl_on_Product_unit @ top_to1996260823553986621t_unit @ id_Product_unit ).
% refl_Id
thf(fact_4206_refl__Id,axiom,
refl_on_o @ top_top_set_o @ id_o2 ).
% refl_Id
thf(fact_4207_refl__Id,axiom,
refl_on_nat @ top_top_set_nat @ id_nat ).
% refl_Id
thf(fact_4208_refl__Id,axiom,
refl_on_int @ top_top_set_int @ id_int ).
% refl_Id
thf(fact_4209_boolean__algebra_Ocomplement__unique,axiom,
! [A2: product_unit,X: product_unit,Y: product_unit] :
( ( ( inf_inf_Product_unit @ A2 @ X )
= bot_bot_Product_unit )
=> ( ( ( sup_sup_Product_unit @ A2 @ X )
= top_top_Product_unit )
=> ( ( ( inf_inf_Product_unit @ A2 @ Y )
= bot_bot_Product_unit )
=> ( ( ( sup_sup_Product_unit @ A2 @ Y )
= top_top_Product_unit )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4210_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Pr4532377907799695533_nat_o,X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A2 @ X )
= bot_bo7824918357723582233_nat_o )
=> ( ( ( sup_su5209123915105501825_nat_o @ A2 @ X )
= top_to4869887016220417533_nat_o )
=> ( ( ( inf_in1906310914598751387_nat_o @ A2 @ Y )
= bot_bo7824918357723582233_nat_o )
=> ( ( ( sup_su5209123915105501825_nat_o @ A2 @ Y )
= top_to4869887016220417533_nat_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4211_boolean__algebra_Ocomplement__unique,axiom,
! [A2: assn,X: assn,Y: assn] :
( ( ( inf_inf_assn @ A2 @ X )
= bot_bot_assn )
=> ( ( ( sup_sup_assn @ A2 @ X )
= top_top_assn )
=> ( ( ( inf_inf_assn @ A2 @ Y )
= bot_bot_assn )
=> ( ( ( sup_sup_assn @ A2 @ Y )
= top_top_assn )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4212_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ X )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ A2 @ X )
= top_to1996260823553986621t_unit )
=> ( ( ( inf_in4660618365625256667t_unit @ A2 @ Y )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ A2 @ Y )
= top_to1996260823553986621t_unit )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4213_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_o,X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ A2 @ X )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A2 @ X )
= top_top_set_o )
=> ( ( ( inf_inf_set_o @ A2 @ Y )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ A2 @ Y )
= top_top_set_o )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4214_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4215_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_int,X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ A2 @ X )
= bot_bot_set_int )
=> ( ( ( sup_sup_set_int @ A2 @ X )
= top_top_set_int )
=> ( ( ( inf_inf_set_int @ A2 @ Y )
= bot_bot_set_int )
=> ( ( ( sup_sup_set_int @ A2 @ Y )
= top_top_set_int )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4216_union__fold__insert,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( sup_sup_set_int @ A @ B4 )
= ( finite1723285575846220376et_int @ insert_int @ B4 @ A ) ) ) ).
% union_fold_insert
thf(fact_4217_union__fold__insert,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( sup_sup_set_o @ A @ B4 )
= ( finite_fold_o_set_o @ insert_o @ B4 @ A ) ) ) ).
% union_fold_insert
thf(fact_4218_union__fold__insert,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( sup_su793286257634532545t_unit @ A @ B4 )
= ( finite2740360166580943592t_unit @ insert_Product_unit @ B4 @ A ) ) ) ).
% union_fold_insert
thf(fact_4219_union__fold__insert,axiom,
! [A: set_Code_integer,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( sup_su848401254843788991nteger @ A @ B4 )
= ( finite7486036041278456378nteger @ insert_Code_integer @ B4 @ A ) ) ) ).
% union_fold_insert
thf(fact_4220_union__fold__insert,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( sup_sup_set_nat @ A @ B4 )
= ( finite5529483035118572448et_nat @ insert_nat @ B4 @ A ) ) ) ).
% union_fold_insert
thf(fact_4221_finite__linorder__min__induct,axiom,
! [A: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [B2: product_unit,A9: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A9 )
=> ( ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ A9 )
=> ( ord_le361264281704409273t_unit @ B2 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_Product_unit @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4222_finite__linorder__min__induct,axiom,
! [A: set_o,P: set_o > $o] :
( ( finite_finite_o @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [B2: $o,A9: set_o] :
( ( finite_finite_o @ A9 )
=> ( ! [X6: $o] :
( ( member_o @ X6 @ A9 )
=> ( ord_less_o @ B2 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_o @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4223_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B2: nat,A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A9 )
=> ( ord_less_nat @ B2 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_nat @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4224_finite__linorder__min__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B2: int,A9: set_int] :
( ( finite_finite_int @ A9 )
=> ( ! [X6: int] :
( ( member_int @ X6 @ A9 )
=> ( ord_less_int @ B2 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_int @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4225_finite__linorder__min__induct,axiom,
! [A: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ A )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [B2: code_integer,A9: set_Code_integer] :
( ( finite6017078050557962740nteger @ A9 )
=> ( ! [X6: code_integer] :
( ( member_Code_integer @ X6 @ A9 )
=> ( ord_le6747313008572928689nteger @ B2 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_Code_integer @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4226_finite__linorder__min__induct,axiom,
! [A: set_rat,P: set_rat > $o] :
( ( finite_finite_rat @ A )
=> ( ( P @ bot_bot_set_rat )
=> ( ! [B2: rat,A9: set_rat] :
( ( finite_finite_rat @ A9 )
=> ( ! [X6: rat] :
( ( member_rat @ X6 @ A9 )
=> ( ord_less_rat @ B2 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_rat @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4227_finite__linorder__max__induct,axiom,
! [A: set_Product_unit,P: set_Product_unit > $o] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( P @ bot_bo3957492148770167129t_unit )
=> ( ! [B2: product_unit,A9: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A9 )
=> ( ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ A9 )
=> ( ord_le361264281704409273t_unit @ X6 @ B2 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_Product_unit @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4228_finite__linorder__max__induct,axiom,
! [A: set_o,P: set_o > $o] :
( ( finite_finite_o @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [B2: $o,A9: set_o] :
( ( finite_finite_o @ A9 )
=> ( ! [X6: $o] :
( ( member_o @ X6 @ A9 )
=> ( ord_less_o @ X6 @ B2 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_o @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4229_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B2: nat,A9: set_nat] :
( ( finite_finite_nat @ A9 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A9 )
=> ( ord_less_nat @ X6 @ B2 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_nat @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4230_finite__linorder__max__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B2: int,A9: set_int] :
( ( finite_finite_int @ A9 )
=> ( ! [X6: int] :
( ( member_int @ X6 @ A9 )
=> ( ord_less_int @ X6 @ B2 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_int @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4231_finite__linorder__max__induct,axiom,
! [A: set_Code_integer,P: set_Code_integer > $o] :
( ( finite6017078050557962740nteger @ A )
=> ( ( P @ bot_bo3990330152332043303nteger )
=> ( ! [B2: code_integer,A9: set_Code_integer] :
( ( finite6017078050557962740nteger @ A9 )
=> ( ! [X6: code_integer] :
( ( member_Code_integer @ X6 @ A9 )
=> ( ord_le6747313008572928689nteger @ X6 @ B2 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_Code_integer @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4232_finite__linorder__max__induct,axiom,
! [A: set_rat,P: set_rat > $o] :
( ( finite_finite_rat @ A )
=> ( ( P @ bot_bot_set_rat )
=> ( ! [B2: rat,A9: set_rat] :
( ( finite_finite_rat @ A9 )
=> ( ! [X6: rat] :
( ( member_rat @ X6 @ A9 )
=> ( ord_less_rat @ X6 @ B2 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert_rat @ B2 @ A9 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4233_finite__Sup__less__iff,axiom,
! [X4: set_int,A2: int] :
( ( finite_finite_int @ X4 )
=> ( ( X4 != bot_bot_set_int )
=> ( ( ord_less_int @ ( complete_Sup_Sup_int @ X4 ) @ A2 )
= ( ! [X2: int] :
( ( member_int @ X2 @ X4 )
=> ( ord_less_int @ X2 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4234_finite__Sup__less__iff,axiom,
! [X4: set_nat,A2: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_nat @ X2 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4235_finite__Sup__less__iff,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ ( comple4687483117567863418t_unit @ X4 ) @ A2 )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ X4 )
=> ( ord_le361264281704409273t_unit @ X2 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4236_Max__eq__iff,axiom,
! [A: set_Code_integer,M4: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ( lattic4901227151466704046nteger @ A )
= M4 )
= ( ( member_Code_integer @ M4 @ A )
& ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le3102999989581377725nteger @ X2 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4237_Max__eq__iff,axiom,
! [A: set_Product_unit,M4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ( lattic1210392890313158908t_unit @ A )
= M4 )
= ( ( member_Product_unit @ M4 @ A )
& ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ X2 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4238_Max__eq__iff,axiom,
! [A: set_o,M4: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ( lattic1921953407002678535_Max_o @ A )
= M4 )
= ( ( member_o @ M4 @ A )
& ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4239_Max__eq__iff,axiom,
! [A: set_rat,M4: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ( lattic7630753665789217321ax_rat @ A )
= M4 )
= ( ( member_rat @ M4 @ A )
& ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X2 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4240_Max__eq__iff,axiom,
! [A: set_int,M4: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ( lattic8263393255366662781ax_int @ A )
= M4 )
= ( ( member_int @ M4 @ A )
& ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ X2 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4241_Max__eq__iff,axiom,
! [A: set_nat,M4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A )
= M4 )
= ( ( member_nat @ M4 @ A )
& ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4242_Max__ge__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le3102999989581377725nteger @ X @ ( lattic4901227151466704046nteger @ A ) )
= ( ? [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
& ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4243_Max__ge__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1210392890313158908t_unit @ A ) )
= ( ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
& ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4244_Max__ge__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic1921953407002678535_Max_o @ A ) )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4245_Max__ge__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ X @ ( lattic7630753665789217321ax_rat @ A ) )
= ( ? [X2: rat] :
( ( member_rat @ X2 @ A )
& ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4246_Max__ge__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4247_Max__ge__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4248_eq__Max__iff,axiom,
! [A: set_Code_integer,M4: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( M4
= ( lattic4901227151466704046nteger @ A ) )
= ( ( member_Code_integer @ M4 @ A )
& ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le3102999989581377725nteger @ X2 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4249_eq__Max__iff,axiom,
! [A: set_Product_unit,M4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( M4
= ( lattic1210392890313158908t_unit @ A ) )
= ( ( member_Product_unit @ M4 @ A )
& ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ X2 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4250_eq__Max__iff,axiom,
! [A: set_o,M4: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( M4
= ( lattic1921953407002678535_Max_o @ A ) )
= ( ( member_o @ M4 @ A )
& ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4251_eq__Max__iff,axiom,
! [A: set_rat,M4: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( M4
= ( lattic7630753665789217321ax_rat @ A ) )
= ( ( member_rat @ M4 @ A )
& ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X2 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4252_eq__Max__iff,axiom,
! [A: set_int,M4: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( M4
= ( lattic8263393255366662781ax_int @ A ) )
= ( ( member_int @ M4 @ A )
& ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ X2 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4253_eq__Max__iff,axiom,
! [A: set_nat,M4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M4
= ( lattic8265883725875713057ax_nat @ A ) )
= ( ( member_nat @ M4 @ A )
& ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4254_Max_OboundedE,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
=> ! [A10: code_integer] :
( ( member_Code_integer @ A10 @ A )
=> ( ord_le3102999989581377725nteger @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_4255_Max_OboundedE,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
=> ! [A10: product_unit] :
( ( member_Product_unit @ A10 @ A )
=> ( ord_le3221252021190050221t_unit @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_4256_Max_OboundedE,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
=> ! [A10: $o] :
( ( member_o @ A10 @ A )
=> ( ord_less_eq_o @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_4257_Max_OboundedE,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
=> ! [A10: rat] :
( ( member_rat @ A10 @ A )
=> ( ord_less_eq_rat @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_4258_Max_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
=> ! [A10: int] :
( ( member_int @ A10 @ A )
=> ( ord_less_eq_int @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_4259_Max_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A )
=> ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_4260_Max_OboundedI,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ A )
=> ( ord_le3102999989581377725nteger @ A4 @ X ) )
=> ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_4261_Max_OboundedI,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ord_le3221252021190050221t_unit @ A4 @ X ) )
=> ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_4262_Max_OboundedI,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_o @ A4 @ X ) )
=> ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_4263_Max_OboundedI,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ! [A4: rat] :
( ( member_rat @ A4 @ A )
=> ( ord_less_eq_rat @ A4 @ X ) )
=> ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_4264_Max_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ A4 @ X ) )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_4265_Max_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_4266_Max__insert2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ A )
=> ( ord_less_eq_o @ B2 @ A2 ) )
=> ( ( lattic1921953407002678535_Max_o @ ( insert_o @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_4267_Max__insert2,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ A )
=> ( ord_le3221252021190050221t_unit @ B2 @ A2 ) )
=> ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_4268_Max__insert2,axiom,
! [A: set_Code_integer,A2: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ A )
=> ( ord_le3102999989581377725nteger @ B2 @ A2 ) )
=> ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_4269_Max__insert2,axiom,
! [A: set_rat,A2: rat] :
( ( finite_finite_rat @ A )
=> ( ! [B2: rat] :
( ( member_rat @ B2 @ A )
=> ( ord_less_eq_rat @ B2 @ A2 ) )
=> ( ( lattic7630753665789217321ax_rat @ ( insert_rat @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_4270_Max__insert2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ! [B2: int] :
( ( member_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ A2 ) )
=> ( ( lattic8263393255366662781ax_int @ ( insert_int @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_4271_Max__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A2 ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_4272_remove__subset,axiom,
! [X: product_unit,S: set_Product_unit] :
( ( member_Product_unit @ X @ S )
=> ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ S ) ) ).
% remove_subset
thf(fact_4273_remove__subset,axiom,
! [X: $o,S: set_o] :
( ( member_o @ X @ S )
=> ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).
% remove_subset
thf(fact_4274_remove__subset,axiom,
! [X: int,S: set_int] :
( ( member_int @ X @ S )
=> ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int @ X @ bot_bot_set_int ) ) @ S ) ) ).
% remove_subset
thf(fact_4275_remove__subset,axiom,
! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ S )
=> ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).
% remove_subset
thf(fact_4276_remove__subset,axiom,
! [X: nat,S: set_nat] :
( ( member_nat @ X @ S )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).
% remove_subset
thf(fact_4277_cSup__eq__Max,axiom,
! [X4: set_int] :
( ( finite_finite_int @ X4 )
=> ( ( X4 != bot_bot_set_int )
=> ( ( complete_Sup_Sup_int @ X4 )
= ( lattic8263393255366662781ax_int @ X4 ) ) ) ) ).
% cSup_eq_Max
thf(fact_4278_cSup__eq__Max,axiom,
! [X4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit @ X4 )
= ( lattic1210392890313158908t_unit @ X4 ) ) ) ) ).
% cSup_eq_Max
thf(fact_4279_cSup__eq__Max,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X4 )
= ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).
% cSup_eq_Max
thf(fact_4280_Max__Sup,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ A )
= ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% Max_Sup
thf(fact_4281_disjoint__alt__simp3,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) @ A )
= ( ( inf_in4660618365625256667t_unit @ A @ B4 )
!= bot_bo3957492148770167129t_unit ) ) ).
% disjoint_alt_simp3
thf(fact_4282_disjoint__alt__simp3,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_set_o @ ( minus_minus_set_o @ A @ B4 ) @ A )
= ( ( inf_inf_set_o @ A @ B4 )
!= bot_bot_set_o ) ) ).
% disjoint_alt_simp3
thf(fact_4283_disjoint__alt__simp3,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_set_int @ ( minus_minus_set_int @ A @ B4 ) @ A )
= ( ( inf_inf_set_int @ A @ B4 )
!= bot_bot_set_int ) ) ).
% disjoint_alt_simp3
thf(fact_4284_disjoint__alt__simp3,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) @ A )
= ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
!= bot_bo7824918357723582233_nat_o ) ) ).
% disjoint_alt_simp3
thf(fact_4285_disjoint__alt__simp3,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ A )
= ( ( inf_inf_set_nat @ A @ B4 )
!= bot_bot_set_nat ) ) ).
% disjoint_alt_simp3
thf(fact_4286_range__eq__singletonD,axiom,
! [F: code_integer > code_integer,A2: code_integer,X: code_integer] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4287_range__eq__singletonD,axiom,
! [F: product_unit > product_unit,A2: product_unit,X: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4288_range__eq__singletonD,axiom,
! [F: product_unit > $o,A2: $o,X: product_unit] :
( ( ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit )
= ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4289_range__eq__singletonD,axiom,
! [F: product_unit > nat,A2: nat,X: product_unit] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4290_range__eq__singletonD,axiom,
! [F: product_unit > int,A2: int,X: product_unit] :
( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
= ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4291_range__eq__singletonD,axiom,
! [F: $o > product_unit,A2: product_unit,X: $o] :
( ( ( image_o_Product_unit2 @ F @ top_top_set_o )
= ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4292_range__eq__singletonD,axiom,
! [F: $o > $o,A2: $o,X: $o] :
( ( ( image_o_o2 @ F @ top_top_set_o )
= ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4293_range__eq__singletonD,axiom,
! [F: $o > nat,A2: nat,X: $o] :
( ( ( image_o_nat2 @ F @ top_top_set_o )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4294_range__eq__singletonD,axiom,
! [F: $o > int,A2: int,X: $o] :
( ( ( image_o_int2 @ F @ top_top_set_o )
= ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4295_range__eq__singletonD,axiom,
! [F: nat > code_integer,A2: code_integer,X: nat] :
( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
= ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4296_less__cSup__iff,axiom,
! [X4: set_int,Y: int] :
( ( X4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ X4 )
=> ( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X4 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ X4 )
& ( ord_less_int @ Y @ X2 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_4297_less__cSup__iff,axiom,
! [X4: set_nat,Y: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( ord_less_nat @ Y @ X2 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_4298_less__cSup__iff,axiom,
! [X4: set_Product_unit,Y: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ X4 )
=> ( ( ord_le361264281704409273t_unit @ Y @ ( comple4687483117567863418t_unit @ X4 ) )
= ( ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ X4 )
& ( ord_le361264281704409273t_unit @ Y @ X2 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_4299_underS__Field3,axiom,
! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
( ( ( field_Product_unit @ R2 )
!= bot_bo3957492148770167129t_unit )
=> ( ord_le8056459307392131481t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( field_Product_unit @ R2 ) ) ) ).
% underS_Field3
thf(fact_4300_underS__Field3,axiom,
! [R2: set_Product_prod_o_o,A2: $o] :
( ( ( field_o @ R2 )
!= bot_bot_set_o )
=> ( ord_less_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( field_o @ R2 ) ) ) ).
% underS_Field3
thf(fact_4301_underS__Field3,axiom,
! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
( ( ( field_nat @ R2 )
!= bot_bot_set_nat )
=> ( ord_less_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( field_nat @ R2 ) ) ) ).
% underS_Field3
thf(fact_4302_underS__Field3,axiom,
! [R2: set_Pr958786334691620121nt_int,A2: int] :
( ( ( field_int @ R2 )
!= bot_bot_set_int )
=> ( ord_less_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( field_int @ R2 ) ) ) ).
% underS_Field3
thf(fact_4303_underS__Field3,axiom,
! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( ( field_81821745747897882_nat_o @ R2 )
!= bot_bo7824918357723582233_nat_o )
=> ( ord_le2453136405763929_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ).
% underS_Field3
thf(fact_4304_arg__min__if__finite_I2_J,axiom,
! [S: set_Code_integer,F: code_integer > assn] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ~ ? [X6: code_integer] :
( ( member_Code_integer @ X6 @ S )
& ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic4682378690128378784r_assn @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4305_arg__min__if__finite_I2_J,axiom,
! [S: set_Product_unit,F: product_unit > assn] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( S != bot_bo3957492148770167129t_unit )
=> ~ ? [X6: product_unit] :
( ( member_Product_unit @ X6 @ S )
& ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic4376901495776413404t_assn @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4306_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F: $o > assn] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ~ ? [X6: $o] :
( ( member_o @ X6 @ S )
& ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic7708394068118247271o_assn @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4307_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > assn] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic2187264146484958483t_assn @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4308_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: int > assn] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ~ ? [X6: int] :
( ( member_int @ X6 @ S )
& ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic3162799611691135855t_assn @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4309_arg__min__if__finite_I2_J,axiom,
! [S: set_Code_integer,F: code_integer > nat] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ~ ? [X6: code_integer] :
( ( member_Code_integer @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic6074936555552774728er_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4310_arg__min__if__finite_I2_J,axiom,
! [S: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( S != bot_bo3957492148770167129t_unit )
=> ~ ? [X6: product_unit] :
( ( member_Product_unit @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic1360832392136924548it_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4311_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ~ ? [X6: $o] :
( ( member_o @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4312_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4313_arg__min__if__finite_I2_J,axiom,
! [S: set_int,F: int > nat] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ~ ? [X6: int] :
( ( member_int @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4314_Sup__fold__sup,axiom,
! [A: set_set_Product_unit] :
( ( finite1772178364199683094t_unit @ A )
=> ( ( comple3349607820504768346t_unit @ A )
= ( finite4243028863286625800t_unit @ sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% Sup_fold_sup
thf(fact_4315_Sup__fold__sup,axiom,
! [A: set_set_o] :
( ( finite_finite_set_o @ A )
=> ( ( comple90263536869209701_set_o @ A )
= ( finite4337638375924247368_set_o @ sup_sup_set_o @ bot_bot_set_o @ A ) ) ) ).
% Sup_fold_sup
thf(fact_4316_Sup__fold__sup,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( comple7399068483239264473et_nat @ A )
= ( finite677925301803182934et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ A ) ) ) ).
% Sup_fold_sup
thf(fact_4317_Sup__fold__sup,axiom,
! [A: set_set_int] :
( ( finite6197958912794628473et_int @ A )
=> ( ( comple3221217463730067765et_int @ A )
= ( finite2508883933289561102et_int @ sup_sup_set_int @ bot_bot_set_int @ A ) ) ) ).
% Sup_fold_sup
thf(fact_4318_Sup__fold__sup,axiom,
! [A: set_se1353816476878259085_nat_o] :
( ( finite6069017635265841110_nat_o @ A )
=> ( ( comple3382087578465172250_nat_o @ A )
= ( finite8456856254070787592_nat_o @ sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ) ) ).
% Sup_fold_sup
thf(fact_4319_Sup__fold__sup,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( comple4687483117567863418t_unit @ A )
= ( finite8834431295148555784t_unit @ sup_sup_Product_unit @ bot_bot_Product_unit @ A ) ) ) ).
% Sup_fold_sup
thf(fact_4320_SUP__fold__sup,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
= ( finite6602901031542678131t_unit @ ( comp_P6204500334276302049it_nat @ sup_sup_Product_unit @ F ) @ bot_bot_Product_unit @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4321_SUP__fold__sup,axiom,
! [A: set_Code_integer,F: code_integer > product_unit] :
( ( finite6017078050557962740nteger @ A )
=> ( ( comple4687483117567863418t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
= ( finite2281032298429728806t_unit @ ( comp_P4558236313141964142nteger @ sup_sup_Product_unit @ F ) @ bot_bot_Product_unit @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4322_SUP__fold__sup,axiom,
! [A: set_nat,F: nat > set_Product_unit] :
( ( finite_finite_nat @ A )
=> ( ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ F @ A ) )
= ( finite9145781422887121747t_unit @ ( comp_s6943286182646368705it_nat @ sup_su793286257634532545t_unit @ F ) @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4323_SUP__fold__sup,axiom,
! [A: set_Code_integer,F: code_integer > set_Product_unit] :
( ( finite6017078050557962740nteger @ A )
=> ( ( comple3349607820504768346t_unit @ ( image_5870443399019143561t_unit @ F @ A ) )
= ( finite3687622995246107910t_unit @ ( comp_s3199726385246003790nteger @ sup_su793286257634532545t_unit @ F ) @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4324_SUP__fold__sup,axiom,
! [A: set_nat,F: nat > set_o] :
( ( finite_finite_nat @ A )
=> ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A ) )
= ( finite3217087857726763998_set_o @ ( comp_s4424444925706540684_o_nat @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4325_SUP__fold__sup,axiom,
! [A: set_Code_integer,F: code_integer > set_o] :
( ( finite6017078050557962740nteger @ A )
=> ( ( comple90263536869209701_set_o @ ( image_1867029960091478804_set_o @ F @ A ) )
= ( finite1928688952044918289_set_o @ ( comp_s90868338361139993nteger @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4326_SUP__fold__sup,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( finite5529483035118572448et_nat @ ( comp_s2351873599094224870at_nat @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4327_SUP__fold__sup,axiom,
! [A: set_Code_integer,F: code_integer > set_nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( comple7399068483239264473et_nat @ ( image_8829277216472192490et_nat @ F @ A ) )
= ( finite1590314701948642349et_nat @ ( comp_s3169385134670675059nteger @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4328_SUP__fold__sup,axiom,
! [A: set_nat,F: nat > set_int] :
( ( finite_finite_nat @ A )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A ) )
= ( finite1351632015609375740et_int @ ( comp_s6689493721604790522nt_nat @ sup_sup_set_int @ F ) @ bot_bot_set_int @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4329_SUP__fold__sup,axiom,
! [A: set_Code_integer,F: code_integer > set_int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( comple3221217463730067765et_int @ ( image_4651426196962995782et_int @ F @ A ) )
= ( finite6635835719294221449et_int @ ( comp_s3634918272309710215nteger @ sup_sup_set_int @ F ) @ bot_bot_set_int @ A ) ) ) ).
% SUP_fold_sup
thf(fact_4330_finite__subset__Union__chain,axiom,
! [A: set_Code_integer,B9: set_set_Code_integer,A11: set_set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ord_le7084787975880047091nteger @ A @ ( comple739944243200306918nteger @ B9 ) )
=> ( ( B9 != bot_bo9204850693367297629nteger )
=> ( ( pred_c5891848916688490280nteger @ A11 @ ord_le1307284697595431911nteger @ B9 )
=> ~ ! [B5: set_Code_integer] :
( ( member1834298987299537450nteger @ B5 @ B9 )
=> ~ ( ord_le7084787975880047091nteger @ A @ B5 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_4331_finite__subset__Union__chain,axiom,
! [A: set_nat,B9: set_set_nat,A11: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B9 ) )
=> ( ( B9 != bot_bot_set_set_nat )
=> ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ B9 )
=> ~ ! [B5: set_nat] :
( ( member_set_nat @ B5 @ B9 )
=> ~ ( ord_less_eq_set_nat @ A @ B5 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_4332_merge__true__star,axiom,
( ( times_times_assn @ top_top_assn @ top_top_assn )
= top_top_assn ) ).
% merge_true_star
thf(fact_4333_assn__basic__inequalities_I5_J,axiom,
top_top_assn != bot_bot_assn ).
% assn_basic_inequalities(5)
thf(fact_4334_assn__basic__inequalities_I1_J,axiom,
top_top_assn != one_one_assn ).
% assn_basic_inequalities(1)
thf(fact_4335_mod__true,axiom,
! [H2: produc3658429121746597890et_nat] :
( ( rep_assn @ top_top_assn @ H2 )
= ( in_range @ H2 ) ) ).
% mod_true
thf(fact_4336_fun__comp__eq__conv,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( comp_C1593894019821074884nteger @ F @ G )
= Fg )
= ( ! [X2: code_integer] :
( ( F @ ( G @ X2 ) )
= ( Fg @ X2 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_4337_fun__comp__eq__conv,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( comp_C8797469213163452608nteger @ F @ G )
= Fg )
= ( ! [X2: code_integer] :
( ( F @ ( G @ X2 ) )
= ( Fg @ X2 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_4338_comp__cong__right,axiom,
! [X: code_integer > code_integer,Y: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( X = Y )
=> ( ( comp_C1593894019821074884nteger @ F @ X )
= ( comp_C1593894019821074884nteger @ F @ Y ) ) ) ).
% comp_cong_right
thf(fact_4339_comp__cong__right,axiom,
! [X: code_integer > code_integer > code_integer,Y: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( X = Y )
=> ( ( comp_C8797469213163452608nteger @ F @ X )
= ( comp_C8797469213163452608nteger @ F @ Y ) ) ) ).
% comp_cong_right
thf(fact_4340_comp__cong__left,axiom,
! [X: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
( ( X = Y )
=> ( ( comp_C1593894019821074884nteger @ X @ F )
= ( comp_C1593894019821074884nteger @ Y @ F ) ) ) ).
% comp_cong_left
thf(fact_4341_comp__cong__left,axiom,
! [X: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
( ( X = Y )
=> ( ( comp_C8797469213163452608nteger @ X @ F )
= ( comp_C8797469213163452608nteger @ Y @ F ) ) ) ).
% comp_cong_left
thf(fact_4342_comp__apply__eq,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer] :
( ( ( F @ ( G @ X ) )
= ( H2 @ ( K3 @ X ) ) )
=> ( ( comp_C1593894019821074884nteger @ F @ G @ X )
= ( comp_C1593894019821074884nteger @ H2 @ K3 @ X ) ) ) ).
% comp_apply_eq
thf(fact_4343_comp__apply__eq,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer > code_integer] :
( ( ( F @ ( G @ X ) )
= ( H2 @ ( K3 @ X ) ) )
=> ( ( comp_C1593894019821074884nteger @ F @ G @ X )
= ( comp_C8797469213163452608nteger @ H2 @ K3 @ X ) ) ) ).
% comp_apply_eq
thf(fact_4344_comp__apply__eq,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer] :
( ( ( F @ ( G @ X ) )
= ( H2 @ ( K3 @ X ) ) )
=> ( ( comp_C8797469213163452608nteger @ F @ G @ X )
= ( comp_C1593894019821074884nteger @ H2 @ K3 @ X ) ) ) ).
% comp_apply_eq
thf(fact_4345_comp__apply__eq,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer > code_integer] :
( ( ( F @ ( G @ X ) )
= ( H2 @ ( K3 @ X ) ) )
=> ( ( comp_C8797469213163452608nteger @ F @ G @ X )
= ( comp_C8797469213163452608nteger @ H2 @ K3 @ X ) ) ) ).
% comp_apply_eq
thf(fact_4346_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_nat,M: code_integer > code_integer,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
=> ( ( ( comp_C6537567860429791781nteger @ S @ M )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
=> ( ( comp_P8262219321096090349nteger @ ( comp_C3184339336801759213nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4347_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_Code_integer,M: code_integer > code_integer,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
=> ( ( ( comp_C8186931911283765298nteger @ S @ M )
= ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
=> ( ( comp_P4885729792186999034nteger @ ( comp_C6344129587964912378nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
= ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4348_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_Code_integer,M: code_integer > code_integer,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
=> ( ( ( comp_C8186931911283765298nteger @ S @ M )
= ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
=> ( ( comp_P4885729792186999034nteger @ ( comp_C6344129587964912378nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
= ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4349_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_nat,M: code_integer > product_unit,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
=> ( ( ( comp_P5878433733084393551nteger @ S @ M )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
=> ( ( comp_P8262219321096090349nteger @ ( comp_P6101996899264665623nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4350_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_Code_integer,M: code_integer > product_unit,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
=> ( ( ( comp_P4767411901762164316nteger @ S @ M )
= ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
=> ( ( comp_P4885729792186999034nteger @ ( comp_P4635212033913086756nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
= ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4351_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_Code_integer,M: code_integer > product_unit,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
=> ( ( ( comp_P4767411901762164316nteger @ S @ M )
= ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
=> ( ( comp_P4885729792186999034nteger @ ( comp_P4635212033913086756nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
= ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4352_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_nat,M: code_integer > $o,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( ( comp_o172754640513974618nteger @ S @ M )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
=> ( ( comp_P8262219321096090349nteger @ ( comp_o1289984117834210338nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4353_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_Code_integer,M: code_integer > $o,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( ( comp_o4291842030903409511nteger @ S @ M )
= ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
=> ( ( comp_P4885729792186999034nteger @ ( comp_o7929101295636871983nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
= ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4354_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_Code_integer,M: code_integer > $o,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( ( comp_o4291842030903409511nteger @ S @ M )
= ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
=> ( ( comp_P4885729792186999034nteger @ ( comp_o7929101295636871983nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
= ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4355_type__copy__set__map0,axiom,
! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat,Abs2: nat > produc8923325533196201883nteger > produc8923325533196201883nteger,S: nat > set_nat,M: code_integer > nat,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
( ( type_d6278479724763174947er_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( ( comp_n5963683855531483544nteger @ S @ M )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
=> ( ( comp_P8262219321096090349nteger @ ( comp_n3623798160786426464nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_n2033352843805742135nteger @ Abs2 @ M ) @ G ) )
= ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4356_top__empty__eq,axiom,
( top_to2462227307520033056at_o_o
= ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ top_to4869887016220417533_nat_o ) ) ) ).
% top_empty_eq
thf(fact_4357_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_4358_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_4359_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_4360_top__empty__eq,axiom,
( top_top_int_o
= ( ^ [X2: int] : ( member_int @ X2 @ top_top_set_int ) ) ) ).
% top_empty_eq
thf(fact_4361_top__set__def,axiom,
( top_to4869887016220417533_nat_o
= ( collec939566748876313656_nat_o @ top_to2462227307520033056at_o_o ) ) ).
% top_set_def
thf(fact_4362_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_4363_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_4364_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_4365_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_4366_pred__on_Ochain__empty,axiom,
! [A: set_Product_unit,P: product_unit > product_unit > $o] : ( pred_c6915491785556283576t_unit @ A @ P @ bot_bo3957492148770167129t_unit ) ).
% pred_on.chain_empty
thf(fact_4367_pred__on_Ochain__empty,axiom,
! [A: set_o,P: $o > $o > $o] : ( pred_chain_o @ A @ P @ bot_bot_set_o ) ).
% pred_on.chain_empty
thf(fact_4368_pred__on_Ochain__empty,axiom,
! [A: set_nat,P: nat > nat > $o] : ( pred_chain_nat @ A @ P @ bot_bot_set_nat ) ).
% pred_on.chain_empty
thf(fact_4369_pred__on_Ochain__empty,axiom,
! [A: set_int,P: int > int > $o] : ( pred_chain_int @ A @ P @ bot_bot_set_int ) ).
% pred_on.chain_empty
thf(fact_4370_pred__on_Ochain__empty,axiom,
! [A: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] : ( pred_c177362287423414904_nat_o @ A @ P @ bot_bo7824918357723582233_nat_o ) ).
% pred_on.chain_empty
thf(fact_4371_surj__fun__eq,axiom,
! [F: code_integer > code_integer > code_integer,X4: set_Code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( image_2177946750938214461nteger @ F @ X4 )
= top_to3804862627464599871nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ X4 )
=> ( ( comp_C8797469213163452608nteger @ G1 @ F @ X3 )
= ( comp_C8797469213163452608nteger @ G2 @ F @ X3 ) ) )
=> ( G1 = G2 ) ) ) ).
% surj_fun_eq
thf(fact_4372_surj__fun__eq,axiom,
! [F: code_integer > code_integer,X4: set_Code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( image_4470545334726330049nteger @ F @ X4 )
= top_to4645266643341252675nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ X4 )
=> ( ( comp_C1593894019821074884nteger @ G1 @ F @ X3 )
= ( comp_C1593894019821074884nteger @ G2 @ F @ X3 ) ) )
=> ( G1 = G2 ) ) ) ).
% surj_fun_eq
thf(fact_4373_type__copy__map__comp0,axiom,
! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,M: code_integer > code_integer,M1: code_integer > code_integer,M22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
=> ( ( M
= ( comp_C7449957260575251196nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C7449957260575251196nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4374_type__copy__map__comp0,axiom,
! [Rep2: code_integer > product_unit,Abs2: product_unit > code_integer,M: code_integer > code_integer,M1: product_unit > code_integer,M22: code_integer > product_unit,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d9006111508436661314t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
=> ( ( M
= ( comp_P6790823133229852966nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ ( comp_C5854136495726678886t_unit @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_P6790823133229852966nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4375_type__copy__map__comp0,axiom,
! [Rep2: code_integer > $o,Abs2: $o > code_integer,M: code_integer > code_integer,M1: $o > code_integer,M22: code_integer > $o,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d6812736842639505741eger_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( M
= ( comp_o1085144040659434033nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ ( comp_C5733264393235242609eger_o @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_o1085144040659434033nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4376_type__copy__map__comp0,axiom,
! [Rep2: code_integer > nat,Abs2: nat > code_integer,M: code_integer > code_integer,M1: nat > code_integer,M22: code_integer > nat,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d8368584451094728027er_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( M
= ( comp_n6876073255676942959nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_n2033352843805742135nteger @ ( comp_C836733381939181879er_nat @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_n6876073255676942959nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4377_type__copy__map__comp0,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,M: code_integer > code_integer,M1: int > code_integer,M22: code_integer > int,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( M
= ( comp_i6014793192326455499nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_i2874689474467862163nteger @ ( comp_C834242911430131603er_int @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_i6014793192326455499nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4378_type__copy__map__comp0,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,M: code_integer > code_integer,M1: $o > code_integer,M22: code_integer > $o,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( M
= ( comp_o1085144040659434033nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_P4511551582283981294nteger @ ( comp_o6961622296474427249t_unit @ ( comp_C5733264393235242609eger_o @ F @ M1 ) @ Rep2 ) @ ( comp_C8146212659816229034nteger @ ( comp_o6635994240230588853nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4379_type__copy__map__comp0,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,M: code_integer > code_integer,M1: nat > code_integer,M22: code_integer > nat,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( M
= ( comp_n6876073255676942959nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C456341964420721295nteger @ ( comp_n856947995562459650atural @ ( comp_C836733381939181879er_nat @ F @ M1 ) @ Rep2 ) @ ( comp_C1891057325546811591nteger @ ( comp_n1317173320648503354nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4380_type__copy__map__comp0,axiom,
! [Rep2: code_integer > code_integer > code_integer,Abs2: ( code_integer > code_integer ) > code_integer,M: code_integer > code_integer,M1: ( code_integer > code_integer ) > code_integer,M22: code_integer > code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d669523880323286244nteger @ Rep2 @ Abs2 @ top_to3804862627464599871nteger )
=> ( ( M
= ( comp_C5076291335850163704nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C5076291335850163704nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4381_type__copy__map__comp0,axiom,
! [Rep2: ( code_integer > code_integer ) > code_integer,Abs2: code_integer > code_integer > code_integer,M: code_integer > code_integer,M1: code_integer > code_integer,M22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d5515310227294319076nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
=> ( ( M
= ( comp_C7449957260575251196nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
= ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ ( comp_C1593894019821074884nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C2931745410781319416nteger @ ( comp_C2931745410781319416nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4382_type__copy__map__comp0,axiom,
! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,M: code_integer > code_integer > code_integer,M1: code_integer > code_integer > code_integer,M22: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
=> ( ( M
= ( comp_C2931745410781319416nteger @ M1 @ M22 ) )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M ) @ G )
= ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C7449957260575251196nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4383_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_integer > int,Abs4: int > code_integer,M: int > int,M1: int > int,M22: int > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
=> ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
=> ( ( ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M ) @ Rep4 )
= ( comp_C7449957260575251196nteger @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_int_int @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4384_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > int,M1: int > int,M22: $o > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
=> ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
=> ( ( ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs3 @ M ) @ Rep4 )
= ( comp_C2290379119555921966t_unit @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_int_o @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4385_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > int,M1: int > int,M22: nat > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
=> ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
=> ( ( ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs3 @ M ) @ Rep4 )
= ( comp_C6273552412331968711atural @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_int_nat @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4386_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: code_integer > int,Abs4: int > code_integer,M: int > $o,M1: int > $o,M22: int > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
=> ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
=> ( ( ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M ) @ Rep4 )
= ( comp_C8146212659816229034nteger @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_o_int @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4387_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > $o,M1: int > $o,M22: $o > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
=> ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
=> ( ( ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs3 @ M ) @ Rep4 )
= ( comp_C2408090164605745152t_unit @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_o_o @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4388_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > $o,M1: int > $o,M22: nat > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
=> ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
=> ( ( ( comp_n8360577435903631554atural @ ( comp_o7051512559447311656it_nat @ Abs3 @ M ) @ Rep4 )
= ( comp_C6969807811572946549atural @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_o_nat @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4389_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: code_integer > int,Abs4: int > code_integer,M: int > nat,M1: int > nat,M22: int > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
=> ( ( ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M ) @ Rep4 )
= ( comp_C1891057325546811591nteger @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_nat_int @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4390_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > nat,M1: int > nat,M22: $o > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
=> ( ( ( comp_o2643362028263920430t_unit @ ( comp_n2857367870785791611ural_o @ Abs3 @ M ) @ Rep4 )
= ( comp_C4153580447849560611t_unit @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_nat_o @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4391_type__copy__map__comp0__undo,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > nat,M1: int > nat,M22: nat > int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
=> ( ( ( comp_n140768472405220869atural @ ( comp_n9163001964490475437al_nat @ Abs3 @ M ) @ Rep4 )
= ( comp_C714652477303529106atural @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_int_nat_nat @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4392_type__copy__map__comp0__undo,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_integer > int,Abs4: int > code_integer,M: int > int,M1: $o > int,M22: int > $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
=> ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
=> ( ( ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M ) @ Rep4 )
= ( comp_P6790823133229852966nteger @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs2 @ M22 ) @ Rep4 ) ) )
=> ( ( comp_o_int_int @ M1 @ M22 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4393_wf__bounded__measure,axiom,
! [R2: set_Pr1261947904930325089at_nat,Ub: nat > nat,F: nat > nat] :
( ! [A4: nat,B2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
=> ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
=> ( wf_nat @ R2 ) ) ).
% wf_bounded_measure
thf(fact_4394_wf__bounded__measure,axiom,
! [R2: set_Pr958786334691620121nt_int,Ub: int > nat,F: int > nat] :
( ! [A4: int,B2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
=> ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
& ( ord_less_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
=> ( wf_int @ R2 ) ) ).
% wf_bounded_measure
thf(fact_4395_less__assn__def,axiom,
( ord_less_assn
= ( ^ [A6: assn,B6: assn] :
( ( ord_less_eq_assn @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% less_assn_def
thf(fact_4396_top__assn__def,axiom,
( top_top_assn
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_4397_pred__on_Ochain__extend,axiom,
! [A: set_Product_unit,P: product_unit > product_unit > $o,C5: set_Product_unit,Z: product_unit] :
( ( pred_c6915491785556283576t_unit @ A @ P @ C5 )
=> ( ( member_Product_unit @ Z @ A )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ C5 )
=> ( sup_su3411595859437804162unit_o @ P
@ ^ [Y4: product_unit,Z4: product_unit] : ( Y4 = Z4 )
@ X3
@ Z ) )
=> ( pred_c6915491785556283576t_unit @ A @ P @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ Z @ bot_bo3957492148770167129t_unit ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_4398_pred__on_Ochain__extend,axiom,
! [A: set_o,P: $o > $o > $o,C5: set_o,Z: $o] :
( ( pred_chain_o @ A @ P @ C5 )
=> ( ( member_o @ Z @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ C5 )
=> ( sup_sup_o_o_o @ P
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ X3
@ Z ) )
=> ( pred_chain_o @ A @ P @ ( sup_sup_set_o @ ( insert_o @ Z @ bot_bot_set_o ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_4399_pred__on_Ochain__extend,axiom,
! [A: set_nat,P: nat > nat > $o,C5: set_nat,Z: nat] :
( ( pred_chain_nat @ A @ P @ C5 )
=> ( ( member_nat @ Z @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ C5 )
=> ( sup_sup_nat_nat_o @ P
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ X3
@ Z ) )
=> ( pred_chain_nat @ A @ P @ ( sup_sup_set_nat @ ( insert_nat @ Z @ bot_bot_set_nat ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_4400_pred__on_Ochain__extend,axiom,
! [A: set_int,P: int > int > $o,C5: set_int,Z: int] :
( ( pred_chain_int @ A @ P @ C5 )
=> ( ( member_int @ Z @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ C5 )
=> ( sup_sup_int_int_o @ P
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ X3
@ Z ) )
=> ( pred_chain_int @ A @ P @ ( sup_sup_set_int @ ( insert_int @ Z @ bot_bot_set_int ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_4401_pred__on_Ochain__extend,axiom,
! [A: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,C5: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
( ( pred_c177362287423414904_nat_o @ A @ P @ C5 )
=> ( ( member6576561426505652726_nat_o @ Z @ A )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ C5 )
=> ( sup_su1942872429929737346at_o_o @ P
@ ^ [Y4: produc3658429121746597890et_nat > $o,Z4: produc3658429121746597890et_nat > $o] : ( Y4 = Z4 )
@ X3
@ Z ) )
=> ( pred_c177362287423414904_nat_o @ A @ P @ ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ Z @ bot_bo7824918357723582233_nat_o ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_4402_subset__Zorn__nonempty,axiom,
! [A11: set_set_nat] :
( ( A11 != bot_bot_set_set_nat )
=> ( ! [C7: set_set_nat] :
( ( C7 != bot_bot_set_set_nat )
=> ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ C7 )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ C7 ) @ A11 ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A11 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ A11 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_4403_in__measure,axiom,
! [X: nat,Y: nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_4404_in__measure,axiom,
! [X: int,Y: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_4405_top_Oordering__top__axioms,axiom,
orderi2935705153647960990t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit ).
% top.ordering_top_axioms
thf(fact_4406_top_Oordering__top__axioms,axiom,
ordering_top_set_o @ ord_less_eq_set_o @ ord_less_set_o @ top_top_set_o ).
% top.ordering_top_axioms
thf(fact_4407_top_Oordering__top__axioms,axiom,
ordering_top_set_int @ ord_less_eq_set_int @ ord_less_set_int @ top_top_set_int ).
% top.ordering_top_axioms
thf(fact_4408_top_Oordering__top__axioms,axiom,
ordering_top_assn @ ord_less_eq_assn @ ord_less_assn @ top_top_assn ).
% top.ordering_top_axioms
thf(fact_4409_top_Oordering__top__axioms,axiom,
orderi2807725203697798511er_nat @ ord_le2510731241096832064er_nat @ ord_less_filter_nat @ top_top_filter_nat ).
% top.ordering_top_axioms
thf(fact_4410_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_4411_Max_Oinsert__remove,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
= bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
= X ) )
& ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
!= bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
= ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4412_Max_Oinsert__remove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
= ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4413_Max_Oinsert__remove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4414_Max_Oinsert__remove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
= ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4415_Max_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4416_Max_Oremove,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
= bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ A )
= X ) )
& ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
!= bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ A )
= ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4417_Max_Oremove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ A )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ A )
= ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4418_Max_Oremove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( lattic1921953407002678535_Max_o @ A )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4419_Max_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ A )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ A )
= ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4420_Max_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4421_inj__on__Un,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
( ( inj_on1158064722259180589nteger @ F @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( ( inj_on1158064722259180589nteger @ F @ A )
& ( inj_on1158064722259180589nteger @ F @ B4 )
& ( ( inf_in1364745209274528805nteger @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ B4 @ A ) ) )
= bot_bo3990330152332043303nteger ) ) ) ).
% inj_on_Un
thf(fact_4422_inj__on__Un,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_nat] :
( ( inj_on_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on_nat_set_nat @ F @ A )
& ( inj_on_nat_set_nat @ F @ B4 )
& ( ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bot_set_set_nat ) ) ) ).
% inj_on_Un
thf(fact_4423_inj__on__Un,axiom,
! [F: nat > code_integer,A: set_nat,B4: set_nat] :
( ( inj_on2245834603434889120nteger @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on2245834603434889120nteger @ F @ A )
& ( inj_on2245834603434889120nteger @ F @ B4 )
& ( ( inf_in1364745209274528805nteger @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bo3990330152332043303nteger ) ) ) ).
% inj_on_Un
thf(fact_4424_inj__on__Un,axiom,
! [F: nat > product_unit,A: set_nat,B4: set_nat] :
( ( inj_on7061601236592826506t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on7061601236592826506t_unit @ F @ A )
& ( inj_on7061601236592826506t_unit @ F @ B4 )
& ( ( inf_in4660618365625256667t_unit @ ( image_8730104196221521654t_unit @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_8730104196221521654t_unit @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% inj_on_Un
thf(fact_4425_inj__on__Un,axiom,
! [F: nat > $o,A: set_nat,B4: set_nat] :
( ( inj_on_nat_o @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on_nat_o @ F @ A )
& ( inj_on_nat_o @ F @ B4 )
& ( ( inf_inf_set_o @ ( image_nat_o2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_o2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bot_set_o ) ) ) ).
% inj_on_Un
thf(fact_4426_inj__on__Un,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on_nat_nat @ F @ A )
& ( inj_on_nat_nat @ F @ B4 )
& ( ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_4427_inj__on__Un,axiom,
! [F: nat > int,A: set_nat,B4: set_nat] :
( ( inj_on_nat_int @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on_nat_int @ F @ A )
& ( inj_on_nat_int @ F @ B4 )
& ( ( inf_inf_set_int @ ( image_nat_int2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_int2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bot_set_int ) ) ) ).
% inj_on_Un
thf(fact_4428_inj__on__Un,axiom,
! [F: nat > produc3658429121746597890et_nat > $o,A: set_nat,B4: set_nat] :
( ( inj_on7779444506608719434_nat_o @ F @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on7779444506608719434_nat_o @ F @ A )
& ( inj_on7779444506608719434_nat_o @ F @ B4 )
& ( ( inf_in1906310914598751387_nat_o @ ( image_4682239675366380726_nat_o @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_4682239675366380726_nat_o @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bo7824918357723582233_nat_o ) ) ) ).
% inj_on_Un
thf(fact_4429_Max__insert,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
= ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A ) ) ) ) ) ).
% Max_insert
thf(fact_4430_Max__insert,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
= ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ).
% Max_insert
thf(fact_4431_Max__insert,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
= ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ).
% Max_insert
thf(fact_4432_Max__insert,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
= ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ).
% Max_insert
thf(fact_4433_Max__insert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ).
% Max_insert
thf(fact_4434_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: code_integer > code_integer] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_4470545334726330049nteger @ G @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( image_7082141833288935621nteger @ ( produc1802105962441033582nteger @ F @ G ) @ top_to2363841527343591755nteger )
= top_to2363841527343591755nteger ) ) ) ).
% map_prod_surj
thf(fact_4435_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: product_unit > product_unit] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( image_3456600171786721233t_unit @ ( produc3346700480963112818t_unit @ F @ G ) @ top_to3244471725241573579t_unit )
= top_to3244471725241573579t_unit ) ) ) ).
% map_prod_surj
thf(fact_4436_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: product_unit > $o] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_Product_unit_o2 @ G @ top_to1996260823553986621t_unit )
= top_top_set_o )
=> ( ( image_7060974388358999388eger_o @ ( produc8116347370398923901unit_o @ F @ G ) @ top_to3244471725241573579t_unit )
= top_to2658285827425157974eger_o ) ) ) ).
% map_prod_surj
thf(fact_4437_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: product_unit > nat] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( image_8236373689398998434er_nat @ ( produc4022400886476617259it_nat @ F @ G ) @ top_to3244471725241573579t_unit )
= top_to8760411888770933182er_nat ) ) ) ).
% map_prod_surj
thf(fact_4438_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: product_unit > int] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit )
= top_top_set_int )
=> ( ( image_4058522669889801726er_int @ ( produc4019910415967566983it_int @ F @ G ) @ top_to3244471725241573579t_unit )
= top_to6270328743839206426er_int ) ) ) ).
% map_prod_surj
thf(fact_4439_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: $o > product_unit] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_o_Product_unit2 @ G @ top_top_set_o )
= top_to1996260823553986621t_unit )
=> ( ( image_2425804817718344284t_unit @ ( produc3530376873132787239t_unit @ F @ G ) @ top_to2658285827425157974eger_o )
= top_to3244471725241573579t_unit ) ) ) ).
% map_prod_surj
thf(fact_4440_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: $o > $o] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_o_o2 @ G @ top_top_set_o )
= top_top_set_o )
=> ( ( image_227735840630799463eger_o @ ( produc1084426483498010290er_o_o @ F @ G ) @ top_to2658285827425157974eger_o )
= top_to2658285827425157974eger_o ) ) ) ).
% map_prod_surj
thf(fact_4441_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: $o > nat] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_o_nat2 @ G @ top_top_set_o )
= top_top_set_nat )
=> ( ( image_1252617143230709591er_nat @ ( produc7550393335539917622_o_nat @ F @ G ) @ top_to2658285827425157974eger_o )
= top_to8760411888770933182er_nat ) ) ) ).
% map_prod_surj
thf(fact_4442_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: $o > int] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_o_int2 @ G @ top_top_set_o )
= top_top_set_int )
=> ( ( image_6298138160576288691er_int @ ( produc7547902865030867346_o_int @ F @ G ) @ top_to2658285827425157974eger_o )
= top_to6270328743839206426er_int ) ) ) ).
% map_prod_surj
thf(fact_4443_map__prod__surj,axiom,
! [F: code_integer > code_integer,G: nat > code_integer] :
( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger )
=> ( ( ( image_1215581382706833972nteger @ G @ top_top_set_nat )
= top_to4645266643341252675nteger )
=> ( ( image_4921214744877245202nteger @ ( produc1622059938646302433nteger @ F @ G ) @ top_to8760411888770933182er_nat )
= top_to2363841527343591755nteger ) ) ) ).
% map_prod_surj
thf(fact_4444_inj__on__insert,axiom,
! [F: code_integer > code_integer,A2: code_integer,A: set_Code_integer] :
( ( inj_on1158064722259180589nteger @ F @ ( insert_Code_integer @ A2 @ A ) )
= ( ( inj_on1158064722259180589nteger @ F @ A )
& ~ ( member_Code_integer @ ( F @ A2 ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4445_inj__on__insert,axiom,
! [F: product_unit > nat,A2: product_unit,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ ( insert_Product_unit @ A2 @ A ) )
= ( ( inj_on8430439091780834860it_nat @ F @ A )
& ~ ( member_nat @ ( F @ A2 ) @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4446_inj__on__insert,axiom,
! [F: product_unit > int,A2: product_unit,A: set_Product_unit] :
( ( inj_on8427948621271784584it_int @ F @ ( insert_Product_unit @ A2 @ A ) )
= ( ( inj_on8427948621271784584it_int @ F @ A )
& ~ ( member_int @ ( F @ A2 ) @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4447_inj__on__insert,axiom,
! [F: product_unit > $o,A2: product_unit,A: set_Product_unit] :
( ( inj_on8854722734121536060unit_o @ F @ ( insert_Product_unit @ A2 @ A ) )
= ( ( inj_on8854722734121536060unit_o @ F @ A )
& ~ ( member_o @ ( F @ A2 ) @ ( image_Product_unit_o2 @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4448_inj__on__insert,axiom,
! [F: product_unit > product_unit,A2: product_unit,A: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ ( insert_Product_unit @ A2 @ A ) )
= ( ( inj_on8151373323710067377t_unit @ F @ A )
& ~ ( member_Product_unit @ ( F @ A2 ) @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4449_inj__on__insert,axiom,
! [F: $o > nat,A2: $o,A: set_o] :
( ( inj_on_o_nat @ F @ ( insert_o @ A2 @ A ) )
= ( ( inj_on_o_nat @ F @ A )
& ~ ( member_nat @ ( F @ A2 ) @ ( image_o_nat2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4450_inj__on__insert,axiom,
! [F: $o > int,A2: $o,A: set_o] :
( ( inj_on_o_int @ F @ ( insert_o @ A2 @ A ) )
= ( ( inj_on_o_int @ F @ A )
& ~ ( member_int @ ( F @ A2 ) @ ( image_o_int2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4451_inj__on__insert,axiom,
! [F: $o > $o,A2: $o,A: set_o] :
( ( inj_on_o_o @ F @ ( insert_o @ A2 @ A ) )
= ( ( inj_on_o_o @ F @ A )
& ~ ( member_o @ ( F @ A2 ) @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4452_inj__on__insert,axiom,
! [F: $o > product_unit,A2: $o,A: set_o] :
( ( inj_on4268752236855399398t_unit @ F @ ( insert_o @ A2 @ A ) )
= ( ( inj_on4268752236855399398t_unit @ F @ A )
& ~ ( member_Product_unit @ ( F @ A2 ) @ ( image_o_Product_unit2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4453_inj__on__insert,axiom,
! [F: int > nat,A2: int,A: set_int] :
( ( inj_on_int_nat @ F @ ( insert_int @ A2 @ A ) )
= ( ( inj_on_int_nat @ F @ A )
& ~ ( member_nat @ ( F @ A2 ) @ ( image_int_nat2 @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4454_max__bot,axiom,
! [X: set_Product_unit] :
( ( ord_ma508538259134630082t_unit @ bot_bo3957492148770167129t_unit @ X )
= X ) ).
% max_bot
thf(fact_4455_max__bot,axiom,
! [X: set_o] :
( ( ord_max_set_o @ bot_bot_set_o @ X )
= X ) ).
% max_bot
thf(fact_4456_max__bot,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% max_bot
thf(fact_4457_max__bot,axiom,
! [X: set_int] :
( ( ord_max_set_int @ bot_bot_set_int @ X )
= X ) ).
% max_bot
thf(fact_4458_max__bot,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( ord_ma8206644353667601538_nat_o @ bot_bo7824918357723582233_nat_o @ X )
= X ) ).
% max_bot
thf(fact_4459_max__bot,axiom,
! [X: nat] :
( ( ord_max_nat @ bot_bot_nat @ X )
= X ) ).
% max_bot
thf(fact_4460_max__bot2,axiom,
! [X: set_Product_unit] :
( ( ord_ma508538259134630082t_unit @ X @ bot_bo3957492148770167129t_unit )
= X ) ).
% max_bot2
thf(fact_4461_max__bot2,axiom,
! [X: set_o] :
( ( ord_max_set_o @ X @ bot_bot_set_o )
= X ) ).
% max_bot2
thf(fact_4462_max__bot2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% max_bot2
thf(fact_4463_max__bot2,axiom,
! [X: set_int] :
( ( ord_max_set_int @ X @ bot_bot_set_int )
= X ) ).
% max_bot2
thf(fact_4464_max__bot2,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( ord_ma8206644353667601538_nat_o @ X @ bot_bo7824918357723582233_nat_o )
= X ) ).
% max_bot2
thf(fact_4465_max__bot2,axiom,
! [X: nat] :
( ( ord_max_nat @ X @ bot_bot_nat )
= X ) ).
% max_bot2
thf(fact_4466_max__top,axiom,
! [X: assn] :
( ( ord_max_assn @ top_top_assn @ X )
= top_top_assn ) ).
% max_top
thf(fact_4467_max__top,axiom,
! [X: set_Product_unit] :
( ( ord_ma508538259134630082t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% max_top
thf(fact_4468_max__top,axiom,
! [X: set_o] :
( ( ord_max_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% max_top
thf(fact_4469_max__top,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% max_top
thf(fact_4470_max__top,axiom,
! [X: set_int] :
( ( ord_max_set_int @ top_top_set_int @ X )
= top_top_set_int ) ).
% max_top
thf(fact_4471_max__top2,axiom,
! [X: assn] :
( ( ord_max_assn @ X @ top_top_assn )
= top_top_assn ) ).
% max_top2
thf(fact_4472_max__top2,axiom,
! [X: set_Product_unit] :
( ( ord_ma508538259134630082t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% max_top2
thf(fact_4473_max__top2,axiom,
! [X: set_o] :
( ( ord_max_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% max_top2
thf(fact_4474_max__top2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% max_top2
thf(fact_4475_max__top2,axiom,
! [X: set_int] :
( ( ord_max_set_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% max_top2
thf(fact_4476_map__prod__simp,axiom,
! [F: nat > nat,G: nat > nat,A2: nat,B: nat] :
( ( produc6977886695330630970at_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
= ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4477_map__prod__simp,axiom,
! [F: nat > int,G: nat > int,A2: nat,B: nat] :
( ( produc7347049785058425330at_int @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
= ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4478_map__prod__simp,axiom,
! [F: int > nat,G: int > nat,A2: int,B: int] :
( ( produc5667422475138734578nt_nat @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
= ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4479_map__prod__simp,axiom,
! [F: int > int,G: int > int,A2: int,B: int] :
( ( produc6036585564866528938nt_int @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
= ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4480_map__prod__simp,axiom,
! [F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,A2: heap_e7401611519738050253t_unit,B: set_nat] :
( ( produc8285742784527409791at_nat @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
= ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4481_map__prod__simp,axiom,
! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,A2: heap_e7401611519738050253t_unit,B: set_nat] :
( ( produc5973434381925408567at_int @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
= ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4482_map__prod__simp,axiom,
! [F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat,A2: nat,B: nat] :
( ( produc2064153442785501et_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
= ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4483_map__prod__simp,axiom,
! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,A2: int,B: int] :
( ( produc3152090231653401565et_nat @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
= ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4484_map__prod__simp,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,A2: heap_e7401611519738050253t_unit,B: set_nat] :
( ( produc8234977478398262212et_nat @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
= ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4485_map__prod__simp,axiom,
! [F: nat > produc3658429121746597890et_nat > $o,G: nat > produc3658429121746597890et_nat,A2: nat,B: nat] :
( ( produc79747477514316377et_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
= ( produc5001842942810119800et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_4486_snd__comp__map__prod,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat] :
( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc8234977478398262212et_nat @ F @ G ) )
= ( comp_s5696514601250091763et_nat @ G @ produc8586169260539613262et_nat ) ) ).
% snd_comp_map_prod
thf(fact_4487_snd__comp__map__prod,axiom,
! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat] :
( ( comp_P8606889026643445772nt_int @ produc8586169260539613262et_nat @ ( produc3152090231653401565et_nat @ F @ G ) )
= ( comp_i6200299785623974754nt_int @ G @ product_snd_int_int ) ) ).
% snd_comp_map_prod
thf(fact_4488_snd__comp__map__prod,axiom,
! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int] :
( ( comp_P8308861049717174982et_nat @ product_snd_int_int @ ( produc5973434381925408567at_int @ F @ G ) )
= ( comp_s1601710177542341017et_nat @ G @ produc8586169260539613262et_nat ) ) ).
% snd_comp_map_prod
thf(fact_4489_snd__comp__map__prod,axiom,
! [F: int > int,G: int > int] :
( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G ) )
= ( comp_i4330622015127193916nt_int @ G @ product_snd_int_int ) ) ).
% snd_comp_map_prod
thf(fact_4490_snd__map__prod,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,X: produc3658429121746597890et_nat] :
( ( produc8586169260539613262et_nat @ ( produc8234977478398262212et_nat @ F @ G @ X ) )
= ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% snd_map_prod
thf(fact_4491_snd__map__prod,axiom,
! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,X: product_prod_int_int] :
( ( produc8586169260539613262et_nat @ ( produc3152090231653401565et_nat @ F @ G @ X ) )
= ( G @ ( product_snd_int_int @ X ) ) ) ).
% snd_map_prod
thf(fact_4492_snd__map__prod,axiom,
! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,X: produc3658429121746597890et_nat] :
( ( product_snd_int_int @ ( produc5973434381925408567at_int @ F @ G @ X ) )
= ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% snd_map_prod
thf(fact_4493_snd__map__prod,axiom,
! [F: int > int,G: int > int,X: product_prod_int_int] :
( ( product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
= ( G @ ( product_snd_int_int @ X ) ) ) ).
% snd_map_prod
thf(fact_4494_map__prod__imageI,axiom,
! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4495_map__prod__imageI,axiom,
! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > int,G: nat > int] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4496_map__prod__imageI,axiom,
! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > nat,G: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4497_map__prod__imageI,axiom,
! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > int,G: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4498_map__prod__imageI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_7757775281999735736at_nat @ ( produc8285742784527409791at_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4499_map__prod__imageI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > int,G: set_nat > int] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_4579277974391593584nt_int @ ( produc5973434381925408567at_int @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4500_map__prod__imageI,axiom,
! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_9068448769188195518et_nat @ ( produc2064153442785501et_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4501_map__prod__imageI,axiom,
! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > heap_e7401611519738050253t_unit,G: int > set_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2127501072796355446et_nat @ ( produc3152090231653401565et_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4502_map__prod__imageI,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
=> ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_7426123528223433739et_nat @ ( produc8234977478398262212et_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4503_map__prod__imageI,axiom,
! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > produc3658429121746597890et_nat > $o,G: nat > produc3658429121746597890et_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
=> ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5744643625575355962et_nat @ ( produc79747477514316377et_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_4504_exists__leI,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [N5: nat] :
( ( ord_less_nat @ N5 @ N2 )
=> ~ ( P @ N5 ) )
=> ( P @ N2 ) )
=> ? [N6: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
& ( P @ N6 ) ) ) ).
% exists_leI
thf(fact_4505_max__def,axiom,
( ord_max_filter_nat
= ( ^ [A6: filter_nat,B6: filter_nat] : ( if_filter_nat @ ( ord_le2510731241096832064er_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% max_def
thf(fact_4506_max__def,axiom,
( ord_max_set_nat
= ( ^ [A6: set_nat,B6: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% max_def
thf(fact_4507_max__def,axiom,
( ord_max_rat
= ( ^ [A6: rat,B6: rat] : ( if_rat @ ( ord_less_eq_rat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% max_def
thf(fact_4508_max__def,axiom,
( ord_max_nat
= ( ^ [A6: nat,B6: nat] : ( if_nat @ ( ord_less_eq_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% max_def
thf(fact_4509_max__def,axiom,
( ord_max_int
= ( ^ [A6: int,B6: int] : ( if_int @ ( ord_less_eq_int @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% max_def
thf(fact_4510_max__absorb1,axiom,
! [Y: filter_nat,X: filter_nat] :
( ( ord_le2510731241096832064er_nat @ Y @ X )
=> ( ( ord_max_filter_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_4511_max__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_max_set_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_4512_max__absorb1,axiom,
! [Y: rat,X: rat] :
( ( ord_less_eq_rat @ Y @ X )
=> ( ( ord_max_rat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_4513_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_4514_max__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_max_int @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_4515_max__absorb2,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( ord_le2510731241096832064er_nat @ X @ Y )
=> ( ( ord_max_filter_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_4516_max__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_max_set_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_4517_max__absorb2,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ Y )
=> ( ( ord_max_rat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_4518_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_4519_max__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_max_int @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_4520_max__diff__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_4521_max__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% max_diff_distrib_left
thf(fact_4522_map__prod_Ocomp,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer] :
( ( comp_P234064951390806312nteger @ ( produc5081366107664765182nteger @ F @ G ) @ ( produc1802105962441033582nteger @ H2 @ I2 ) )
= ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) ) ) ).
% map_prod.comp
thf(fact_4523_map__prod_Ocomp,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer > code_integer] :
( ( comp_P6312383820422015140nteger @ ( produc3090791198996863994nteger @ F @ G ) @ ( produc6023007152014423402nteger @ H2 @ I2 ) )
= ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) ) ) ).
% map_prod.comp
thf(fact_4524_map__prod_Ocomp,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer] :
( ( comp_P1414880308642389412nteger @ ( produc6158254626097298938nteger @ F @ G ) @ ( produc3789967387199524714nteger @ H2 @ I2 ) )
= ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) ) ) ).
% map_prod.comp
thf(fact_4525_map__prod_Ocomp,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer > code_integer] :
( ( comp_P5544053077415136032nteger @ ( produc4724772261320961782nteger @ F @ G ) @ ( produc7818937854452556646nteger @ H2 @ I2 ) )
= ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) ) ) ).
% map_prod.comp
thf(fact_4526_map__prod_Ocompositionality,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer,Prod: produc8923325533196201883nteger] :
( ( produc5081366107664765182nteger @ F @ G @ ( produc1802105962441033582nteger @ H2 @ I2 @ Prod ) )
= ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_4527_map__prod_Ocompositionality,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer > code_integer,Prod: produc8923325533196201883nteger] :
( ( produc3090791198996863994nteger @ F @ G @ ( produc6023007152014423402nteger @ H2 @ I2 @ Prod ) )
= ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_4528_map__prod_Ocompositionality,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer,Prod: produc8923325533196201883nteger] :
( ( produc6158254626097298938nteger @ F @ G @ ( produc3789967387199524714nteger @ H2 @ I2 @ Prod ) )
= ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_4529_map__prod_Ocompositionality,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer > code_integer,Prod: produc8923325533196201883nteger] :
( ( produc4724772261320961782nteger @ F @ G @ ( produc7818937854452556646nteger @ H2 @ I2 @ Prod ) )
= ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_4530_map__prod__compose,axiom,
! [F1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer] :
( ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F1 @ F22 ) @ ( comp_C1593894019821074884nteger @ G1 @ G2 ) )
= ( comp_P234064951390806312nteger @ ( produc5081366107664765182nteger @ F1 @ G1 ) @ ( produc1802105962441033582nteger @ F22 @ G2 ) ) ) ).
% map_prod_compose
thf(fact_4531_map__prod__compose,axiom,
! [F1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer] :
( ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F1 @ F22 ) @ ( comp_C8797469213163452608nteger @ G1 @ G2 ) )
= ( comp_P6312383820422015140nteger @ ( produc3090791198996863994nteger @ F1 @ G1 ) @ ( produc6023007152014423402nteger @ F22 @ G2 ) ) ) ).
% map_prod_compose
thf(fact_4532_map__prod__compose,axiom,
! [F1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer > code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer] :
( ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F1 @ F22 ) @ ( comp_C1593894019821074884nteger @ G1 @ G2 ) )
= ( comp_P1414880308642389412nteger @ ( produc6158254626097298938nteger @ F1 @ G1 ) @ ( produc3789967387199524714nteger @ F22 @ G2 ) ) ) ).
% map_prod_compose
thf(fact_4533_map__prod__compose,axiom,
! [F1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer > code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer] :
( ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F1 @ F22 ) @ ( comp_C8797469213163452608nteger @ G1 @ G2 ) )
= ( comp_P5544053077415136032nteger @ ( produc4724772261320961782nteger @ F1 @ G1 ) @ ( produc7818937854452556646nteger @ F22 @ G2 ) ) ) ).
% map_prod_compose
thf(fact_4534_inj__img__insertE,axiom,
! [F: code_integer > code_integer,A: set_Code_integer,X: code_integer,B4: set_Code_integer] :
( ( inj_on1158064722259180589nteger @ F @ A )
=> ( ~ ( member_Code_integer @ X @ B4 )
=> ( ( ( insert_Code_integer @ X @ B4 )
= ( image_4470545334726330049nteger @ F @ A ) )
=> ~ ! [X10: code_integer,A7: set_Code_integer] :
( ~ ( member_Code_integer @ X10 @ A7 )
=> ( ( A
= ( insert_Code_integer @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_4470545334726330049nteger @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4535_inj__img__insertE,axiom,
! [F: nat > code_integer,A: set_nat,X: code_integer,B4: set_Code_integer] :
( ( inj_on2245834603434889120nteger @ F @ A )
=> ( ~ ( member_Code_integer @ X @ B4 )
=> ( ( ( insert_Code_integer @ X @ B4 )
= ( image_1215581382706833972nteger @ F @ A ) )
=> ~ ! [X10: nat,A7: set_nat] :
( ~ ( member_nat @ X10 @ A7 )
=> ( ( A
= ( insert_nat @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_1215581382706833972nteger @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4536_inj__img__insertE,axiom,
! [F: nat > nat,A: set_nat,X: nat,B4: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ B4 )
= ( image_nat_nat2 @ F @ A ) )
=> ~ ! [X10: nat,A7: set_nat] :
( ~ ( member_nat @ X10 @ A7 )
=> ( ( A
= ( insert_nat @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_nat_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4537_inj__img__insertE,axiom,
! [F: int > nat,A: set_int,X: nat,B4: set_nat] :
( ( inj_on_int_nat @ F @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ B4 )
= ( image_int_nat2 @ F @ A ) )
=> ~ ! [X10: int,A7: set_int] :
( ~ ( member_int @ X10 @ A7 )
=> ( ( A
= ( insert_int @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_int_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4538_inj__img__insertE,axiom,
! [F: $o > nat,A: set_o,X: nat,B4: set_nat] :
( ( inj_on_o_nat @ F @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ B4 )
= ( image_o_nat2 @ F @ A ) )
=> ~ ! [X10: $o,A7: set_o] :
( ~ ( member_o @ X10 @ A7 )
=> ( ( A
= ( insert_o @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_o_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4539_inj__img__insertE,axiom,
! [F: product_unit > nat,A: set_Product_unit,X: nat,B4: set_nat] :
( ( inj_on8430439091780834860it_nat @ F @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ B4 )
= ( image_875570014554754200it_nat @ F @ A ) )
=> ~ ! [X10: product_unit,A7: set_Product_unit] :
( ~ ( member_Product_unit @ X10 @ A7 )
=> ( ( A
= ( insert_Product_unit @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_875570014554754200it_nat @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4540_inj__img__insertE,axiom,
! [F: nat > int,A: set_nat,X: int,B4: set_int] :
( ( inj_on_nat_int @ F @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ B4 )
= ( image_nat_int2 @ F @ A ) )
=> ~ ! [X10: nat,A7: set_nat] :
( ~ ( member_nat @ X10 @ A7 )
=> ( ( A
= ( insert_nat @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_nat_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4541_inj__img__insertE,axiom,
! [F: int > int,A: set_int,X: int,B4: set_int] :
( ( inj_on_int_int @ F @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ B4 )
= ( image_int_int2 @ F @ A ) )
=> ~ ! [X10: int,A7: set_int] :
( ~ ( member_int @ X10 @ A7 )
=> ( ( A
= ( insert_int @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_int_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4542_inj__img__insertE,axiom,
! [F: $o > int,A: set_o,X: int,B4: set_int] :
( ( inj_on_o_int @ F @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ B4 )
= ( image_o_int2 @ F @ A ) )
=> ~ ! [X10: $o,A7: set_o] :
( ~ ( member_o @ X10 @ A7 )
=> ( ( A
= ( insert_o @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_o_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4543_inj__img__insertE,axiom,
! [F: product_unit > int,A: set_Product_unit,X: int,B4: set_int] :
( ( inj_on8427948621271784584it_int @ F @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ B4 )
= ( image_873079544045703924it_int @ F @ A ) )
=> ~ ! [X10: product_unit,A7: set_Product_unit] :
( ~ ( member_Product_unit @ X10 @ A7 )
=> ( ( A
= ( insert_Product_unit @ X10 @ A7 ) )
=> ( ( X
= ( F @ X10 ) )
=> ( B4
!= ( image_873079544045703924it_int @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4544_type__copy__wit,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,X: nat,S: int > set_nat,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( member_nat @ X @ ( comp_i5102403792180996084nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4545_type__copy__wit,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,X: int,S: int > set_int,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( member_int @ X @ ( comp_i1887841406321795152nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_int @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4546_type__copy__wit,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,X: $o,S: int > set_o,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( member_o @ X @ ( comp_i5479505385629679622nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_o @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4547_type__copy__wit,axiom,
! [Rep2: code_integer > int,Abs2: int > code_integer,X: product_unit,S: int > set_Product_unit,Y: int] :
( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
=> ( ( member_Product_unit @ X @ ( comp_i3319814986458681467nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_Product_unit @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4548_type__copy__wit,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: nat,S: $o > set_nat,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( member_nat @ X @ ( comp_o9190160103024335696t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4549_type__copy__wit,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: int,S: $o > set_int,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( member_int @ X @ ( comp_o1521574261646630388t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_int @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4550_type__copy__wit,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: $o,S: $o > set_o,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( member_o @ X @ ( comp_o6895741221744486154t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_o @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4551_type__copy__wit,axiom,
! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: product_unit,S: $o > set_Product_unit,Y: $o] :
( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( member_Product_unit @ X @ ( comp_o2086126948152544725t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_Product_unit @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4552_type__copy__wit,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,X: nat,S: nat > set_nat,Y: nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( member_nat @ X @ ( comp_n4787279007288201059atural @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4553_type__copy__wit,axiom,
! [Rep2: code_natural > nat,Abs2: nat > code_natural,X: int,S: nat > set_int,Y: nat] :
( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
=> ( ( member_int @ X @ ( comp_n1572716621429000127atural @ S @ Rep2 @ ( Abs2 @ Y ) ) )
=> ( member_int @ X @ ( S @ Y ) ) ) ) ).
% type_copy_wit
thf(fact_4554_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: nat > nat,G: nat > nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4555_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: int > nat,G: int > nat,R3: set_Pr958786334691620121nt_int] :
( ( member8440522571783428010at_nat @ C @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: int,Y2: int] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4556_prod__fun__imageE,axiom,
! [C: product_prod_int_int,F: nat > int,G: nat > int,R3: set_Pr1261947904930325089at_nat] :
( ( member5262025264175285858nt_int @ C @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( C
= ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4557_prod__fun__imageE,axiom,
! [C: product_prod_int_int,F: int > int,G: int > int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) )
=> ~ ! [X3: int,Y2: int] :
( ( C
= ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4558_prod__fun__imageE,axiom,
! [C: produc3658429121746597890et_nat,F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat,R3: set_Pr1261947904930325089at_nat] :
( ( member6260224972018164377et_nat @ C @ ( image_9068448769188195518et_nat @ ( produc2064153442785501et_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( C
= ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4559_prod__fun__imageE,axiom,
! [C: produc3658429121746597890et_nat,F: int > heap_e7401611519738050253t_unit,G: int > set_nat,R3: set_Pr958786334691620121nt_int] :
( ( member6260224972018164377et_nat @ C @ ( image_2127501072796355446et_nat @ ( produc3152090231653401565et_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: int,Y2: int] :
( ( C
= ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4560_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,R3: set_Pr3948176798113811640et_nat] :
( ( member8440522571783428010at_nat @ C @ ( image_7757775281999735736at_nat @ ( produc8285742784527409791at_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4561_prod__fun__imageE,axiom,
! [C: product_prod_int_int,F: heap_e7401611519738050253t_unit > int,G: set_nat > int,R3: set_Pr3948176798113811640et_nat] :
( ( member5262025264175285858nt_int @ C @ ( image_4579277974391593584nt_int @ ( produc5973434381925408567at_int @ F @ G ) @ R3 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( ( C
= ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4562_prod__fun__imageE,axiom,
! [C: produc3658429121746597890et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,R3: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ C @ ( image_7426123528223433739et_nat @ ( produc8234977478398262212et_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( ( C
= ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4563_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: ( produc3658429121746597890et_nat > $o ) > nat,G: produc3658429121746597890et_nat > nat,R3: set_Pr3286484037609594932et_nat] :
( ( member8440522571783428010at_nat @ C @ ( image_7073998564157519676at_nat @ ( produc2784369760062397435at_nat @ F @ G ) @ R3 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
=> ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_4564_inj__on__iff__surj,axiom,
! [A: set_Code_integer,A8: set_Code_integer] :
( ( A != bot_bo3990330152332043303nteger )
=> ( ( ? [F4: code_integer > code_integer] :
( ( inj_on1158064722259180589nteger @ F4 @ A )
& ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: code_integer > code_integer] :
( ( image_4470545334726330049nteger @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4565_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > set_nat] :
( ( inj_on_nat_set_nat @ F4 @ A )
& ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: set_nat > nat] :
( ( image_set_nat_nat @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4566_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_Code_integer] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > code_integer] :
( ( inj_on2245834603434889120nteger @ F4 @ A )
& ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: code_integer > nat] :
( ( image_951025933927791156er_nat @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4567_inj__on__iff__surj,axiom,
! [A: set_set_nat,A8: set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( ? [F4: set_nat > nat] :
( ( inj_on_set_nat_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > set_nat] :
( ( image_nat_set_nat @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4568_inj__on__iff__surj,axiom,
! [A: set_Code_integer,A8: set_nat] :
( ( A != bot_bo3990330152332043303nteger )
=> ( ( ? [F4: code_integer > nat] :
( ( inj_on1981279154655846304er_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_951025933927791156er_nat @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > code_integer] :
( ( image_1215581382706833972nteger @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4569_inj__on__iff__surj,axiom,
! [A: set_Product_unit,A8: set_nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ? [F4: product_unit > nat] :
( ( inj_on8430439091780834860it_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > product_unit] :
( ( image_8730104196221521654t_unit @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4570_inj__on__iff__surj,axiom,
! [A: set_o,A8: set_nat] :
( ( A != bot_bot_set_o )
=> ( ( ? [F4: $o > nat] :
( ( inj_on_o_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_o_nat2 @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > $o] :
( ( image_nat_o2 @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4571_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F4: nat > nat] :
( ( inj_on_nat_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > nat] :
( ( image_nat_nat2 @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4572_inj__on__iff__surj,axiom,
! [A: set_int,A8: set_nat] :
( ( A != bot_bot_set_int )
=> ( ( ? [F4: int > nat] :
( ( inj_on_int_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_int_nat2 @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > int] :
( ( image_nat_int2 @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4573_inj__on__iff__surj,axiom,
! [A: set_Pr4532377907799695533_nat_o,A8: set_nat] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( ? [F4: ( produc3658429121746597890et_nat > $o ) > nat] :
( ( inj_on5271586719803267564_o_nat @ F4 @ A )
& ( ord_less_eq_set_nat @ ( image_2174381888560928856_o_nat @ F4 @ A ) @ A8 ) ) )
= ( ? [G3: nat > produc3658429121746597890et_nat > $o] :
( ( image_4682239675366380726_nat_o @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_4574_Sup__insert__finite,axiom,
! [S: set_int,X: int] :
( ( finite_finite_int @ S )
=> ( ( ( S = bot_bot_set_int )
=> ( ( complete_Sup_Sup_int @ ( insert_int @ X @ S ) )
= X ) )
& ( ( S != bot_bot_set_int )
=> ( ( complete_Sup_Sup_int @ ( insert_int @ X @ S ) )
= ( ord_max_int @ X @ ( complete_Sup_Sup_int @ S ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4575_Sup__insert__finite,axiom,
! [S: set_nat,X: nat] :
( ( finite_finite_nat @ S )
=> ( ( ( S = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ S ) )
= X ) )
& ( ( S != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ S ) )
= ( ord_max_nat @ X @ ( complete_Sup_Sup_nat @ S ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4576_Sup__insert__finite,axiom,
! [S: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( ( S = bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ S ) )
= X ) )
& ( ( S != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ S ) )
= ( ord_max_Product_unit @ X @ ( comple4687483117567863418t_unit @ S ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4577_hom__Max__commute,axiom,
! [H2: code_integer > code_integer,N: set_Code_integer] :
( ! [X3: code_integer,Y2: code_integer] :
( ( H2 @ ( ord_max_Code_integer @ X3 @ Y2 ) )
= ( ord_max_Code_integer @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite6017078050557962740nteger @ N )
=> ( ( N != bot_bo3990330152332043303nteger )
=> ( ( H2 @ ( lattic4901227151466704046nteger @ N ) )
= ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ H2 @ N ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4578_hom__Max__commute,axiom,
! [H2: product_unit > product_unit,N: set_Product_unit] :
( ! [X3: product_unit,Y2: product_unit] :
( ( H2 @ ( ord_max_Product_unit @ X3 @ Y2 ) )
= ( ord_max_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite4290736615968046902t_unit @ N )
=> ( ( N != bot_bo3957492148770167129t_unit )
=> ( ( H2 @ ( lattic1210392890313158908t_unit @ N ) )
= ( lattic1210392890313158908t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4579_hom__Max__commute,axiom,
! [H2: $o > $o,N: set_o] :
( ! [X3: $o,Y2: $o] :
( ( H2 @ ( ord_max_o @ X3 @ Y2 ) )
= ( ord_max_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_o @ N )
=> ( ( N != bot_bot_set_o )
=> ( ( H2 @ ( lattic1921953407002678535_Max_o @ N ) )
= ( lattic1921953407002678535_Max_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4580_hom__Max__commute,axiom,
! [H2: int > int,N: set_int] :
( ! [X3: int,Y2: int] :
( ( H2 @ ( ord_max_int @ X3 @ Y2 ) )
= ( ord_max_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_int @ N )
=> ( ( N != bot_bot_set_int )
=> ( ( H2 @ ( lattic8263393255366662781ax_int @ N ) )
= ( lattic8263393255366662781ax_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4581_hom__Max__commute,axiom,
! [H2: nat > nat,N: set_nat] :
( ! [X3: nat,Y2: nat] :
( ( H2 @ ( ord_max_nat @ X3 @ Y2 ) )
= ( ord_max_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite_nat @ N )
=> ( ( N != bot_bot_set_nat )
=> ( ( H2 @ ( lattic8265883725875713057ax_nat @ N ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4582_Max_Osubset,axiom,
! [A: set_Code_integer,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( B4 != bot_bo3990330152332043303nteger )
=> ( ( ord_le7084787975880047091nteger @ B4 @ A )
=> ( ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ B4 ) @ ( lattic4901227151466704046nteger @ A ) )
= ( lattic4901227151466704046nteger @ A ) ) ) ) ) ).
% Max.subset
thf(fact_4583_Max_Osubset,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ A )
=> ( ( ord_max_Product_unit @ ( lattic1210392890313158908t_unit @ B4 ) @ ( lattic1210392890313158908t_unit @ A ) )
= ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ).
% Max.subset
thf(fact_4584_Max_Osubset,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( B4 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ B4 @ A )
=> ( ( ord_max_o @ ( lattic1921953407002678535_Max_o @ B4 ) @ ( lattic1921953407002678535_Max_o @ A ) )
= ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ).
% Max.subset
thf(fact_4585_Max_Osubset,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( B4 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ B4 @ A )
=> ( ( ord_max_int @ ( lattic8263393255366662781ax_int @ B4 ) @ ( lattic8263393255366662781ax_int @ A ) )
= ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ).
% Max.subset
thf(fact_4586_Max_Osubset,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B4 ) @ ( lattic8265883725875713057ax_nat @ A ) )
= ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ).
% Max.subset
thf(fact_4587_Max_Oinsert__not__elem,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
= ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4588_Max_Oinsert__not__elem,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
= ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4589_Max_Oinsert__not__elem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
= ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4590_Max_Oinsert__not__elem,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
= ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4591_Max_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4592_Max_Oclosed,axiom,
! [A: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer,Y2: code_integer] : ( member_Code_integer @ ( ord_max_Code_integer @ X3 @ Y2 ) @ ( insert_Code_integer @ X3 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) )
=> ( member_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ A ) ) ) ) ).
% Max.closed
thf(fact_4593_Max_Oclosed,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( ord_max_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ A ) ) ) ) ).
% Max.closed
thf(fact_4594_Max_Oclosed,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] : ( member_o @ ( ord_max_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
=> ( member_o @ ( lattic1921953407002678535_Max_o @ A ) @ A ) ) ) ) ).
% Max.closed
thf(fact_4595_Max_Oclosed,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [X3: int,Y2: int] : ( member_int @ ( ord_max_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
=> ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ) ).
% Max.closed
thf(fact_4596_Max_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat @ ( ord_max_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ) ).
% Max.closed
thf(fact_4597_Max_Ounion,axiom,
! [A: set_Code_integer,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( B4 != bot_bo3990330152332043303nteger )
=> ( ( lattic4901227151466704046nteger @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4598_Max_Ounion,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( lattic1210392890313158908t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( ord_max_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ ( lattic1210392890313158908t_unit @ B4 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4599_Max_Ounion,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ( B4 != bot_bot_set_o )
=> ( ( lattic1921953407002678535_Max_o @ ( sup_sup_set_o @ A @ B4 ) )
= ( ord_max_o @ ( lattic1921953407002678535_Max_o @ A ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4600_Max_Ounion,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ( B4 != bot_bot_set_int )
=> ( ( lattic8263393255366662781ax_int @ ( sup_sup_set_int @ A @ B4 ) )
= ( ord_max_int @ ( lattic8263393255366662781ax_int @ A ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4601_Max_Ounion,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4602_Max_Oeq__fold,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
= ( finite_fold_o_o @ ord_max_o @ X @ A ) ) ) ).
% Max.eq_fold
thf(fact_4603_Max_Oeq__fold,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
= ( finite8834431295148555784t_unit @ ord_max_Product_unit @ X @ A ) ) ) ).
% Max.eq_fold
thf(fact_4604_Max_Oeq__fold,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
= ( finite6454954857057555716nteger @ ord_max_Code_integer @ X @ A ) ) ) ).
% Max.eq_fold
thf(fact_4605_Max_Oeq__fold,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
= ( finite_fold_int_int @ ord_max_int @ X @ A ) ) ) ).
% Max.eq_fold
thf(fact_4606_Max_Oeq__fold,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
= ( finite_fold_nat_nat @ ord_max_nat @ X @ A ) ) ) ).
% Max.eq_fold
thf(fact_4607_finite__range__prod,axiom,
! [F: code_integer > produc2591615631378489102nteger] :
( ( finite_finite_nat @ ( image_951025933927791156er_nat @ ( comp_P5725896914063526938nteger @ produc8897218493789310642nteger @ F ) @ top_to4645266643341252675nteger ) )
=> ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P6122662217367900327nteger @ produc1781891006948753140nteger @ F ) @ top_to4645266643341252675nteger ) )
=> ( finite1154042800025562295nteger @ ( image_3408592110290467754nteger @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% finite_range_prod
thf(fact_4608_finite__range__prod,axiom,
! [F: code_integer > produc6433947425880249742er_nat] :
( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7101464685127059239nteger @ produc8632663045010267826er_nat @ F ) @ top_to4645266643341252675nteger ) )
=> ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ ( comp_P5964733648194011802nteger @ produc1517335558169710324er_nat @ F ) @ top_to4645266643341252675nteger ) )
=> ( finite4996374594527322935er_nat @ ( image_7250923904792228394er_nat @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% finite_range_prod
thf(fact_4609_finite__range__prod,axiom,
! [F: code_integer > produc8923325533196201883nteger] :
( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7111796090210262324nteger @ produc8508995932063986495nteger @ F ) @ top_to4645266643341252675nteger ) )
=> ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7111796090210262324nteger @ produc6174133586879617921nteger @ F ) @ top_to4645266643341252675nteger ) )
=> ( finite2154694888287295812nteger @ ( image_8049062792287394871nteger @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% finite_range_prod
thf(fact_4610_finite__range__prod,axiom,
! [F: product_unit > product_prod_nat_nat] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_fst_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
=> ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_snd_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
=> ( finite6177210948735845034at_nat @ ( image_7281959405765017145at_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% finite_range_prod
thf(fact_4611_finite__range__prod,axiom,
! [F: product_unit > produc2591615631378489102nteger] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P7621350044865347728t_unit @ produc8897218493789310642nteger @ F ) @ top_to1996260823553986621t_unit ) )
=> ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P7320554561093743171t_unit @ produc1781891006948753140nteger @ F ) @ top_to1996260823553986621t_unit ) )
=> ( finite1154042800025562295nteger @ ( image_4946983642536848454nteger @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% finite_range_prod
thf(fact_4612_finite__range__prod,axiom,
! [F: product_unit > produc6433947425880249742er_nat] :
( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P7581484221039074243t_unit @ produc8632663045010267826er_nat @ F ) @ top_to1996260823553986621t_unit ) )
=> ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P6821003776282389008t_unit @ produc1517335558169710324er_nat @ F ) @ top_to1996260823553986621t_unit ) )
=> ( finite4996374594527322935er_nat @ ( image_8789315437038609094er_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% finite_range_prod
thf(fact_4613_finite__range__prod,axiom,
! [F: product_unit > produc8923325533196201883nteger] :
( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc8508995932063986495nteger @ F ) @ top_to1996260823553986621t_unit ) )
=> ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc6174133586879617921nteger @ F ) @ top_to1996260823553986621t_unit ) )
=> ( finite2154694888287295812nteger @ ( image_8089323762101378259nteger @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% finite_range_prod
thf(fact_4614_finite__range__prod,axiom,
! [F: product_unit > product_prod_int_int] :
( ( finite_finite_int @ ( image_873079544045703924it_int @ ( comp_P6796073005956588617t_unit @ product_fst_int_int @ F ) @ top_to1996260823553986621t_unit ) )
=> ( ( finite_finite_int @ ( image_873079544045703924it_int @ ( comp_P6796073005956588617t_unit @ product_snd_int_int @ F ) @ top_to1996260823553986621t_unit ) )
=> ( finite2998713641127702882nt_int @ ( image_4103462098156874993nt_int @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% finite_range_prod
thf(fact_4615_finite__range__prod,axiom,
! [F: $o > product_prod_nat_nat] :
( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P4394204175247225320_nat_o @ product_fst_nat_nat @ F ) @ top_top_set_o ) )
=> ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P4394204175247225320_nat_o @ product_snd_nat_nat @ F ) @ top_top_set_o ) )
=> ( finite6177210948735845034at_nat @ ( image_3855930084881510382at_nat @ F @ top_top_set_o ) ) ) ) ).
% finite_range_prod
thf(fact_4616_finite__range__prod,axiom,
! [F: $o > produc2591615631378489102nteger] :
( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P7179991235626930843_nat_o @ produc8897218493789310642nteger @ F ) @ top_top_set_o ) )
=> ( ( finite6017078050557962740nteger @ ( image_o_Code_integer @ ( comp_P7177471617286032590eger_o @ produc1781891006948753140nteger @ F ) @ top_top_set_o ) )
=> ( finite1154042800025562295nteger @ ( image_8407096251609565691nteger @ F @ top_top_set_o ) ) ) ) ).
% finite_range_prod
thf(fact_4617_inj__apsnd,axiom,
! [F: code_integer > code_integer] :
( ( inj_on20428373856169009nteger @ ( produc6499014454317279255nteger @ F ) @ top_to2363841527343591755nteger )
= ( inj_on1158064722259180589nteger @ F @ top_to4645266643341252675nteger ) ) ).
% inj_apsnd
thf(fact_4618_total__inv__image,axiom,
! [F: product_unit > product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit )
=> ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
=> ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im8770536620514187154t_unit @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4619_total__inv__image,axiom,
! [F: product_unit > $o,R2: set_Product_prod_o_o] :
( ( inj_on8854722734121536060unit_o @ F @ top_to1996260823553986621t_unit )
=> ( ( total_on_o @ top_top_set_o @ R2 )
=> ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im4602722609573551687t_unit @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4620_total__inv__image,axiom,
! [F: product_unit > nat,R2: set_Pr1261947904930325089at_nat] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( total_on_nat @ top_top_set_nat @ R2 )
=> ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im7624054837341743209t_unit @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4621_total__inv__image,axiom,
! [F: product_unit > int,R2: set_Pr958786334691620121nt_int] :
( ( inj_on8427948621271784584it_int @ F @ top_to1996260823553986621t_unit )
=> ( ( total_on_int @ top_top_set_int @ R2 )
=> ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im4740476784543923469t_unit @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4622_total__inv__image,axiom,
! [F: $o > product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( inj_on4268752236855399398t_unit @ F @ top_top_set_o )
=> ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
=> ( total_on_o @ top_top_set_o @ ( inv_im9188693106839688349unit_o @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4623_total__inv__image,axiom,
! [F: $o > $o,R2: set_Product_prod_o_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( total_on_o @ top_top_set_o @ R2 )
=> ( total_on_o @ top_top_set_o @ ( inv_image_o_o @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4624_total__inv__image,axiom,
! [F: $o > nat,R2: set_Pr1261947904930325089at_nat] :
( ( inj_on_o_nat @ F @ top_top_set_o )
=> ( ( total_on_nat @ top_top_set_nat @ R2 )
=> ( total_on_o @ top_top_set_o @ ( inv_image_nat_o @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4625_total__inv__image,axiom,
! [F: $o > int,R2: set_Pr958786334691620121nt_int] :
( ( inj_on_o_int @ F @ top_top_set_o )
=> ( ( total_on_int @ top_top_set_int @ R2 )
=> ( total_on_o @ top_top_set_o @ ( inv_image_int_o @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4626_total__inv__image,axiom,
! [F: nat > product_unit,R2: set_Pr5094982260447487303t_unit] :
( ( inj_on7061601236592826506t_unit @ F @ top_top_set_nat )
=> ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
=> ( total_on_nat @ top_top_set_nat @ ( inv_im8992892692529751563it_nat @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4627_total__inv__image,axiom,
! [F: nat > $o,R2: set_Product_prod_o_o] :
( ( inj_on_nat_o @ F @ top_top_set_nat )
=> ( ( total_on_o @ top_top_set_o @ R2 )
=> ( total_on_nat @ top_top_set_nat @ ( inv_image_o_nat @ R2 @ F ) ) ) ) ).
% total_inv_image
thf(fact_4628_card__Diff1__less,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_4629_card__Diff1__less,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_4630_card__Diff1__less,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_4631_card__Diff1__less,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_4632_card__Diff1__less,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_4633_card__Diff1__less,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_4634_card__Diff2__less,axiom,
! [A: set_Code_integer,X: code_integer,Y: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( member_Code_integer @ Y @ A )
=> ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) @ ( insert_Code_integer @ Y @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4635_card__Diff2__less,axiom,
! [A: set_Product_unit,X: product_unit,Y: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( member_Product_unit @ Y @ A )
=> ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4636_card__Diff2__less,axiom,
! [A: set_o,X: $o,Y: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( member_o @ Y @ A )
=> ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4637_card__Diff2__less,axiom,
! [A: set_int,X: int,Y: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4638_card__Diff2__less,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( member6576561426505652726_nat_o @ Y @ A )
=> ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4639_card__Diff2__less,axiom,
! [A: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4640_card__Diff1__less__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) )
= ( ( finite6017078050557962740nteger @ A )
& ( member_Code_integer @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_4641_card__Diff1__less__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) )
= ( ( finite4290736615968046902t_unit @ A )
& ( member_Product_unit @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_4642_card__Diff1__less__iff,axiom,
! [A: set_o,X: $o] :
( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) )
= ( ( finite_finite_o @ A )
& ( member_o @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_4643_card__Diff1__less__iff,axiom,
! [A: set_int,X: int] :
( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) )
= ( ( finite_finite_int @ A )
& ( member_int @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_4644_card__Diff1__less__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) )
= ( ( finite3252695134891459830_nat_o @ A )
& ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_4645_card__Diff1__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
= ( ( finite_finite_nat @ A )
& ( member_nat @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_4646_Inter__in__chain,axiom,
! [B9: set_set_nat,A11: set_set_nat] :
( ( finite1152437895449049373et_nat @ B9 )
=> ( ( B9 != bot_bot_set_set_nat )
=> ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ B9 )
=> ( member_set_nat @ ( comple7806235888213564991et_nat @ B9 ) @ B9 ) ) ) ) ).
% Inter_in_chain
thf(fact_4647_card__eq__UNIV2,axiom,
! [S: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= ( finite410649719033368117t_unit @ S ) )
= ( S = top_to1996260823553986621t_unit ) ) ).
% card_eq_UNIV2
thf(fact_4648_card__eq__UNIV2,axiom,
! [S: set_o] :
( ( ( finite_card_o @ top_top_set_o )
= ( finite_card_o @ S ) )
= ( S = top_top_set_o ) ) ).
% card_eq_UNIV2
thf(fact_4649_card__eq__UNIV,axiom,
! [S: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ S )
= ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
= ( S = top_to1996260823553986621t_unit ) ) ).
% card_eq_UNIV
thf(fact_4650_card__eq__UNIV,axiom,
! [S: set_o] :
( ( ( finite_card_o @ S )
= ( finite_card_o @ top_top_set_o ) )
= ( S = top_top_set_o ) ) ).
% card_eq_UNIV
thf(fact_4651_fst__map__prod,axiom,
! [F: int > int,G: int > int,X: product_prod_int_int] :
( ( product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
= ( F @ ( product_fst_int_int @ X ) ) ) ).
% fst_map_prod
thf(fact_4652_apfst__conv,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( produc4172003873565521026et_nat @ F @ ( produc7507926704131184380et_nat @ X @ Y ) )
= ( produc7507926704131184380et_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_4653_apfst__conv,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
= ( product_Pair_nat_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_4654_apfst__conv,axiom,
! [F: int > int,X: int,Y: int] :
( ( produc2355237562570658114nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
= ( product_Pair_int_int @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_4655_apfst__conv,axiom,
! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( produc3198230245617193214et_nat @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( produc5001842942810119800et_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_4656_apfst__conv,axiom,
! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( produc1442628905706551930et_nat @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( produc2245416461498447860et_nat @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_4657_apsnd__conv,axiom,
! [F: set_nat > set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( produc3286488329990770625t_unit @ F @ ( produc7507926704131184380et_nat @ X @ Y ) )
= ( produc7507926704131184380et_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4658_apsnd__conv,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
= ( product_Pair_nat_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4659_apsnd__conv,axiom,
! [F: int > int,X: int,Y: int] :
( ( produc4463282112584876420nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
= ( product_Pair_int_int @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4660_apsnd__conv,axiom,
! [F: produc3658429121746597890et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( produc8750854537940449737_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4661_apsnd__conv,axiom,
! [F: produc3658429121746597890et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( produc7714581247149323085_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
= ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4662_apsnd__conv,axiom,
! [F: produc3925858234332021118et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( produc1515462096303866701_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4663_apsnd__conv,axiom,
! [F: produc3925858234332021118et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( produc969530845752564945_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
= ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4664_apsnd__conv,axiom,
! [F: code_integer > code_integer,X: code_integer,Y: code_integer] :
( ( produc6499014454317279255nteger @ F @ ( produc1086072967326762835nteger @ X @ Y ) )
= ( produc1086072967326762835nteger @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_4665_fst__apfst,axiom,
! [F: int > int,X: product_prod_int_int] :
( ( product_fst_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
= ( F @ ( product_fst_int_int @ X ) ) ) ).
% fst_apfst
thf(fact_4666_snd__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: produc3658429121746597890et_nat] :
( ( produc8586169260539613262et_nat @ ( produc4172003873565521026et_nat @ F @ X ) )
= ( produc8586169260539613262et_nat @ X ) ) ).
% snd_apfst
thf(fact_4667_snd__apfst,axiom,
! [F: int > int,X: product_prod_int_int] :
( ( product_snd_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
= ( product_snd_int_int @ X ) ) ).
% snd_apfst
thf(fact_4668_in__inv__image,axiom,
! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_4669_in__inv__image,axiom,
! [X: nat,Y: nat,R2: set_Pr958786334691620121nt_int,F: nat > int] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_int_nat @ R2 @ F ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_4670_in__inv__image,axiom,
! [X: int,Y: int,R2: set_Pr1261947904930325089at_nat,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_nat_int @ R2 @ F ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_4671_in__inv__image,axiom,
! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,F: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_int_int @ R2 @ F ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_4672_fst__apsnd,axiom,
! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
( ( produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F @ X ) )
= ( produc8508995932063986495nteger @ X ) ) ).
% fst_apsnd
thf(fact_4673_fst__apsnd,axiom,
! [F: int > int,X: product_prod_int_int] :
( ( product_fst_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
= ( product_fst_int_int @ X ) ) ).
% fst_apsnd
thf(fact_4674_apsnd__eq__conv,axiom,
! [F: code_integer > code_integer,X: produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( ( produc6499014454317279255nteger @ F @ X )
= ( produc6499014454317279255nteger @ G @ X ) )
= ( ( F @ ( produc6174133586879617921nteger @ X ) )
= ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_4675_snd__apsnd,axiom,
! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
( ( produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F @ X ) )
= ( F @ ( produc6174133586879617921nteger @ X ) ) ) ).
% snd_apsnd
thf(fact_4676_snd__apsnd,axiom,
! [F: set_nat > set_nat,X: produc3658429121746597890et_nat] :
( ( produc8586169260539613262et_nat @ ( produc3286488329990770625t_unit @ F @ X ) )
= ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% snd_apsnd
thf(fact_4677_snd__apsnd,axiom,
! [F: int > int,X: product_prod_int_int] :
( ( product_snd_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
= ( F @ ( product_snd_int_int @ X ) ) ) ).
% snd_apsnd
thf(fact_4678_Inf__eq__bot__iff,axiom,
! [A: set_Product_unit] :
( ( ( comple2584293577114468500t_unit @ A )
= bot_bot_Product_unit )
= ( ! [X2: product_unit] :
( ( ord_le361264281704409273t_unit @ bot_bot_Product_unit @ X2 )
=> ? [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ A )
& ( ord_le361264281704409273t_unit @ Y3 @ X2 ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4679_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_4680_Inf__empty,axiom,
( ( comple90455603224345172_nat_o @ bot_bo7824918357723582233_nat_o )
= top_to5213382387105117981_nat_o ) ).
% Inf_empty
thf(fact_4681_Inf__empty,axiom,
( ( comple7216939928495492468t_unit @ bot_bo6284674557190688313t_unit )
= top_to1996260823553986621t_unit ) ).
% Inf_empty
thf(fact_4682_Inf__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inf_empty
thf(fact_4683_Inf__empty,axiom,
( ( comple3628384868704368283et_int @ bot_bot_set_set_int )
= top_top_set_int ) ).
% Inf_empty
thf(fact_4684_Inf__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inf_empty
thf(fact_4685_Inf__empty,axiom,
( ( comple2584293577114468500t_unit @ bot_bo3957492148770167129t_unit )
= top_top_Product_unit ) ).
% Inf_empty
thf(fact_4686_Inf__UNIV,axiom,
( ( comple7216939928495492468t_unit @ top_to1767297665138865437t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Inf_UNIV
thf(fact_4687_Inf__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inf_UNIV
thf(fact_4688_Inf__UNIV,axiom,
( ( comple3628384868704368283et_int @ top_top_set_set_int )
= bot_bot_set_int ) ).
% Inf_UNIV
thf(fact_4689_Inf__UNIV,axiom,
( ( comple79274577958421812_nat_o @ top_to8515903933492314333_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Inf_UNIV
thf(fact_4690_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_4691_Inf__UNIV,axiom,
( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
= bot_bot_set_nat ) ).
% Inf_UNIV
thf(fact_4692_Inf__UNIV,axiom,
( ( comple2584293577114468500t_unit @ top_to1996260823553986621t_unit )
= bot_bot_Product_unit ) ).
% Inf_UNIV
thf(fact_4693_cInf__singleton,axiom,
! [X: $o] :
( ( complete_Inf_Inf_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cInf_singleton
thf(fact_4694_cInf__singleton,axiom,
! [X: int] :
( ( complete_Inf_Inf_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% cInf_singleton
thf(fact_4695_cInf__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o] :
( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
= X ) ).
% cInf_singleton
thf(fact_4696_cInf__singleton,axiom,
! [X: nat] :
( ( complete_Inf_Inf_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cInf_singleton
thf(fact_4697_cInf__singleton,axiom,
! [X: set_nat] :
( ( comple7806235888213564991et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= X ) ).
% cInf_singleton
thf(fact_4698_cInf__singleton,axiom,
! [X: product_unit] :
( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% cInf_singleton
thf(fact_4699_Inf__insert,axiom,
! [A2: $o,A: set_o] :
( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ A ) )
= ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% Inf_insert
thf(fact_4700_Inf__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ A ) )
= ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% Inf_insert
thf(fact_4701_Inf__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ A ) )
= ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ A ) ) ) ).
% Inf_insert
thf(fact_4702_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4703_prod_Ocollapse,axiom,
! [Prod: produc3925858234332021118et_nat] :
( ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4704_prod_Ocollapse,axiom,
! [Prod: produc2732055786443039994et_nat] :
( ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4705_prod_Ocollapse,axiom,
! [Prod: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4706_prod_Ocollapse,axiom,
! [Prod: product_prod_int_int] :
( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4707_card__ge__UNIV,axiom,
! [S: set_Product_unit] :
( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite410649719033368117t_unit @ S ) )
= ( S = top_to1996260823553986621t_unit ) ) ).
% card_ge_UNIV
thf(fact_4708_card__ge__UNIV,axiom,
! [S: set_o] :
( ( ord_less_eq_nat @ ( finite_card_o @ top_top_set_o ) @ ( finite_card_o @ S ) )
= ( S = top_top_set_o ) ) ).
% card_ge_UNIV
thf(fact_4709_card__Diff__insert,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
=> ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) ) )
= ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_4710_card__Diff__insert,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( member_int @ A2 @ A )
=> ( ~ ( member_int @ A2 @ B4 )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) ) )
= ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_4711_card__Diff__insert,axiom,
! [A2: $o,A: set_o,B4: set_o] :
( ( member_o @ A2 @ A )
=> ( ~ ( member_o @ A2 @ B4 )
=> ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) ) )
= ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_4712_card__Diff__insert,axiom,
! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ A2 @ A )
=> ( ~ ( member_Product_unit @ A2 @ B4 )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_4713_card__Diff__insert,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ A2 @ B4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_4714_img__fst,axiom,
! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
=> ( member242389114213480694t_unit @ A2 @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) ) ) ).
% img_fst
thf(fact_4715_img__fst,axiom,
! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
=> ( member_nat @ A2 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) ) ) ).
% img_fst
thf(fact_4716_img__fst,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
=> ( member6576561426505652726_nat_o @ A2 @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) ) ) ).
% img_fst
thf(fact_4717_img__fst,axiom,
! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
=> ( member6576561426505652726_nat_o @ A2 @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) ) ) ).
% img_fst
thf(fact_4718_img__fst,axiom,
! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
=> ( member_int @ A2 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) ) ) ).
% img_fst
thf(fact_4719_fst__swap,axiom,
! [X: produc3658429121746597890et_nat] :
( ( produc6012101397789842562t_unit @ ( produc6300697153377358172et_nat @ X ) )
= ( produc8586169260539613262et_nat @ X ) ) ).
% fst_swap
thf(fact_4720_fst__swap,axiom,
! [X: product_prod_int_int] :
( ( product_fst_int_int @ ( product_swap_int_int @ X ) )
= ( product_snd_int_int @ X ) ) ).
% fst_swap
thf(fact_4721_snd__swap,axiom,
! [X: produc7348995623955488188t_unit] :
( ( produc8586169260539613262et_nat @ ( produc1264744871843189710t_unit @ X ) )
= ( produc6012101397789842562t_unit @ X ) ) ).
% snd_swap
thf(fact_4722_snd__swap,axiom,
! [X: product_prod_int_int] :
( ( product_snd_int_int @ ( product_swap_int_int @ X ) )
= ( product_fst_int_int @ X ) ) ).
% snd_swap
thf(fact_4723_snd__comp__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc4172003873565521026et_nat @ F ) )
= produc8586169260539613262et_nat ) ).
% snd_comp_apfst
thf(fact_4724_snd__comp__apfst,axiom,
! [F: int > int] :
( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc2355237562570658114nt_int @ F ) )
= product_snd_int_int ) ).
% snd_comp_apfst
thf(fact_4725_fst__comp__apsnd,axiom,
! [F: code_integer > code_integer] :
( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F ) )
= produc8508995932063986495nteger ) ).
% fst_comp_apsnd
thf(fact_4726_fst__comp__apsnd,axiom,
! [F: int > int] :
( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc4463282112584876420nt_int @ F ) )
= product_fst_int_int ) ).
% fst_comp_apsnd
thf(fact_4727_range__fst,axiom,
( ( image_5042161079198086560nt_int @ product_fst_int_int @ top_to4366644338036079209nt_int )
= top_top_set_int ) ).
% range_fst
thf(fact_4728_fst__comp__map__prod,axiom,
! [F: int > int,G: int > int] :
( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G ) )
= ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).
% fst_comp_map_prod
thf(fact_4729_fst__comp__apfst,axiom,
! [F: int > int] :
( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc2355237562570658114nt_int @ F ) )
= ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).
% fst_comp_apfst
thf(fact_4730_snd__comp__apsnd,axiom,
! [F: code_integer > code_integer] :
( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F ) )
= ( comp_C1043215000170864188nteger @ F @ produc6174133586879617921nteger ) ) ).
% snd_comp_apsnd
thf(fact_4731_snd__comp__apsnd,axiom,
! [F: set_nat > set_nat] :
( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc3286488329990770625t_unit @ F ) )
= ( comp_s5696514601250091763et_nat @ F @ produc8586169260539613262et_nat ) ) ).
% snd_comp_apsnd
thf(fact_4732_snd__comp__apsnd,axiom,
! [F: int > int] :
( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc4463282112584876420nt_int @ F ) )
= ( comp_i4330622015127193916nt_int @ F @ product_snd_int_int ) ) ).
% snd_comp_apsnd
thf(fact_4733_apfst__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,X: produc3658429121746597890et_nat] :
( ( produc4172003873565521026et_nat @ F @ ( produc3286488329990770625t_unit @ G @ X ) )
= ( produc7507926704131184380et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4734_apfst__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,X: produc3658429121746597890et_nat] :
( ( produc8233931525780029747at_nat @ F @ ( produc8442695800884449271t_unit @ G @ X ) )
= ( product_Pair_nat_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4735_apfst__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,X: produc3658429121746597890et_nat] :
( ( produc7422730317258219nt_int @ F @ ( produc7345612820240450715t_unit @ G @ X ) )
= ( product_Pair_int_int @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4736_apfst__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,G: set_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat] :
( ( produc6095824665904122878et_nat @ F @ ( produc7441988914112407621t_unit @ G @ X ) )
= ( produc5001842942810119800et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4737_apfst__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,G: set_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat] :
( ( produc4273547285574235514et_nat @ F @ ( produc7176435795764513225t_unit @ G @ X ) )
= ( produc2245416461498447860et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4738_apfst__apsnd,axiom,
! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,X: product_prod_int_int] :
( ( produc7086134074927288299et_nat @ F @ ( produc1253519502038700766at_int @ G @ X ) )
= ( produc7507926704131184380et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4739_apfst__apsnd,axiom,
! [F: int > nat,G: int > nat,X: product_prod_int_int] :
( ( produc1358374321178653834at_nat @ F @ ( produc3463928400683821864at_int @ G @ X ) )
= ( product_Pair_nat_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4740_apfst__apsnd,axiom,
! [F: int > int,G: int > int,X: product_prod_int_int] :
( ( produc2355237562570658114nt_int @ F @ ( produc4463282112584876420nt_int @ G @ X ) )
= ( product_Pair_int_int @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4741_apfst__apsnd,axiom,
! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3658429121746597890et_nat,X: product_prod_int_int] :
( ( produc5259063908904861543et_nat @ F @ ( produc9121118195053341018at_int @ G @ X ) )
= ( produc5001842942810119800et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4742_apfst__apsnd,axiom,
! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3925858234332021118et_nat,X: product_prod_int_int] :
( ( produc3602029435013415139et_nat @ F @ ( produc2878475440847162070at_int @ G @ X ) )
= ( produc2245416461498447860et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_4743_apsnd__apfst,axiom,
! [F: set_nat > set_nat,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: produc3658429121746597890et_nat] :
( ( produc3286488329990770625t_unit @ F @ ( produc4172003873565521026et_nat @ G @ X ) )
= ( produc7507926704131184380et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4744_apsnd__apfst,axiom,
! [F: set_nat > nat,G: heap_e7401611519738050253t_unit > nat,X: produc3658429121746597890et_nat] :
( ( produc1747169720485888550at_nat @ F @ ( produc2974404638222180585et_nat @ G @ X ) )
= ( product_Pair_nat_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4745_apsnd__apfst,axiom,
! [F: set_nat > int,G: heap_e7401611519738050253t_unit > int,X: produc3658429121746597890et_nat] :
( ( produc2744032961877892830nt_int @ F @ ( produc3346058198459025221et_nat @ G @ X ) )
= ( product_Pair_int_int @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4746_apsnd__apfst,axiom,
! [F: set_nat > produc3658429121746597890et_nat,G: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat] :
( ( produc1718959742329256773_nat_o @ F @ ( produc962520821680634754et_nat @ G @ X ) )
= ( produc5001842942810119800et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4747_apsnd__apfst,axiom,
! [F: set_nat > produc3925858234332021118et_nat,G: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat] :
( ( produc3009152965630989513_nat_o @ F @ ( produc962520821680634754et_nat @ G @ X ) )
= ( produc2245416461498447860et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4748_apsnd__apfst,axiom,
! [F: int > set_nat,G: int > heap_e7401611519738050253t_unit,X: product_prod_int_int] :
( ( produc2122153897363312795t_unit @ F @ ( produc1158806734383251089it_int @ G @ X ) )
= ( produc7507926704131184380et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4749_apsnd__apfst,axiom,
! [F: int > nat,G: int > nat,X: product_prod_int_int] :
( ( produc3466418871192872140at_nat @ F @ ( produc1355883850669603558at_int @ G @ X ) )
= ( product_Pair_nat_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4750_apsnd__apfst,axiom,
! [F: int > int,G: int > int,X: product_prod_int_int] :
( ( produc4463282112584876420nt_int @ F @ ( produc2355237562570658114nt_int @ G @ X ) )
= ( product_Pair_int_int @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4751_apsnd__apfst,axiom,
! [F: int > produc3658429121746597890et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
( ( produc3447581998352395295_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
= ( produc5001842942810119800et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4752_apsnd__apfst,axiom,
! [F: int > produc3925858234332021118et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
( ( produc1783401272518604195_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
= ( produc2245416461498447860et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_4753_apsnd__apfst__commute,axiom,
! [F: code_integer > code_integer,G: code_integer > code_integer,P3: produc8923325533196201883nteger] :
( ( produc6499014454317279255nteger @ F @ ( produc3195571438000033749nteger @ G @ P3 ) )
= ( produc3195571438000033749nteger @ G @ ( produc6499014454317279255nteger @ F @ P3 ) ) ) ).
% apsnd_apfst_commute
thf(fact_4754_fstI,axiom,
! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ Y @ Z ) )
=> ( ( produc1824681642469235216et_nat @ X )
= Y ) ) ).
% fstI
thf(fact_4755_fstI,axiom,
! [X: product_prod_nat_nat,Y: nat,Z: nat] :
( ( X
= ( product_Pair_nat_nat @ Y @ Z ) )
=> ( ( product_fst_nat_nat @ X )
= Y ) ) ).
% fstI
thf(fact_4756_fstI,axiom,
! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
( ( X
= ( produc5001842942810119800et_nat @ Y @ Z ) )
=> ( ( produc995936583742144908et_nat @ X )
= Y ) ) ).
% fstI
thf(fact_4757_fstI,axiom,
! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
( ( X
= ( produc2245416461498447860et_nat @ Y @ Z ) )
=> ( ( produc180342877477747464et_nat @ X )
= Y ) ) ).
% fstI
thf(fact_4758_fstI,axiom,
! [X: product_prod_int_int,Y: int,Z: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z ) )
=> ( ( product_fst_int_int @ X )
= Y ) ) ).
% fstI
thf(fact_4759_fst__eqD,axiom,
! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: heap_e7401611519738050253t_unit] :
( ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_4760_fst__eqD,axiom,
! [X: nat,Y: nat,A2: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_4761_fst__eqD,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o] :
( ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_4762_fst__eqD,axiom,
! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o] :
( ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_4763_fst__eqD,axiom,
! [X: int,Y: int,A2: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_4764_fst__conv,axiom,
! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4765_fst__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4766_fst__conv,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4767_fst__conv,axiom,
! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4768_fst__conv,axiom,
! [X1: int,X22: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4769_eq__fst__iff,axiom,
! [A2: heap_e7401611519738050253t_unit,P3: produc3658429121746597890et_nat] :
( ( A2
= ( produc1824681642469235216et_nat @ P3 ) )
= ( ? [B6: set_nat] :
( P3
= ( produc7507926704131184380et_nat @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_4770_eq__fst__iff,axiom,
! [A2: nat,P3: product_prod_nat_nat] :
( ( A2
= ( product_fst_nat_nat @ P3 ) )
= ( ? [B6: nat] :
( P3
= ( product_Pair_nat_nat @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_4771_eq__fst__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
( ( A2
= ( produc995936583742144908et_nat @ P3 ) )
= ( ? [B6: produc3658429121746597890et_nat] :
( P3
= ( produc5001842942810119800et_nat @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_4772_eq__fst__iff,axiom,
! [A2: produc3658429121746597890et_nat > $o,P3: produc2732055786443039994et_nat] :
( ( A2
= ( produc180342877477747464et_nat @ P3 ) )
= ( ? [B6: produc3925858234332021118et_nat] :
( P3
= ( produc2245416461498447860et_nat @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_4773_eq__fst__iff,axiom,
! [A2: int,P3: product_prod_int_int] :
( ( A2
= ( product_fst_int_int @ P3 ) )
= ( ? [B6: int] :
( P3
= ( product_Pair_int_int @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_4774_fstE,axiom,
! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: heap_e7401611519738050253t_unit > $o] :
( ( X
= ( produc7507926704131184380et_nat @ A2 @ B ) )
=> ( ( P @ ( produc1824681642469235216et_nat @ X ) )
=> ( P @ A2 ) ) ) ).
% fstE
thf(fact_4775_fstE,axiom,
! [X: product_prod_nat_nat,A2: nat,B: nat,P: nat > $o] :
( ( X
= ( product_Pair_nat_nat @ A2 @ B ) )
=> ( ( P @ ( product_fst_nat_nat @ X ) )
=> ( P @ A2 ) ) ) ).
% fstE
thf(fact_4776_fstE,axiom,
! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( X
= ( produc5001842942810119800et_nat @ A2 @ B ) )
=> ( ( P @ ( produc995936583742144908et_nat @ X ) )
=> ( P @ A2 ) ) ) ).
% fstE
thf(fact_4777_fstE,axiom,
! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( X
= ( produc2245416461498447860et_nat @ A2 @ B ) )
=> ( ( P @ ( produc180342877477747464et_nat @ X ) )
=> ( P @ A2 ) ) ) ).
% fstE
thf(fact_4778_fstE,axiom,
! [X: product_prod_int_int,A2: int,B: int,P: int > $o] :
( ( X
= ( product_Pair_int_int @ A2 @ B ) )
=> ( ( P @ ( product_fst_int_int @ X ) )
=> ( P @ A2 ) ) ) ).
% fstE
thf(fact_4779_prod__eq__iff,axiom,
( ( ^ [Y4: produc3658429121746597890et_nat,Z4: produc3658429121746597890et_nat] : ( Y4 = Z4 ) )
= ( ^ [S7: produc3658429121746597890et_nat,T: produc3658429121746597890et_nat] :
( ( ( produc1824681642469235216et_nat @ S7 )
= ( produc1824681642469235216et_nat @ T ) )
& ( ( produc8586169260539613262et_nat @ S7 )
= ( produc8586169260539613262et_nat @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_4780_prod__eq__iff,axiom,
( ( ^ [Y4: product_prod_int_int,Z4: product_prod_int_int] : ( Y4 = Z4 ) )
= ( ^ [S7: product_prod_int_int,T: product_prod_int_int] :
( ( ( product_fst_int_int @ S7 )
= ( product_fst_int_int @ T ) )
& ( ( product_snd_int_int @ S7 )
= ( product_snd_int_int @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_4781_prod__eqI,axiom,
! [P3: produc3658429121746597890et_nat,Q4: produc3658429121746597890et_nat] :
( ( ( produc1824681642469235216et_nat @ P3 )
= ( produc1824681642469235216et_nat @ Q4 ) )
=> ( ( ( produc8586169260539613262et_nat @ P3 )
= ( produc8586169260539613262et_nat @ Q4 ) )
=> ( P3 = Q4 ) ) ) ).
% prod_eqI
thf(fact_4782_prod__eqI,axiom,
! [P3: product_prod_int_int,Q4: product_prod_int_int] :
( ( ( product_fst_int_int @ P3 )
= ( product_fst_int_int @ Q4 ) )
=> ( ( ( product_snd_int_int @ P3 )
= ( product_snd_int_int @ Q4 ) )
=> ( P3 = Q4 ) ) ) ).
% prod_eqI
thf(fact_4783_prod_Oexpand,axiom,
! [Prod: produc3658429121746597890et_nat,Prod2: produc3658429121746597890et_nat] :
( ( ( ( produc1824681642469235216et_nat @ Prod )
= ( produc1824681642469235216et_nat @ Prod2 ) )
& ( ( produc8586169260539613262et_nat @ Prod )
= ( produc8586169260539613262et_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_4784_prod_Oexpand,axiom,
! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
( ( ( ( product_fst_int_int @ Prod )
= ( product_fst_int_int @ Prod2 ) )
& ( ( product_snd_int_int @ Prod )
= ( product_snd_int_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_4785_All__prod__contract,axiom,
! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
( ( ! [A6: heap_e7401611519738050253t_unit,X7: set_nat] : ( P @ A6 @ X7 ) )
= ( ! [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).
% All_prod_contract
thf(fact_4786_All__prod__contract,axiom,
! [P: int > int > $o] :
( ( ! [A6: int,X7: int] : ( P @ A6 @ X7 ) )
= ( ! [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).
% All_prod_contract
thf(fact_4787_Ex__prod__contract,axiom,
! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
( ( ? [A6: heap_e7401611519738050253t_unit,X7: set_nat] : ( P @ A6 @ X7 ) )
= ( ? [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).
% Ex_prod_contract
thf(fact_4788_Ex__prod__contract,axiom,
! [P: int > int > $o] :
( ( ? [A6: int,X7: int] : ( P @ A6 @ X7 ) )
= ( ? [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).
% Ex_prod_contract
thf(fact_4789_apsnd__compose,axiom,
! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
( ( produc9020447175693601247nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
= ( produc9020447175693601247nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_4790_apsnd__compose,axiom,
! [F: code_integer > code_integer,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
( ( produc6499014454317279255nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
= ( produc6499014454317279255nteger @ ( comp_C7449957260575251196nteger @ F @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_4791_refl__on__INTER,axiom,
! [S: set_nat,A: nat > set_nat,R2: nat > set_Pr1261947904930325089at_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( refl_on_nat @ ( A @ X3 ) @ ( R2 @ X3 ) ) )
=> ( refl_on_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ S ) ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ R2 @ S ) ) ) ) ).
% refl_on_INTER
thf(fact_4792_cInf__eq__non__empty,axiom,
! [X4: set_o,A2: $o] :
( ( X4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X4 )
=> ( ord_less_eq_o @ A2 @ X3 ) )
=> ( ! [Y2: $o] :
( ! [X6: $o] :
( ( member_o @ X6 @ X4 )
=> ( ord_less_eq_o @ Y2 @ X6 ) )
=> ( ord_less_eq_o @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_o @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4793_cInf__eq__non__empty,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X4 )
=> ( ord_le729326519192465773_nat_o @ A2 @ X3 ) )
=> ( ! [Y2: produc3658429121746597890et_nat > $o] :
( ! [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ X4 )
=> ( ord_le729326519192465773_nat_o @ Y2 @ X6 ) )
=> ( ord_le729326519192465773_nat_o @ Y2 @ A2 ) )
=> ( ( comple90455603224345172_nat_o @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4794_cInf__eq__non__empty,axiom,
! [X4: set_filter_nat,A2: filter_nat] :
( ( X4 != bot_bo498966703094740906er_nat )
=> ( ! [X3: filter_nat] :
( ( member_filter_nat @ X3 @ X4 )
=> ( ord_le2510731241096832064er_nat @ A2 @ X3 ) )
=> ( ! [Y2: filter_nat] :
( ! [X6: filter_nat] :
( ( member_filter_nat @ X6 @ X4 )
=> ( ord_le2510731241096832064er_nat @ Y2 @ X6 ) )
=> ( ord_le2510731241096832064er_nat @ Y2 @ A2 ) )
=> ( ( comple6612786467464916249er_nat @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4795_cInf__eq__non__empty,axiom,
! [X4: set_int,A2: int] :
( ( X4 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ( ord_less_eq_int @ A2 @ X3 ) )
=> ( ! [Y2: int] :
( ! [X6: int] :
( ( member_int @ X6 @ X4 )
=> ( ord_less_eq_int @ Y2 @ X6 ) )
=> ( ord_less_eq_int @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_int @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4796_cInf__eq__non__empty,axiom,
! [X4: set_nat,A2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ord_less_eq_nat @ A2 @ X3 ) )
=> ( ! [Y2: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ X4 )
=> ( ord_less_eq_nat @ Y2 @ X6 ) )
=> ( ord_less_eq_nat @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4797_cInf__eq__non__empty,axiom,
! [X4: set_set_nat,A2: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X4 )
=> ( ord_less_eq_set_nat @ A2 @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ X4 )
=> ( ord_less_eq_set_nat @ Y2 @ X6 ) )
=> ( ord_less_eq_set_nat @ Y2 @ A2 ) )
=> ( ( comple7806235888213564991et_nat @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4798_cInf__eq__non__empty,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ A2 @ X3 ) )
=> ( ! [Y2: product_unit] :
( ! [X6: product_unit] :
( ( member_Product_unit @ X6 @ X4 )
=> ( ord_le3221252021190050221t_unit @ Y2 @ X6 ) )
=> ( ord_le3221252021190050221t_unit @ Y2 @ A2 ) )
=> ( ( comple2584293577114468500t_unit @ X4 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4799_cInf__greatest,axiom,
! [X4: set_o,Z: $o] :
( ( X4 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X4 )
=> ( ord_less_eq_o @ Z @ X3 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4800_cInf__greatest,axiom,
! [X4: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ X4 )
=> ( ord_le729326519192465773_nat_o @ Z @ X3 ) )
=> ( ord_le729326519192465773_nat_o @ Z @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4801_cInf__greatest,axiom,
! [X4: set_filter_nat,Z: filter_nat] :
( ( X4 != bot_bo498966703094740906er_nat )
=> ( ! [X3: filter_nat] :
( ( member_filter_nat @ X3 @ X4 )
=> ( ord_le2510731241096832064er_nat @ Z @ X3 ) )
=> ( ord_le2510731241096832064er_nat @ Z @ ( comple6612786467464916249er_nat @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4802_cInf__greatest,axiom,
! [X4: set_int,Z: int] :
( ( X4 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X4 )
=> ( ord_less_eq_int @ Z @ X3 ) )
=> ( ord_less_eq_int @ Z @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4803_cInf__greatest,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ord_less_eq_nat @ Z @ X3 ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4804_cInf__greatest,axiom,
! [X4: set_set_nat,Z: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X4 )
=> ( ord_less_eq_set_nat @ Z @ X3 ) )
=> ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4805_cInf__greatest,axiom,
! [X4: set_Product_unit,Z: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ Z @ X3 ) )
=> ( ord_le3221252021190050221t_unit @ Z @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ).
% cInf_greatest
thf(fact_4806_Inf__less__eq,axiom,
! [A: set_o,U: $o] :
( ! [V3: $o] :
( ( member_o @ V3 @ A )
=> ( ord_less_eq_o @ V3 @ U ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_4807_Inf__less__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o,U: produc3658429121746597890et_nat > $o] :
( ! [V3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ V3 @ A )
=> ( ord_le729326519192465773_nat_o @ V3 @ U ) )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_4808_Inf__less__eq,axiom,
! [A: set_filter_nat,U: filter_nat] :
( ! [V3: filter_nat] :
( ( member_filter_nat @ V3 @ A )
=> ( ord_le2510731241096832064er_nat @ V3 @ U ) )
=> ( ( A != bot_bo498966703094740906er_nat )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_4809_Inf__less__eq,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V3: set_nat] :
( ( member_set_nat @ V3 @ A )
=> ( ord_less_eq_set_nat @ V3 @ U ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_4810_Inf__less__eq,axiom,
! [A: set_Product_unit,U: product_unit] :
( ! [V3: product_unit] :
( ( member_Product_unit @ V3 @ A )
=> ( ord_le3221252021190050221t_unit @ V3 @ U ) )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_4811_cInf__lessD,axiom,
! [X4: set_int,Z: int] :
( ( X4 != bot_bot_set_int )
=> ( ( ord_less_int @ ( complete_Inf_Inf_int @ X4 ) @ Z )
=> ? [X3: int] :
( ( member_int @ X3 @ X4 )
& ( ord_less_int @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_4812_cInf__lessD,axiom,
! [X4: set_nat,Z: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X4 ) @ Z )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( ord_less_nat @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_4813_cInf__lessD,axiom,
! [X4: set_Product_unit,Z: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ ( comple2584293577114468500t_unit @ X4 ) @ Z )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ X4 )
& ( ord_le361264281704409273t_unit @ X3 @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_4814_INF__eq__const,axiom,
! [I: set_Product_unit,F: product_unit > set_nat,X: set_nat] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4815_INF__eq__const,axiom,
! [I: set_o,F: $o > set_nat,X: set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4816_INF__eq__const,axiom,
! [I: set_nat,F: nat > set_nat,X: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4817_INF__eq__const,axiom,
! [I: set_int,F: int > set_nat,X: set_nat] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4818_INF__eq__const,axiom,
! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > set_nat,X: set_nat] :
( ( I != bot_bo7824918357723582233_nat_o )
=> ( ! [I3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7806235888213564991et_nat @ ( image_6888641473154130958et_nat @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4819_INF__eq__const,axiom,
! [I: set_Product_unit,F: product_unit > product_unit,X: product_unit] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4820_INF__eq__const,axiom,
! [I: set_o,F: $o > product_unit,X: product_unit] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4821_INF__eq__const,axiom,
! [I: set_nat,F: nat > product_unit,X: product_unit] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4822_INF__eq__const,axiom,
! [I: set_int,F: int > product_unit,X: product_unit] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4823_INF__eq__const,axiom,
! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > product_unit,X: product_unit] :
( ( I != bot_bo7824918357723582233_nat_o )
=> ( ! [I3: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8451000719036923781t_unit @ F @ I ) )
= X ) ) ) ).
% INF_eq_const
thf(fact_4824_card__1__singletonE,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= one_one_nat )
=> ~ ! [X3: product_unit] :
( A
!= ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% card_1_singletonE
thf(fact_4825_card__1__singletonE,axiom,
! [A: set_o] :
( ( ( finite_card_o @ A )
= one_one_nat )
=> ~ ! [X3: $o] :
( A
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% card_1_singletonE
thf(fact_4826_card__1__singletonE,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= one_one_nat )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_4827_card__1__singletonE,axiom,
! [A: set_int] :
( ( ( finite_card_int @ A )
= one_one_nat )
=> ~ ! [X3: int] :
( A
!= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% card_1_singletonE
thf(fact_4828_card__1__singletonE,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ( finite1363419556375932405_nat_o @ A )
= one_one_nat )
=> ~ ! [X3: produc3658429121746597890et_nat > $o] :
( A
!= ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% card_1_singletonE
thf(fact_4829_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,A2: product_prod_nat_nat] :
( ( P @ X @ Y )
=> ( ( A2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ( P @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4830_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3925858234332021118et_nat] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc5001842942810119800et_nat @ X @ Y ) )
=> ( P @ ( produc995936583742144908et_nat @ A2 ) @ ( produc4011572625026189258et_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4831_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc2732055786443039994et_nat] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc2245416461498447860et_nat @ X @ Y ) )
=> ( P @ ( produc180342877477747464et_nat @ A2 ) @ ( produc5374455773327741254et_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4832_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: heap_e7401611519738050253t_unit > set_nat > $o,X: heap_e7401611519738050253t_unit,Y: set_nat,A2: produc3658429121746597890et_nat] :
( ( P @ X @ Y )
=> ( ( A2
= ( produc7507926704131184380et_nat @ X @ Y ) )
=> ( P @ ( produc1824681642469235216et_nat @ A2 ) @ ( produc8586169260539613262et_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4833_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > int > $o,X: int,Y: int,A2: product_prod_int_int] :
( ( P @ X @ Y )
=> ( ( A2
= ( product_Pair_int_int @ X @ Y ) )
=> ( P @ ( product_fst_int_int @ A2 ) @ ( product_snd_int_int @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4834_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4835_prod_Oexhaust__sel,axiom,
! [Prod: produc3925858234332021118et_nat] :
( Prod
= ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4836_prod_Oexhaust__sel,axiom,
! [Prod: produc2732055786443039994et_nat] :
( Prod
= ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4837_prod_Oexhaust__sel,axiom,
! [Prod: produc3658429121746597890et_nat] :
( Prod
= ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4838_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_int_int] :
( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4839_surjective__pairing,axiom,
! [T4: product_prod_nat_nat] :
( T4
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T4 ) @ ( product_snd_nat_nat @ T4 ) ) ) ).
% surjective_pairing
thf(fact_4840_surjective__pairing,axiom,
! [T4: produc3925858234332021118et_nat] :
( T4
= ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ T4 ) @ ( produc4011572625026189258et_nat @ T4 ) ) ) ).
% surjective_pairing
thf(fact_4841_surjective__pairing,axiom,
! [T4: produc2732055786443039994et_nat] :
( T4
= ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ T4 ) @ ( produc5374455773327741254et_nat @ T4 ) ) ) ).
% surjective_pairing
thf(fact_4842_surjective__pairing,axiom,
! [T4: produc3658429121746597890et_nat] :
( T4
= ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ T4 ) @ ( produc8586169260539613262et_nat @ T4 ) ) ) ).
% surjective_pairing
thf(fact_4843_surjective__pairing,axiom,
! [T4: product_prod_int_int] :
( T4
= ( product_Pair_int_int @ ( product_fst_int_int @ T4 ) @ ( product_snd_int_int @ T4 ) ) ) ).
% surjective_pairing
thf(fact_4844_card__insert__le,axiom,
! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).
% card_insert_le
thf(fact_4845_card__insert__le,axiom,
! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ ( insert_int @ X @ A ) ) ) ).
% card_insert_le
thf(fact_4846_card__insert__le,axiom,
! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ ( insert_o @ X @ A ) ) ) ).
% card_insert_le
thf(fact_4847_card__insert__le,axiom,
! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) ) ).
% card_insert_le
thf(fact_4848_in__fst__imageE,axiom,
! [X: heap_e7401611519738050253t_unit,S: set_Pr3948176798113811640et_nat] :
( ( member242389114213480694t_unit @ X @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) )
=> ~ ! [Y2: set_nat] :
~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4849_in__fst__imageE,axiom,
! [X: nat,S: set_Pr1261947904930325089at_nat] :
( ( member_nat @ X @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) )
=> ~ ! [Y2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4850_in__fst__imageE,axiom,
! [X: produc3658429121746597890et_nat > $o,S: set_Pr3286484037609594932et_nat] :
( ( member6576561426505652726_nat_o @ X @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) )
=> ~ ! [Y2: produc3658429121746597890et_nat] :
~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4851_in__fst__imageE,axiom,
! [X: produc3658429121746597890et_nat > $o,S: set_Pr8536935166611901872et_nat] :
( ( member6576561426505652726_nat_o @ X @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) )
=> ~ ! [Y2: produc3925858234332021118et_nat] :
~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4852_in__fst__imageE,axiom,
! [X: int,S: set_Pr958786334691620121nt_int] :
( ( member_int @ X @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) )
=> ~ ! [Y2: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4853_Inter__subset,axiom,
! [A: set_set_nat,B4: set_nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A )
=> ( ord_less_eq_set_nat @ X8 @ B4 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B4 ) ) ) ).
% Inter_subset
thf(fact_4854_Inter__empty,axiom,
( ( comple7216939928495492468t_unit @ bot_bo6284674557190688313t_unit )
= top_to1996260823553986621t_unit ) ).
% Inter_empty
thf(fact_4855_Inter__empty,axiom,
( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
= top_top_set_o ) ).
% Inter_empty
thf(fact_4856_Inter__empty,axiom,
( ( comple3628384868704368283et_int @ bot_bot_set_set_int )
= top_top_set_int ) ).
% Inter_empty
thf(fact_4857_Inter__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inter_empty
thf(fact_4858_fst__eq__Domain,axiom,
! [R3: set_Pr958786334691620121nt_int] :
( ( image_5042161079198086560nt_int @ product_fst_int_int @ R3 )
= ( domain_int_int @ R3 ) ) ).
% fst_eq_Domain
thf(fact_4859_Domain__fst,axiom,
( domain_int_int
= ( image_5042161079198086560nt_int @ product_fst_int_int ) ) ).
% Domain_fst
thf(fact_4860_card__partition,axiom,
! [C5: set_set_Code_integer,K3: nat] :
( ( finite6931041176100689706nteger @ C5 )
=> ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ C5 ) )
=> ( ! [C2: set_Code_integer] :
( ( member1834298987299537450nteger @ C2 @ C5 )
=> ( ( finite4902975817058060853nteger @ C2 )
= K3 ) )
=> ( ! [C1: set_Code_integer,C22: set_Code_integer] :
( ( member1834298987299537450nteger @ C1 @ C5 )
=> ( ( member1834298987299537450nteger @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_in1364745209274528805nteger @ C1 @ C22 )
= bot_bo3990330152332043303nteger ) ) ) )
=> ( ( times_times_nat @ K3 @ ( finite1340570857867686763nteger @ C5 ) )
= ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4861_card__partition,axiom,
! [C5: set_set_Product_unit,K3: nat] :
( ( finite1772178364199683094t_unit @ C5 )
=> ( ( finite4290736615968046902t_unit @ ( comple3349607820504768346t_unit @ C5 ) )
=> ( ! [C2: set_Product_unit] :
( ( member5877623283571906838t_unit @ C2 @ C5 )
=> ( ( finite410649719033368117t_unit @ C2 )
= K3 ) )
=> ( ! [C1: set_Product_unit,C22: set_Product_unit] :
( ( member5877623283571906838t_unit @ C1 @ C5 )
=> ( ( member5877623283571906838t_unit @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_in4660618365625256667t_unit @ C1 @ C22 )
= bot_bo3957492148770167129t_unit ) ) ) )
=> ( ( times_times_nat @ K3 @ ( finite22775419188893461t_unit @ C5 ) )
= ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4862_card__partition,axiom,
! [C5: set_set_o,K3: nat] :
( ( finite_finite_set_o @ C5 )
=> ( ( finite_finite_o @ ( comple90263536869209701_set_o @ C5 ) )
=> ( ! [C2: set_o] :
( ( member_set_o @ C2 @ C5 )
=> ( ( finite_card_o @ C2 )
= K3 ) )
=> ( ! [C1: set_o,C22: set_o] :
( ( member_set_o @ C1 @ C5 )
=> ( ( member_set_o @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_o @ C1 @ C22 )
= bot_bot_set_o ) ) ) )
=> ( ( times_times_nat @ K3 @ ( finite_card_set_o @ C5 ) )
= ( finite_card_o @ ( comple90263536869209701_set_o @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4863_card__partition,axiom,
! [C5: set_set_nat,K3: nat] :
( ( finite1152437895449049373et_nat @ C5 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C5 ) )
=> ( ! [C2: set_nat] :
( ( member_set_nat @ C2 @ C5 )
=> ( ( finite_card_nat @ C2 )
= K3 ) )
=> ( ! [C1: set_nat,C22: set_nat] :
( ( member_set_nat @ C1 @ C5 )
=> ( ( member_set_nat @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_nat @ C1 @ C22 )
= bot_bot_set_nat ) ) ) )
=> ( ( times_times_nat @ K3 @ ( finite_card_set_nat @ C5 ) )
= ( finite_card_nat @ ( comple7399068483239264473et_nat @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4864_card__partition,axiom,
! [C5: set_set_int,K3: nat] :
( ( finite6197958912794628473et_int @ C5 )
=> ( ( finite_finite_int @ ( comple3221217463730067765et_int @ C5 ) )
=> ( ! [C2: set_int] :
( ( member_set_int @ C2 @ C5 )
=> ( ( finite_card_int @ C2 )
= K3 ) )
=> ( ! [C1: set_int,C22: set_int] :
( ( member_set_int @ C1 @ C5 )
=> ( ( member_set_int @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_int @ C1 @ C22 )
= bot_bot_set_int ) ) ) )
=> ( ( times_times_nat @ K3 @ ( finite_card_set_int @ C5 ) )
= ( finite_card_int @ ( comple3221217463730067765et_int @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4865_card__partition,axiom,
! [C5: set_se1353816476878259085_nat_o,K3: nat] :
( ( finite6069017635265841110_nat_o @ C5 )
=> ( ( finite3252695134891459830_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) )
=> ( ! [C2: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ C2 @ C5 )
=> ( ( finite1363419556375932405_nat_o @ C2 )
= K3 ) )
=> ( ! [C1: set_Pr4532377907799695533_nat_o,C22: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ C1 @ C5 )
=> ( ( member1452989533377600726_nat_o @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_in1906310914598751387_nat_o @ C1 @ C22 )
= bot_bo7824918357723582233_nat_o ) ) ) )
=> ( ( times_times_nat @ K3 @ ( finite4731484169140532437_nat_o @ C5 ) )
= ( finite1363419556375932405_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4866_Inter__UNIV,axiom,
( ( comple7216939928495492468t_unit @ top_to1767297665138865437t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Inter_UNIV
thf(fact_4867_Inter__UNIV,axiom,
( ( comple3063163877087187839_set_o @ top_top_set_set_o )
= bot_bot_set_o ) ).
% Inter_UNIV
thf(fact_4868_Inter__UNIV,axiom,
( ( comple3628384868704368283et_int @ top_top_set_set_int )
= bot_bot_set_int ) ).
% Inter_UNIV
thf(fact_4869_Inter__UNIV,axiom,
( ( comple79274577958421812_nat_o @ top_to8515903933492314333_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Inter_UNIV
thf(fact_4870_Inter__UNIV,axiom,
( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
= bot_bot_set_nat ) ).
% Inter_UNIV
thf(fact_4871_inv__on__f__range,axiom,
! [Y: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
( ( member_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( member_Code_integer @ ( inv_on4869000451854154612nteger @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4872_inv__on__f__range,axiom,
! [Y: code_integer,F: nat > code_integer,A: set_nat] :
( ( member_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( member_nat @ ( inv_on1369624593536199399nteger @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4873_inv__on__f__range,axiom,
! [Y: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Y @ ( image_nat_nat2 @ F @ A ) )
=> ( member_nat @ ( inv_on_nat_nat @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4874_inv__on__f__range,axiom,
! [Y: nat,F: int > nat,A: set_int] :
( ( member_nat @ Y @ ( image_int_nat2 @ F @ A ) )
=> ( member_int @ ( inv_on_int_nat @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4875_inv__on__f__range,axiom,
! [Y: nat,F: $o > nat,A: set_o] :
( ( member_nat @ Y @ ( image_o_nat2 @ F @ A ) )
=> ( member_o @ ( inv_on_o_nat @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4876_inv__on__f__range,axiom,
! [Y: nat,F: product_unit > nat,A: set_Product_unit] :
( ( member_nat @ Y @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( member_Product_unit @ ( inv_on7727615724625388965it_nat @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4877_inv__on__f__range,axiom,
! [Y: int,F: nat > int,A: set_nat] :
( ( member_int @ Y @ ( image_nat_int2 @ F @ A ) )
=> ( member_nat @ ( inv_on_nat_int @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4878_inv__on__f__range,axiom,
! [Y: int,F: int > int,A: set_int] :
( ( member_int @ Y @ ( image_int_int2 @ F @ A ) )
=> ( member_int @ ( inv_on_int_int @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4879_inv__on__f__range,axiom,
! [Y: int,F: $o > int,A: set_o] :
( ( member_int @ Y @ ( image_o_int2 @ F @ A ) )
=> ( member_o @ ( inv_on_o_int @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4880_inv__on__f__range,axiom,
! [Y: int,F: product_unit > int,A: set_Product_unit] :
( ( member_int @ Y @ ( image_873079544045703924it_int @ F @ A ) )
=> ( member_Product_unit @ ( inv_on7725125254116338689it_int @ F @ A @ Y ) @ A ) ) ).
% inv_on_f_range
thf(fact_4881_f__inv__on__f,axiom,
! [Y: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Y @ ( image_nat_set_nat @ F @ A ) )
=> ( ( F @ ( inv_on_nat_set_nat @ F @ A @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_4882_f__inv__on__f,axiom,
! [Y: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
( ( member_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ A ) )
=> ( ( F @ ( inv_on4869000451854154612nteger @ F @ A @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_4883_f__inv__on__f,axiom,
! [Y: code_integer,F: nat > code_integer,A: set_nat] :
( ( member_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ A ) )
=> ( ( F @ ( inv_on1369624593536199399nteger @ F @ A @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_4884_f__inv__on__f,axiom,
! [Y: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Y @ ( image_nat_nat2 @ F @ A ) )
=> ( ( F @ ( inv_on_nat_nat @ F @ A @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_4885_cINF__greatest,axiom,
! [A: set_Product_unit,M4: int,F: product_unit > int] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4886_cINF__greatest,axiom,
! [A: set_o,M4: int,F: $o > int] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4887_cINF__greatest,axiom,
! [A: set_nat,M4: int,F: nat > int] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4888_cINF__greatest,axiom,
! [A: set_int,M4: int,F: int > int] :
( ( A != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4889_cINF__greatest,axiom,
! [A: set_Product_unit,M4: nat,F: product_unit > nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4890_cINF__greatest,axiom,
! [A: set_o,M4: nat,F: $o > nat] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4891_cINF__greatest,axiom,
! [A: set_nat,M4: nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4892_cINF__greatest,axiom,
! [A: set_int,M4: nat,F: int > nat] :
( ( A != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4893_cINF__greatest,axiom,
! [A: set_Product_unit,M4: product_unit,F: product_unit > product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ M4 @ ( F @ X3 ) ) )
=> ( ord_le3221252021190050221t_unit @ M4 @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4894_cINF__greatest,axiom,
! [A: set_o,M4: product_unit,F: $o > product_unit] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_le3221252021190050221t_unit @ M4 @ ( F @ X3 ) ) )
=> ( ord_le3221252021190050221t_unit @ M4 @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ).
% cINF_greatest
thf(fact_4895_INF__eq__iff,axiom,
! [I: set_Product_unit,F: product_unit > product_unit,C: product_unit] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
= C )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4896_INF__eq__iff,axiom,
! [I: set_o,F: $o > product_unit,C: product_unit] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4897_INF__eq__iff,axiom,
! [I: set_nat,F: nat > product_unit,C: product_unit] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4898_INF__eq__iff,axiom,
! [I: set_int,F: int > product_unit,C: product_unit] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
= C )
= ( ! [X2: int] :
( ( member_int @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4899_INF__eq__iff,axiom,
! [I: set_Product_unit,F: product_unit > filter_nat,C: filter_nat] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6612786467464916249er_nat @ ( image_1853478764106424488er_nat @ F @ I ) )
= C )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4900_INF__eq__iff,axiom,
! [I: set_o,F: $o > filter_nat,C: filter_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6612786467464916249er_nat @ ( image_o_filter_nat @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4901_INF__eq__iff,axiom,
! [I: set_nat,F: nat > filter_nat,C: filter_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4902_INF__eq__iff,axiom,
! [I: set_int,F: int > filter_nat,C: filter_nat] :
( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple6612786467464916249er_nat @ ( image_int_filter_nat @ F @ I ) )
= C )
= ( ! [X2: int] :
( ( member_int @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4903_INF__eq__iff,axiom,
! [I: set_Product_unit,F: product_unit > set_nat,C: set_nat] :
( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
= C )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4904_INF__eq__iff,axiom,
! [I: set_o,F: $o > set_nat,C: set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4905_finite__less__Inf__iff,axiom,
! [X4: set_int,A2: int] :
( ( finite_finite_int @ X4 )
=> ( ( X4 != bot_bot_set_int )
=> ( ( ord_less_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ X4 )
=> ( ord_less_int @ A2 @ X2 ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4906_finite__less__Inf__iff,axiom,
! [X4: set_nat,A2: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_nat @ A2 @ X2 ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4907_finite__less__Inf__iff,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ X4 )
=> ( ord_le361264281704409273t_unit @ A2 @ X2 ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4908_Inf__le__Sup,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Inf_le_Sup
thf(fact_4909_Inf__le__Sup,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ).
% Inf_le_Sup
thf(fact_4910_Inf__le__Sup,axiom,
! [A: set_filter_nat] :
( ( A != bot_bo498966703094740906er_nat )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple2454491259644266675er_nat @ A ) ) ) ).
% Inf_le_Sup
thf(fact_4911_Inf__le__Sup,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Inf_le_Sup
thf(fact_4912_Inf__le__Sup,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple4687483117567863418t_unit @ A ) ) ) ).
% Inf_le_Sup
thf(fact_4913_finite__Inf__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y2 @ A )
=> ( member_o @ ( inf_inf_o @ X3 @ Y2 ) @ A ) ) )
=> ( member_o @ ( complete_Inf_Inf_o @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_4914_finite__Inf__in,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X3 @ A )
=> ( ( member6576561426505652726_nat_o @ Y2 @ A )
=> ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) @ A ) ) )
=> ( member6576561426505652726_nat_o @ ( comple90455603224345172_nat_o @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_4915_finite__Inf__in,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( member_set_nat @ Y2 @ A )
=> ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ A ) ) )
=> ( member_set_nat @ ( comple7806235888213564991et_nat @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_4916_finite__Inf__in,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit,Y2: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( member_Product_unit @ Y2 @ A )
=> ( member_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ A ) ) )
=> ( member_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ A ) ) ) ) ).
% finite_Inf_in
thf(fact_4917_card__1__singletonI,axiom,
! [S: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ S )
=> ( ( ( finite4902975817058060853nteger @ S )
= one_one_nat )
=> ( ( member_Code_integer @ X @ S )
=> ( S
= ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ).
% card_1_singletonI
thf(fact_4918_card__1__singletonI,axiom,
! [S: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( ( finite410649719033368117t_unit @ S )
= one_one_nat )
=> ( ( member_Product_unit @ X @ S )
=> ( S
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% card_1_singletonI
thf(fact_4919_card__1__singletonI,axiom,
! [S: set_o,X: $o] :
( ( finite_finite_o @ S )
=> ( ( ( finite_card_o @ S )
= one_one_nat )
=> ( ( member_o @ X @ S )
=> ( S
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% card_1_singletonI
thf(fact_4920_card__1__singletonI,axiom,
! [S: set_nat,X: nat] :
( ( finite_finite_nat @ S )
=> ( ( ( finite_card_nat @ S )
= one_one_nat )
=> ( ( member_nat @ X @ S )
=> ( S
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% card_1_singletonI
thf(fact_4921_card__1__singletonI,axiom,
! [S: set_int,X: int] :
( ( finite_finite_int @ S )
=> ( ( ( finite_card_int @ S )
= one_one_nat )
=> ( ( member_int @ X @ S )
=> ( S
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% card_1_singletonI
thf(fact_4922_card__1__singletonI,axiom,
! [S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ S )
=> ( ( ( finite1363419556375932405_nat_o @ S )
= one_one_nat )
=> ( ( member6576561426505652726_nat_o @ X @ S )
=> ( S
= ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% card_1_singletonI
thf(fact_4923_card__Diff__singleton__if,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4924_card__Diff__singleton__if,axiom,
! [X: $o,A: set_o] :
( ( ( member_o @ X @ A )
=> ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
= ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
= ( finite_card_o @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4925_card__Diff__singleton__if,axiom,
! [X: int,A: set_int] :
( ( ( member_int @ X @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4926_card__Diff__singleton__if,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
= ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
= ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4927_card__Diff__singleton__if,axiom,
! [X: nat,A: set_nat] :
( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4928_card__Diff__singleton,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_4929_card__Diff__singleton,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
= ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_4930_card__Diff__singleton,axiom,
! [X: int,A: set_int] :
( ( member_int @ X @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_4931_card__Diff__singleton,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
= ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_4932_card__Diff__singleton,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_4933_fst__image__mp,axiom,
! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
( ( ord_le4903435234951335757t_unit @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ A ) @ B4 )
=> ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
=> ( member242389114213480694t_unit @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_4934_fst__image__mp,axiom,
! [A: set_Pr3286484037609594932et_nat,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
( ( ord_le2965882846123202637_nat_o @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ A ) @ B4 )
=> ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
=> ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_4935_fst__image__mp,axiom,
! [A: set_Pr8536935166611901872et_nat,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
( ( ord_le2965882846123202637_nat_o @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ A ) @ B4 )
=> ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
=> ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_4936_fst__image__mp,axiom,
! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ A ) @ B4 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
=> ( member_int @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_4937_fst__image__mp,axiom,
! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A ) @ B4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
=> ( member_nat @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_4938_fst__in__Field,axiom,
! [R3: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ R3 ) @ ( field_int @ R3 ) ) ).
% fst_in_Field
thf(fact_4939_fst__in__Field,axiom,
! [R3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ R3 ) @ ( field_nat @ R3 ) ) ).
% fst_in_Field
thf(fact_4940_Inf__fin__Inf,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ A )
= ( complete_Inf_Inf_o @ A ) ) ) ) ).
% Inf_fin_Inf
thf(fact_4941_Inf__fin__Inf,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( lattic956194824204696298_nat_o @ A )
= ( comple90455603224345172_nat_o @ A ) ) ) ) ).
% Inf_fin_Inf
thf(fact_4942_Inf__fin__Inf,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A )
= ( comple7806235888213564991et_nat @ A ) ) ) ) ).
% Inf_fin_Inf
thf(fact_4943_Inf__fin__Inf,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A )
= ( comple2584293577114468500t_unit @ A ) ) ) ) ).
% Inf_fin_Inf
thf(fact_4944_cInf__eq__Inf__fin,axiom,
! [X4: set_o] :
( ( finite_finite_o @ X4 )
=> ( ( X4 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_o @ X4 )
= ( lattic4107685809792843317_fin_o @ X4 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4945_cInf__eq__Inf__fin,axiom,
! [X4: set_int] :
( ( finite_finite_int @ X4 )
=> ( ( X4 != bot_bot_set_int )
=> ( ( complete_Inf_Inf_int @ X4 )
= ( lattic5235898064620869839in_int @ X4 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4946_cInf__eq__Inf__fin,axiom,
! [X4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ X4 )
=> ( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ( comple90455603224345172_nat_o @ X4 )
= ( lattic956194824204696298_nat_o @ X4 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4947_cInf__eq__Inf__fin,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ X4 )
= ( lattic5238388535129920115in_nat @ X4 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4948_cInf__eq__Inf__fin,axiom,
! [X4: set_set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( ( X4 != bot_bot_set_set_nat )
=> ( ( comple7806235888213564991et_nat @ X4 )
= ( lattic3014633134055518761et_nat @ X4 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4949_cInf__eq__Inf__fin,axiom,
! [X4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit @ X4 )
= ( lattic1263872656861969706t_unit @ X4 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4950_prod__set__defs_I1_J,axiom,
( basic_fsts_int_int
= ( ^ [P5: product_prod_int_int] : ( insert_int @ ( product_fst_int_int @ P5 ) @ bot_bot_set_int ) ) ) ).
% prod_set_defs(1)
thf(fact_4951_prod_Oswap__def,axiom,
( produc1264744871843189710t_unit
= ( ^ [P5: produc7348995623955488188t_unit] : ( produc7507926704131184380et_nat @ ( produc3550216979005444800t_unit @ P5 ) @ ( produc6012101397789842562t_unit @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4952_prod_Oswap__def,axiom,
( product_swap_nat_nat
= ( ^ [P5: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P5 ) @ ( product_fst_nat_nat @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4953_prod_Oswap__def,axiom,
( produc1078497009909708626_nat_o
= ( ^ [P5: produc6907119039986491712_nat_o] : ( produc5001842942810119800et_nat @ ( produc1838968430882301508_nat_o @ P5 ) @ ( produc8046704426453032966_nat_o @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4954_prod_Oswap__def,axiom,
( produc2580581461503396054_nat_o
= ( ^ [P5: produc4962441903510821572_nat_o] : ( produc2245416461498447860et_nat @ ( produc6803565743942592456_nat_o @ P5 ) @ ( produc1609452848092598666_nat_o @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4955_prod_Oswap__def,axiom,
( produc6300697153377358172et_nat
= ( ^ [P5: produc3658429121746597890et_nat] : ( produc2471974422597015918t_unit @ ( produc8586169260539613262et_nat @ P5 ) @ ( produc1824681642469235216et_nat @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4956_prod_Oswap__def,axiom,
( product_swap_int_int
= ( ^ [P5: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P5 ) @ ( product_fst_int_int @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4957_equiv__proj,axiom,
! [A: set_int,R3: set_Pr958786334691620121nt_int,Z: product_prod_int_int] :
( ( equiv_equiv_int @ A @ R3 )
=> ( ( member5262025264175285858nt_int @ Z @ R3 )
=> ( ( comp_i5103216804979976198nt_int @ ( equiv_proj_int_int @ R3 ) @ product_fst_int_int @ Z )
= ( comp_i5103216804979976198nt_int @ ( equiv_proj_int_int @ R3 ) @ product_snd_int_int @ Z ) ) ) ) ).
% equiv_proj
thf(fact_4958_INF__le__SUP,axiom,
! [A: set_Product_unit,F: product_unit > product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4959_INF__le__SUP,axiom,
! [A: set_o,F: $o > product_unit] :
( ( A != bot_bot_set_o )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4960_INF__le__SUP,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( A != bot_bot_set_nat )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4961_INF__le__SUP,axiom,
! [A: set_int,F: int > product_unit] :
( ( A != bot_bot_set_int )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4962_INF__le__SUP,axiom,
! [A: set_Product_unit,F: product_unit > filter_nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_1853478764106424488er_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_1853478764106424488er_nat @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4963_INF__le__SUP,axiom,
! [A: set_o,F: $o > filter_nat] :
( ( A != bot_bot_set_o )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_o_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_o_filter_nat @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4964_INF__le__SUP,axiom,
! [A: set_nat,F: nat > filter_nat] :
( ( A != bot_bot_set_nat )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4965_INF__le__SUP,axiom,
! [A: set_int,F: int > filter_nat] :
( ( A != bot_bot_set_int )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_int_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_int_filter_nat @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4966_INF__le__SUP,axiom,
! [A: set_Product_unit,F: product_unit > set_nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4967_INF__le__SUP,axiom,
! [A: set_o,F: $o > set_nat] :
( ( A != bot_bot_set_o )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% INF_le_SUP
thf(fact_4968_card__Diff1__le,axiom,
! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ).
% card_Diff1_le
thf(fact_4969_card__Diff1__le,axiom,
! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ).
% card_Diff1_le
thf(fact_4970_card__Diff1__le,axiom,
! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ).
% card_Diff1_le
thf(fact_4971_card__Diff1__le,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] : ( ord_less_eq_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ).
% card_Diff1_le
thf(fact_4972_card__Diff1__le,axiom,
! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).
% card_Diff1_le
thf(fact_4973_exI__realizer,axiom,
! [P: nat > nat > $o,Y: nat,X: nat] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4974_exI__realizer,axiom,
! [P: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o] :
( ( P @ Y @ X )
=> ( P @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4975_exI__realizer,axiom,
! [P: produc3925858234332021118et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o] :
( ( P @ Y @ X )
=> ( P @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4976_exI__realizer,axiom,
! [P: set_nat > heap_e7401611519738050253t_unit > $o,Y: set_nat,X: heap_e7401611519738050253t_unit] :
( ( P @ Y @ X )
=> ( P @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4977_exI__realizer,axiom,
! [P: int > int > $o,Y: int,X: int] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4978_conjI__realizer,axiom,
! [P: nat > $o,P3: nat,Q2: nat > $o,Q4: nat] :
( ( P @ P3 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q4 ) ) )
& ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4979_conjI__realizer,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,P3: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,Q4: produc3658429121746597890et_nat] :
( ( P @ P3 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ P3 @ Q4 ) ) )
& ( Q2 @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4980_conjI__realizer,axiom,
! [P: ( produc3658429121746597890et_nat > $o ) > $o,P3: produc3658429121746597890et_nat > $o,Q2: produc3925858234332021118et_nat > $o,Q4: produc3925858234332021118et_nat] :
( ( P @ P3 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ P3 @ Q4 ) ) )
& ( Q2 @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4981_conjI__realizer,axiom,
! [P: heap_e7401611519738050253t_unit > $o,P3: heap_e7401611519738050253t_unit,Q2: set_nat > $o,Q4: set_nat] :
( ( P @ P3 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ P3 @ Q4 ) ) )
& ( Q2 @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4982_conjI__realizer,axiom,
! [P: int > $o,P3: int,Q2: int > $o,Q4: int] :
( ( P @ P3 )
=> ( ( Q2 @ Q4 )
=> ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P3 @ Q4 ) ) )
& ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4983_apfst__convE,axiom,
! [Q4: produc3658429121746597890et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,P3: produc3658429121746597890et_nat] :
( ( Q4
= ( produc4172003873565521026et_nat @ F @ P3 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
( ( P3
= ( produc7507926704131184380et_nat @ X3 @ Y2 ) )
=> ( Q4
!= ( produc7507926704131184380et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_4984_apfst__convE,axiom,
! [Q4: product_prod_nat_nat,F: nat > nat,P3: product_prod_nat_nat] :
( ( Q4
= ( produc986720760941809198at_nat @ F @ P3 ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( P3
= ( product_Pair_nat_nat @ X3 @ Y2 ) )
=> ( Q4
!= ( product_Pair_nat_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_4985_apfst__convE,axiom,
! [Q4: product_prod_int_int,F: int > int,P3: product_prod_int_int] :
( ( Q4
= ( produc2355237562570658114nt_int @ F @ P3 ) )
=> ~ ! [X3: int,Y2: int] :
( ( P3
= ( product_Pair_int_int @ X3 @ Y2 ) )
=> ( Q4
!= ( product_Pair_int_int @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_4986_apfst__convE,axiom,
! [Q4: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
( ( Q4
= ( produc3198230245617193214et_nat @ F @ P3 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
( ( P3
= ( produc5001842942810119800et_nat @ X3 @ Y2 ) )
=> ( Q4
!= ( produc5001842942810119800et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_4987_apfst__convE,axiom,
! [Q4: produc2732055786443039994et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc2732055786443039994et_nat] :
( ( Q4
= ( produc1442628905706551930et_nat @ F @ P3 ) )
=> ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
( ( P3
= ( produc2245416461498447860et_nat @ X3 @ Y2 ) )
=> ( Q4
!= ( produc2245416461498447860et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_4988_exE__realizer_H,axiom,
! [P: set_nat > heap_e7401611519738050253t_unit > $o,P3: produc3658429121746597890et_nat] :
( ( P @ ( produc8586169260539613262et_nat @ P3 ) @ ( produc1824681642469235216et_nat @ P3 ) )
=> ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_4989_exE__realizer_H,axiom,
! [P: int > int > $o,P3: product_prod_int_int] :
( ( P @ ( product_snd_int_int @ P3 ) @ ( product_fst_int_int @ P3 ) )
=> ~ ! [X3: int,Y2: int] :
~ ( P @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_4990_snd__sndOp,axiom,
! [P: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,Q2: heap_e7401611519738050253t_unit > set_nat > $o] :
( produc8586169260539613262et_nat
= ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( bNF_sn4039337771197282234et_nat @ P @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_4991_snd__sndOp,axiom,
! [P: int > int > $o,Q2: int > int > $o] :
( product_snd_int_int
= ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( bNF_sn1062102010912252026nt_int @ P @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_4992_fst__fstOp,axiom,
! [P: int > int > $o,Q2: int > int > $o] :
( product_fst_int_int
= ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( bNF_fs8167890477030535480nt_int @ P @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_4993_card__Un__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( finite4902975817058060853nteger @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( plus_plus_nat @ ( finite4902975817058060853nteger @ A ) @ ( finite4902975817058060853nteger @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4994_card__Un__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( finite410649719033368117t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( plus_plus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4995_card__Un__disjoint,axiom,
! [A: set_o,B4: set_o] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
=> ( ( finite_card_o @ ( sup_sup_set_o @ A @ B4 ) )
= ( plus_plus_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4996_card__Un__disjoint,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4997_card__Un__disjoint,axiom,
! [A: set_int,B4: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
=> ( ( finite_card_int @ ( sup_sup_set_int @ A @ B4 ) )
= ( plus_plus_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4998_card__Un__disjoint,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( finite3252695134891459830_nat_o @ B4 )
=> ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
=> ( ( finite1363419556375932405_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( plus_plus_nat @ ( finite1363419556375932405_nat_o @ A ) @ ( finite1363419556375932405_nat_o @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_4999_card__Suc__Diff1,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) )
= ( finite4902975817058060853nteger @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_5000_card__Suc__Diff1,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_5001_card__Suc__Diff1,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
= ( finite_card_o @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_5002_card__Suc__Diff1,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_5003_card__Suc__Diff1,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) )
= ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_5004_card__Suc__Diff1,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_5005_add__left__cancel,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A2 @ B )
= ( plus_p5714425477246183910nteger @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_5006_add__left__cancel,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_5007_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_5008_add__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_5009_add__right__cancel,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ( plus_p5714425477246183910nteger @ B @ A2 )
= ( plus_p5714425477246183910nteger @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_5010_add__right__cancel,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= ( plus_plus_rat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_5011_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_5012_add__right__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_5013_add__le__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
= ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_5014_add__le__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_5015_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_5016_add__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_5017_add__le__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_5018_add__le__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_5019_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_5020_add__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_5021_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_5022_add__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_5023_add__less__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
= ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_5024_add__less__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_rat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_5025_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_5026_add__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_5027_add__less__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_5028_add__less__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_rat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_5029_add__diff__cancel,axiom,
! [A2: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_5030_add__diff__cancel,axiom,
! [A2: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_5031_add__diff__cancel,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel
thf(fact_5032_diff__add__cancel,axiom,
! [A2: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_5033_diff__add__cancel,axiom,
! [A2: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_5034_diff__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
= A2 ) ).
% diff_add_cancel
thf(fact_5035_add__diff__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
= ( minus_8373710615458151222nteger @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_5036_add__diff__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
= ( minus_minus_rat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_5037_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_5038_add__diff__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_5039_add__diff__cancel__left_H,axiom,
! [A2: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_5040_add__diff__cancel__left_H,axiom,
! [A2: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_5041_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_5042_add__diff__cancel__left_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_5043_add__diff__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_5044_add__diff__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_5045_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_5046_add__diff__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_5047_add__diff__cancel__right_H,axiom,
! [A2: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_5048_add__diff__cancel__right_H,axiom,
! [A2: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_5049_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_5050_add__diff__cancel__right_H,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_5051_card__insert__disjoint,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( suc @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_5052_card__insert__disjoint,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( suc @ ( finite_card_int @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_5053_card__insert__disjoint,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( finite_card_o @ ( insert_o @ X @ A ) )
= ( suc @ ( finite_card_o @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_5054_card__insert__disjoint,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_5055_card__insert__disjoint,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_5056_card__insert__disjoint,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
= ( suc @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_5057_Suc__diff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ M4 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ M4 ) )
= ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ).
% Suc_diff
thf(fact_5058_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( I2 = J )
& ( K3 = L ) )
=> ( ( plus_p5714425477246183910nteger @ I2 @ K3 )
= ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_5059_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( I2 = J )
& ( K3 = L ) )
=> ( ( plus_plus_rat @ I2 @ K3 )
= ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_5060_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( I2 = J )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I2 @ K3 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_5061_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( I2 = J )
& ( K3 = L ) )
=> ( ( plus_plus_int @ I2 @ K3 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_5062_group__cancel_Oadd1,axiom,
! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
( ( A
= ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
=> ( ( plus_p5714425477246183910nteger @ A @ B )
= ( plus_p5714425477246183910nteger @ K3 @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_5063_group__cancel_Oadd1,axiom,
! [A: rat,K3: rat,A2: rat,B: rat] :
( ( A
= ( plus_plus_rat @ K3 @ A2 ) )
=> ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_5064_group__cancel_Oadd1,axiom,
! [A: nat,K3: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K3 @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_5065_group__cancel_Oadd1,axiom,
! [A: int,K3: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K3 @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_5066_group__cancel_Oadd2,axiom,
! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
( ( B4
= ( plus_p5714425477246183910nteger @ K3 @ B ) )
=> ( ( plus_p5714425477246183910nteger @ A2 @ B4 )
= ( plus_p5714425477246183910nteger @ K3 @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_5067_group__cancel_Oadd2,axiom,
! [B4: rat,K3: rat,B: rat,A2: rat] :
( ( B4
= ( plus_plus_rat @ K3 @ B ) )
=> ( ( plus_plus_rat @ A2 @ B4 )
= ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_5068_group__cancel_Oadd2,axiom,
! [B4: nat,K3: nat,B: nat,A2: nat] :
( ( B4
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A2 @ B4 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_5069_group__cancel_Oadd2,axiom,
! [B4: int,K3: int,B: int,A2: int] :
( ( B4
= ( plus_plus_int @ K3 @ B ) )
=> ( ( plus_plus_int @ A2 @ B4 )
= ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_5070_add_Oassoc,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
= ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% add.assoc
thf(fact_5071_add_Oassoc,axiom,
! [A2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.assoc
thf(fact_5072_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_5073_add_Oassoc,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_5074_add_Oleft__cancel,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A2 @ B )
= ( plus_p5714425477246183910nteger @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_5075_add_Oleft__cancel,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_5076_add_Oleft__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_5077_add_Oright__cancel,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ( plus_p5714425477246183910nteger @ B @ A2 )
= ( plus_p5714425477246183910nteger @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_5078_add_Oright__cancel,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= ( plus_plus_rat @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_5079_add_Oright__cancel,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_5080_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_p5714425477246183910nteger
= ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ B6 @ A6 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_5081_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ B6 @ A6 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_5082_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B6: nat] : ( plus_plus_nat @ B6 @ A6 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_5083_ab__semigroup__add__class_Oadd_Ocommute,axiom,
( plus_plus_int
= ( ^ [A6: int,B6: int] : ( plus_plus_int @ B6 @ A6 ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_5084_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) )
= ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_5085_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: rat,A2: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A2 @ C ) )
= ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_5086_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_5087_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [B: int,A2: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_5088_add__left__imp__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A2 @ B )
= ( plus_p5714425477246183910nteger @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_5089_add__left__imp__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_5090_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_5091_add__left__imp__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_5092_add__right__imp__eq,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ( plus_p5714425477246183910nteger @ B @ A2 )
= ( plus_p5714425477246183910nteger @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_5093_add__right__imp__eq,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= ( plus_plus_rat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_5094_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_5095_add__right__imp__eq,axiom,
! [B: int,A2: int,C: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_5096_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( ord_le3102999989581377725nteger @ I2 @ J )
& ( K3 = L ) )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_5097_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( K3 = L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_5098_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K3 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_5099_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K3 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_5100_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( I2 = J )
& ( ord_le3102999989581377725nteger @ K3 @ L ) )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_5101_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( I2 = J )
& ( ord_less_eq_rat @ K3 @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_5102_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_5103_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K3 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_5104_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( ord_le3102999989581377725nteger @ I2 @ J )
& ( ord_le3102999989581377725nteger @ K3 @ L ) )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_5105_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_eq_rat @ K3 @ L ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_5106_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_5107_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K3 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_5108_add__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_5109_add__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_5110_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_5111_add__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_5112_add__left__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% add_left_mono
thf(fact_5113_add__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_5114_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_5115_add__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_5116_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_5117_add__right__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% add_right_mono
thf(fact_5118_add__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_5119_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_5120_add__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_5121_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
? [C4: nat] :
( B6
= ( plus_plus_nat @ A6 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_5122_add__le__imp__le__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_5123_add__le__imp__le__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_5124_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_5125_add__le__imp__le__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_5126_add__le__imp__le__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_5127_add__le__imp__le__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_eq_rat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_5128_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_5129_add__le__imp__le__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_5130_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_5131_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K3 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_5132_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( ord_le6747313008572928689nteger @ I2 @ J )
& ( ord_le6747313008572928689nteger @ K3 @ L ) )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_5133_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_rat @ K3 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_5134_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_5135_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K3 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_5136_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( I2 = J )
& ( ord_le6747313008572928689nteger @ K3 @ L ) )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_5137_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( I2 = J )
& ( ord_less_rat @ K3 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_5138_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K3 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_5139_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K3 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_5140_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( ord_le6747313008572928689nteger @ I2 @ J )
& ( K3 = L ) )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_5141_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( K3 = L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_5142_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_5143_add__strict__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_5144_add__strict__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_5145_add__strict__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_5146_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_5147_add__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_5148_add__strict__left__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_5149_add__strict__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_5150_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_5151_add__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_5152_add__strict__right__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_5153_add__strict__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_5154_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_5155_add__less__imp__less__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_5156_add__less__imp__less__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_5157_add__less__imp__less__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_rat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_5158_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_5159_add__less__imp__less__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_5160_add__less__imp__less__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_5161_add__less__imp__less__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_rat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_5162_combine__common__factor,axiom,
! [A2: code_integer,E2: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_5163_combine__common__factor,axiom,
! [A2: rat,E2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_5164_combine__common__factor,axiom,
! [A2: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_5165_combine__common__factor,axiom,
! [A2: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_5166_distrib__right,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% distrib_right
thf(fact_5167_distrib__right,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% distrib_right
thf(fact_5168_distrib__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_5169_distrib__right,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_5170_distrib__left,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_5171_distrib__left,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_5172_distrib__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_5173_distrib__left,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_5174_comm__semiring__class_Odistrib,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_5175_comm__semiring__class_Odistrib,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_5176_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_5177_comm__semiring__class_Odistrib,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_5178_ring__class_Oring__distribs_I1_J,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_5179_ring__class_Oring__distribs_I1_J,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_5180_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_5181_ring__class_Oring__distribs_I2_J,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_5182_ring__class_Oring__distribs_I2_J,axiom,
! [A2: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_5183_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_5184_group__cancel_Osub1,axiom,
! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
( ( A
= ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
=> ( ( minus_8373710615458151222nteger @ A @ B )
= ( plus_p5714425477246183910nteger @ K3 @ ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_5185_group__cancel_Osub1,axiom,
! [A: rat,K3: rat,A2: rat,B: rat] :
( ( A
= ( plus_plus_rat @ K3 @ A2 ) )
=> ( ( minus_minus_rat @ A @ B )
= ( plus_plus_rat @ K3 @ ( minus_minus_rat @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_5186_group__cancel_Osub1,axiom,
! [A: int,K3: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K3 @ A2 ) )
=> ( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ K3 @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_5187_diff__eq__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ( minus_8373710615458151222nteger @ A2 @ B )
= C )
= ( A2
= ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_5188_diff__eq__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ( minus_minus_rat @ A2 @ B )
= C )
= ( A2
= ( plus_plus_rat @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_5189_diff__eq__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ( minus_minus_int @ A2 @ B )
= C )
= ( A2
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_5190_eq__diff__eq,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( A2
= ( minus_8373710615458151222nteger @ C @ B ) )
= ( ( plus_p5714425477246183910nteger @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_5191_eq__diff__eq,axiom,
! [A2: rat,C: rat,B: rat] :
( ( A2
= ( minus_minus_rat @ C @ B ) )
= ( ( plus_plus_rat @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_5192_eq__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( A2
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_5193_add__diff__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ A2 @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_5194_add__diff__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_5195_add__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_5196_diff__diff__eq2,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ ( minus_8373710615458151222nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_5197_diff__diff__eq2,axiom,
! [A2: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_5198_diff__diff__eq2,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_5199_diff__add__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_5200_diff__add__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
= ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_5201_diff__add__eq,axiom,
! [A2: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_5202_diff__add__eq__diff__diff__swap,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_5203_diff__add__eq__diff__diff__swap,axiom,
! [A2: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_5204_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_5205_add__implies__diff,axiom,
! [C: code_integer,B: code_integer,A2: code_integer] :
( ( ( plus_p5714425477246183910nteger @ C @ B )
= A2 )
=> ( C
= ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_5206_add__implies__diff,axiom,
! [C: rat,B: rat,A2: rat] :
( ( ( plus_plus_rat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_rat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_5207_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_5208_add__implies__diff,axiom,
! [C: int,B: int,A2: int] :
( ( ( plus_plus_int @ C @ B )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_5209_diff__diff__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
= ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_5210_diff__diff__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
= ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_5211_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_5212_diff__diff__eq,axiom,
! [A2: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_5213_mlex__bound,axiom,
! [A2: nat,A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A2 @ A )
=> ( ( ord_less_nat @ B @ N )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( times_times_nat @ A @ N ) ) ) ) ).
% mlex_bound
thf(fact_5214_mlex__fst__decrI,axiom,
! [A2: nat,A5: nat,B: nat,N: nat,B3: nat] :
( ( ord_less_nat @ A2 @ A5 )
=> ( ( ord_less_nat @ B @ N )
=> ( ( ord_less_nat @ B3 @ N )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_5215_mlex__snd__decrI,axiom,
! [A2: nat,A5: nat,B: nat,B3: nat,N: nat] :
( ( A2 = A5 )
=> ( ( ord_less_nat @ B @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_5216_mlex__leI,axiom,
! [A2: nat,A5: nat,B: nat,B3: nat,N: nat] :
( ( ord_less_eq_nat @ A2 @ A5 )
=> ( ( ord_less_eq_nat @ B @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ).
% mlex_leI
thf(fact_5217_Inf__nat__def1,axiom,
! [K4: set_nat] :
( ( K4 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% Inf_nat_def1
thf(fact_5218_max__add__distrib__left,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
= ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5219_max__add__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5220_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5221_max__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
= ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_5222_max__add__distrib__right,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
= ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5223_max__add__distrib__right,axiom,
! [X: rat,Y: rat,Z: rat] :
( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5224_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5225_max__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] :
( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
= ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_5226_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( ord_le3102999989581377725nteger @ I2 @ J )
& ( ord_le6747313008572928689nteger @ K3 @ L ) )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_5227_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_rat @ K3 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_5228_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_5229_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K3 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_5230_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
( ( ( ord_le6747313008572928689nteger @ I2 @ J )
& ( ord_le3102999989581377725nteger @ K3 @ L ) )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_5231_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: rat,J: rat,K3: rat,L: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_eq_rat @ K3 @ L ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_5232_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_5233_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K3: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K3 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_5234_add__le__less__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_5235_add__le__less__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_5236_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_5237_add__le__less__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_5238_add__less__le__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_5239_add__less__le__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_5240_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_5241_add__less__le__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_5242_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_5243_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_5244_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_5245_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_5246_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_5247_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_5248_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_5249_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_5250_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_5251_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_5252_le__diff__eq,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B ) )
= ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_5253_le__diff__eq,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_5254_le__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_5255_diff__le__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
= ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_5256_diff__le__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
= ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_5257_diff__le__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_5258_less__add__one,axiom,
! [A2: code_natural] : ( ord_le5570908160329646204atural @ A2 @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) ) ).
% less_add_one
thf(fact_5259_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_5260_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_5261_less__add__one,axiom,
! [A2: code_integer] : ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ).
% less_add_one
thf(fact_5262_less__add__one,axiom,
! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).
% less_add_one
thf(fact_5263_add__mono1,axiom,
! [A2: code_natural,B: code_natural] :
( ( ord_le5570908160329646204atural @ A2 @ B )
=> ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ B @ one_one_Code_natural ) ) ) ).
% add_mono1
thf(fact_5264_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_5265_add__mono1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_5266_add__mono1,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).
% add_mono1
thf(fact_5267_add__mono1,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_5268_diff__less__eq,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_5269_diff__less__eq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
= ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_5270_diff__less__eq,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
= ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_5271_less__diff__eq,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_5272_less__diff__eq,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B ) )
= ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_5273_less__diff__eq,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_5274_square__diff__square__factored,axiom,
! [X: code_integer,Y: code_integer] :
( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_5275_square__diff__square__factored,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_5276_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_5277_eq__add__iff2,axiom,
! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_5278_eq__add__iff2,axiom,
! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_5279_eq__add__iff2,axiom,
! [A2: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_5280_eq__add__iff1,axiom,
! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_5281_eq__add__iff1,axiom,
! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_5282_eq__add__iff1,axiom,
! [A2: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_5283_Suc__n__minus__m__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ M4 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ M4 ) )
= ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_5284_nat__in__between__eq_I1_J,axiom,
! [A2: nat,B: nat] :
( ( ( ord_less_nat @ A2 @ B )
& ( ord_less_eq_nat @ B @ ( suc @ A2 ) ) )
= ( B
= ( suc @ A2 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_5285_nat__in__between__eq_I2_J,axiom,
! [A2: nat,B: nat] :
( ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_nat @ B @ ( suc @ A2 ) ) )
= ( B = A2 ) ) ).
% nat_in_between_eq(2)
thf(fact_5286_wf__no__infinite__down__chainE,axiom,
! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( wf_nat @ R2 )
=> ~ ! [K: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_5287_wf__no__infinite__down__chainE,axiom,
! [R2: set_Pr958786334691620121nt_int,F: nat > int] :
( ( wf_int @ R2 )
=> ~ ! [K: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_5288_wf__iff__no__infinite__down__chain,axiom,
( wf_nat
= ( ^ [R4: set_Pr1261947904930325089at_nat] :
~ ? [F4: nat > nat] :
! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_5289_wf__iff__no__infinite__down__chain,axiom,
( wf_int
= ( ^ [R4: set_Pr958786334691620121nt_int] :
~ ? [F4: nat > int] :
! [I4: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_5290_Inf__INT__eq,axiom,
( comple4605987646318907561at_o_o
= ( ^ [S6: set_Pr6409761941060870310at_o_o,X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ ( comple79274577958421812_nat_o @ ( image_8639947063239330806_nat_o @ collec939566748876313656_nat_o @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_5291_Inf__INT__eq,axiom,
( comple4393433247856107946_int_o
= ( ^ [S6: set_int_o,X2: int] : ( member_int @ X2 @ ( comple3628384868704368283et_int @ ( image_int_o_set_int @ collect_int @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_5292_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S6: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_5293_Inf__INT__eq,axiom,
( comple2092403528524206185unit_o
= ( ^ [S6: set_Product_unit_o,X2: product_unit] : ( member_Product_unit @ X2 @ ( comple7216939928495492468t_unit @ ( image_5138372075374782070t_unit @ collect_Product_unit @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_5294_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S6: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_5295_le__add__iff2,axiom,
! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_5296_le__add__iff2,axiom,
! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_5297_le__add__iff2,axiom,
! [A2: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% le_add_iff2
thf(fact_5298_le__add__iff1,axiom,
! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_5299_le__add__iff1,axiom,
! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_5300_le__add__iff1,axiom,
! [A2: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% le_add_iff1
thf(fact_5301_less__add__iff2,axiom,
! [A2: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_5302_less__add__iff2,axiom,
! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_5303_less__add__iff2,axiom,
! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_5304_less__add__iff1,axiom,
! [A2: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_5305_less__add__iff1,axiom,
! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
= ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_5306_less__add__iff1,axiom,
! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% less_add_iff1
thf(fact_5307_square__diff__one__factored,axiom,
! [X: code_integer] :
( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
= ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).
% square_diff_one_factored
thf(fact_5308_square__diff__one__factored,axiom,
! [X: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% square_diff_one_factored
thf(fact_5309_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_5310_card__Suc__eq__finite,axiom,
! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
( ( ( finite1363419556375932405_nat_o @ A )
= ( suc @ K3 ) )
= ( ? [B6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ B6 @ B7 ) )
& ~ ( member6576561426505652726_nat_o @ B6 @ B7 )
& ( ( finite1363419556375932405_nat_o @ B7 )
= K3 )
& ( finite3252695134891459830_nat_o @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_5311_card__Suc__eq__finite,axiom,
! [A: set_int,K3: nat] :
( ( ( finite_card_int @ A )
= ( suc @ K3 ) )
= ( ? [B6: int,B7: set_int] :
( ( A
= ( insert_int @ B6 @ B7 ) )
& ~ ( member_int @ B6 @ B7 )
& ( ( finite_card_int @ B7 )
= K3 )
& ( finite_finite_int @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_5312_card__Suc__eq__finite,axiom,
! [A: set_o,K3: nat] :
( ( ( finite_card_o @ A )
= ( suc @ K3 ) )
= ( ? [B6: $o,B7: set_o] :
( ( A
= ( insert_o @ B6 @ B7 ) )
& ~ ( member_o @ B6 @ B7 )
& ( ( finite_card_o @ B7 )
= K3 )
& ( finite_finite_o @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_5313_card__Suc__eq__finite,axiom,
! [A: set_Product_unit,K3: nat] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ K3 ) )
= ( ? [B6: product_unit,B7: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B6 @ B7 ) )
& ~ ( member_Product_unit @ B6 @ B7 )
& ( ( finite410649719033368117t_unit @ B7 )
= K3 )
& ( finite4290736615968046902t_unit @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_5314_card__Suc__eq__finite,axiom,
! [A: set_nat,K3: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K3 ) )
= ( ? [B6: nat,B7: set_nat] :
( ( A
= ( insert_nat @ B6 @ B7 ) )
& ~ ( member_nat @ B6 @ B7 )
& ( ( finite_card_nat @ B7 )
= K3 )
& ( finite_finite_nat @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_5315_card__Suc__eq__finite,axiom,
! [A: set_Code_integer,K3: nat] :
( ( ( finite4902975817058060853nteger @ A )
= ( suc @ K3 ) )
= ( ? [B6: code_integer,B7: set_Code_integer] :
( ( A
= ( insert_Code_integer @ B6 @ B7 ) )
& ~ ( member_Code_integer @ B6 @ B7 )
& ( ( finite4902975817058060853nteger @ B7 )
= K3 )
& ( finite6017078050557962740nteger @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_5316_card__insert__if,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( finite1363419556375932405_nat_o @ A ) ) )
& ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( suc @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_5317_card__insert__if,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( finite_card_int @ A ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( suc @ ( finite_card_int @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_5318_card__insert__if,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ X @ A )
=> ( ( finite_card_o @ ( insert_o @ X @ A ) )
= ( finite_card_o @ A ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( finite_card_o @ ( insert_o @ X @ A ) )
= ( suc @ ( finite_card_o @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_5319_card__insert__if,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( finite410649719033368117t_unit @ A ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_5320_card__insert__if,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_5321_card__insert__if,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ X @ A )
=> ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
= ( finite4902975817058060853nteger @ A ) ) )
& ( ~ ( member_Code_integer @ X @ A )
=> ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
= ( suc @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_5322_card__le__Suc__iff,axiom,
! [N2: nat,A: set_Pr4532377907799695533_nat_o] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite1363419556375932405_nat_o @ A ) )
= ( ? [A6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ A6 @ B7 ) )
& ~ ( member6576561426505652726_nat_o @ A6 @ B7 )
& ( ord_less_eq_nat @ N2 @ ( finite1363419556375932405_nat_o @ B7 ) )
& ( finite3252695134891459830_nat_o @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_5323_card__le__Suc__iff,axiom,
! [N2: nat,A: set_int] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_int @ A ) )
= ( ? [A6: int,B7: set_int] :
( ( A
= ( insert_int @ A6 @ B7 ) )
& ~ ( member_int @ A6 @ B7 )
& ( ord_less_eq_nat @ N2 @ ( finite_card_int @ B7 ) )
& ( finite_finite_int @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_5324_card__le__Suc__iff,axiom,
! [N2: nat,A: set_o] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_o @ A ) )
= ( ? [A6: $o,B7: set_o] :
( ( A
= ( insert_o @ A6 @ B7 ) )
& ~ ( member_o @ A6 @ B7 )
& ( ord_less_eq_nat @ N2 @ ( finite_card_o @ B7 ) )
& ( finite_finite_o @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_5325_card__le__Suc__iff,axiom,
! [N2: nat,A: set_Product_unit] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite410649719033368117t_unit @ A ) )
= ( ? [A6: product_unit,B7: set_Product_unit] :
( ( A
= ( insert_Product_unit @ A6 @ B7 ) )
& ~ ( member_Product_unit @ A6 @ B7 )
& ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ B7 ) )
& ( finite4290736615968046902t_unit @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_5326_card__le__Suc__iff,axiom,
! [N2: nat,A: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_nat @ A ) )
= ( ? [A6: nat,B7: set_nat] :
( ( A
= ( insert_nat @ A6 @ B7 ) )
& ~ ( member_nat @ A6 @ B7 )
& ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ B7 ) )
& ( finite_finite_nat @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_5327_card__le__Suc__iff,axiom,
! [N2: nat,A: set_Code_integer] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite4902975817058060853nteger @ A ) )
= ( ? [A6: code_integer,B7: set_Code_integer] :
( ( A
= ( insert_Code_integer @ A6 @ B7 ) )
& ~ ( member_Code_integer @ A6 @ B7 )
& ( ord_less_eq_nat @ N2 @ ( finite4902975817058060853nteger @ B7 ) )
& ( finite6017078050557962740nteger @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_5328_card_Oremove,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( finite4902975817058060853nteger @ A )
= ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% card.remove
thf(fact_5329_card_Oremove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ A )
= ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% card.remove
thf(fact_5330_card_Oremove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( finite_card_o @ A )
= ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% card.remove
thf(fact_5331_card_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( finite_card_int @ A )
= ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% card.remove
thf(fact_5332_card_Oremove,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( member6576561426505652726_nat_o @ X @ A )
=> ( ( finite1363419556375932405_nat_o @ A )
= ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).
% card.remove
thf(fact_5333_card_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ A )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_5334_card_Oinsert__remove,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
= ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_5335_card_Oinsert__remove,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_5336_card_Oinsert__remove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( finite_card_o @ ( insert_o @ X @ A ) )
= ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_5337_card_Oinsert__remove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_5338_card_Oinsert__remove,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_5339_card_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_5340_mult__Suc__right,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ M4 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_5341_nat__mult__eq__1__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_5342_nat__1__eq__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N2 ) )
= ( ( M4 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_5343_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_5344_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_5345_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_5346_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_5347_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_5348_nat__mult__max__right,axiom,
! [M4: nat,N2: nat,Q4: nat] :
( ( times_times_nat @ M4 @ ( ord_max_nat @ N2 @ Q4 ) )
= ( ord_max_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_5349_nat__mult__max__left,axiom,
! [M4: nat,N2: nat,Q4: nat] :
( ( times_times_nat @ ( ord_max_nat @ M4 @ N2 ) @ Q4 )
= ( ord_max_nat @ ( times_times_nat @ M4 @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_5350_Suc__mult__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K3 ) @ M4 )
= ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
= ( M4 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_5351_mult__le__mono2,axiom,
! [I2: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I2 ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_5352_mult__le__mono1,axiom,
! [I2: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_5353_mult__le__mono,axiom,
! [I2: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_5354_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_5355_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_5356_add__mult__distrib,axiom,
! [M4: nat,N2: nat,K3: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N2 ) @ K3 )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% add_mult_distrib
thf(fact_5357_add__mult__distrib2,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K3 @ ( plus_plus_nat @ M4 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_5358_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K3 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib
thf(fact_5359_diff__mult__distrib,axiom,
! [M4: nat,N2: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_5360_diff__mult__distrib2,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M4 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_5361_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_5362_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_5363_Suc__mult__less__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M4 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_5364_Suc__mult__le__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M4 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_5365_mult__Suc,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M4 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% mult_Suc
thf(fact_5366_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_5367_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M4
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_5368_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_5369_discrete,axiom,
( ord_le6747313008572928689nteger
= ( ^ [A6: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A6 @ one_one_Code_integer ) ) ) ) ).
% discrete
thf(fact_5370_discrete,axiom,
( ord_less_nat
= ( ^ [A6: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A6 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_5371_discrete,axiom,
( ord_less_int
= ( ^ [A6: int] : ( ord_less_eq_int @ ( plus_plus_int @ A6 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_5372_fstOp__def,axiom,
( bNF_fs3871268833404974121et_nat
= ( ^ [P2: heap_e7401611519738050253t_unit > set_nat > $o,Q: set_nat > set_nat > $o,Ac: produc3658429121746597890et_nat] : ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Ac ) @ ( bNF_pi4042512755101277815et_nat @ P2 @ Q @ ( produc1824681642469235216et_nat @ Ac ) @ ( produc8586169260539613262et_nat @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_5373_fstOp__def,axiom,
( bNF_fs8167890477030535480nt_int
= ( ^ [P2: int > int > $o,Q: int > int > $o,Ac: product_prod_int_int] : ( product_Pair_int_int @ ( product_fst_int_int @ Ac ) @ ( bNF_pi8853482479757052266nt_int @ P2 @ Q @ ( product_fst_int_int @ Ac ) @ ( product_snd_int_int @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_5374_sndOp__def,axiom,
( bNF_sn4039337771197282234et_nat
= ( ^ [P2: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,Q: heap_e7401611519738050253t_unit > set_nat > $o,Ac: produc3658429121746597890et_nat] : ( produc7507926704131184380et_nat @ ( bNF_pi3450177011487049898et_nat @ P2 @ Q @ ( produc1824681642469235216et_nat @ Ac ) @ ( produc8586169260539613262et_nat @ Ac ) ) @ ( produc8586169260539613262et_nat @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_5375_sndOp__def,axiom,
( bNF_sn1062102010912252026nt_int
= ( ^ [P2: int > int > $o,Q: int > int > $o,Ac: product_prod_int_int] : ( product_Pair_int_int @ ( bNF_pi8853482479757052266nt_int @ P2 @ Q @ ( product_fst_int_int @ Ac ) @ ( product_snd_int_int @ Ac ) ) @ ( product_snd_int_int @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_5376_card__insert__le__m1,axiom,
! [N2: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_5377_card__insert__le__m1,axiom,
! [N2: nat,Y: set_int,X: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_5378_card__insert__le__m1,axiom,
! [N2: nat,Y: set_o,X: $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite_card_o @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_5379_card__insert__le__m1,axiom,
! [N2: nat,Y: set_Product_unit,X: product_unit] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_5380_crossproduct__noteq,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ( A2 != B )
& ( C != D2 ) )
= ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) )
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ D2 ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5381_crossproduct__noteq,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ( A2 != B )
& ( C != D2 ) )
= ( ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) )
!= ( plus_plus_rat @ ( times_times_rat @ A2 @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5382_crossproduct__noteq,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ( A2 != B )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5383_crossproduct__noteq,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ( A2 != B )
& ( C != D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) )
!= ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5384_crossproduct__eq,axiom,
! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_5385_crossproduct__eq,axiom,
! [W: rat,Y: rat,X: rat,Z: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
= ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_5386_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_5387_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_5388_dbl__dec__def,axiom,
( neg_nu7757733837767384882nteger
= ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).
% dbl_dec_def
thf(fact_5389_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_5390_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_5391_card__insert__disjoint_H,axiom,
! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ~ ( member6576561426505652726_nat_o @ X @ A )
=> ( ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
= ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% card_insert_disjoint'
thf(fact_5392_card__insert__disjoint_H,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( minus_minus_nat @ ( finite_card_int @ ( insert_int @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_insert_disjoint'
thf(fact_5393_card__insert__disjoint_H,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( minus_minus_nat @ ( finite_card_o @ ( insert_o @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
= ( finite_card_o @ A ) ) ) ) ).
% card_insert_disjoint'
thf(fact_5394_card__insert__disjoint_H,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_insert_disjoint'
thf(fact_5395_card__insert__disjoint_H,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_nat @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_insert_disjoint'
thf(fact_5396_card__insert__disjoint_H,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( minus_minus_nat @ ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
= ( finite4902975817058060853nteger @ A ) ) ) ) ).
% card_insert_disjoint'
thf(fact_5397_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_5398_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_5399_add_Oright__neutral,axiom,
! [A2: rat] :
( ( plus_plus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% add.right_neutral
thf(fact_5400_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_5401_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_5402_add_Oright__neutral,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
= A2 ) ).
% add.right_neutral
thf(fact_5403_double__zero__sym,axiom,
! [A2: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A2 @ A2 ) )
= ( A2 = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_5404_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_5405_double__zero__sym,axiom,
! [A2: code_integer] :
( ( zero_z3403309356797280102nteger
= ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
= ( A2 = zero_z3403309356797280102nteger ) ) ).
% double_zero_sym
thf(fact_5406_add__cancel__left__left,axiom,
! [B: rat,A2: rat] :
( ( ( plus_plus_rat @ B @ A2 )
= A2 )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_5407_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_5408_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_5409_add__cancel__left__left,axiom,
! [B: code_integer,A2: code_integer] :
( ( ( plus_p5714425477246183910nteger @ B @ A2 )
= A2 )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_left
thf(fact_5410_add__cancel__left__right,axiom,
! [A2: rat,B: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= A2 )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_5411_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_5412_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_5413_add__cancel__left__right,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A2 @ B )
= A2 )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_left_right
thf(fact_5414_add__cancel__right__left,axiom,
! [A2: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ B @ A2 ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_5415_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_5416_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_5417_add__cancel__right__left,axiom,
! [A2: code_integer,B: code_integer] :
( ( A2
= ( plus_p5714425477246183910nteger @ B @ A2 ) )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_left
thf(fact_5418_add__cancel__right__right,axiom,
! [A2: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ A2 @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_5419_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_5420_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_5421_add__cancel__right__right,axiom,
! [A2: code_integer,B: code_integer] :
( ( A2
= ( plus_p5714425477246183910nteger @ A2 @ B ) )
= ( B = zero_z3403309356797280102nteger ) ) ).
% add_cancel_right_right
thf(fact_5422_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_5423_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_5424_add__0,axiom,
! [A2: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A2 )
= A2 ) ).
% add_0
thf(fact_5425_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_5426_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_5427_add__0,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
= A2 ) ).
% add_0
thf(fact_5428_mult__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A2 @ C )
= ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_5429_mult__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_5430_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_5431_mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_5432_mult__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ C @ A2 )
= ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_5433_mult__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_5434_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_5435_mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_5436_mult__eq__0__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A2 @ B )
= zero_z3403309356797280102nteger )
= ( ( A2 = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% mult_eq_0_iff
thf(fact_5437_mult__eq__0__iff,axiom,
! [A2: rat,B: rat] :
( ( ( times_times_rat @ A2 @ B )
= zero_zero_rat )
= ( ( A2 = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_5438_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_5439_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_5440_mult__zero__right,axiom,
! [A2: code_integer] :
( ( times_3573771949741848930nteger @ A2 @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% mult_zero_right
thf(fact_5441_mult__zero__right,axiom,
! [A2: rat] :
( ( times_times_rat @ A2 @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_5442_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_5443_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_5444_mult__zero__left,axiom,
! [A2: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A2 )
= zero_z3403309356797280102nteger ) ).
% mult_zero_left
thf(fact_5445_mult__zero__left,axiom,
! [A2: rat] :
( ( times_times_rat @ zero_zero_rat @ A2 )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_5446_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_5447_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_5448_diff__self,axiom,
! [A2: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ A2 )
= zero_z3403309356797280102nteger ) ).
% diff_self
thf(fact_5449_diff__self,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ A2 )
= zero_zero_rat ) ).
% diff_self
thf(fact_5450_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_5451_diff__0__right,axiom,
! [A2: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
= A2 ) ).
% diff_0_right
thf(fact_5452_diff__0__right,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% diff_0_right
thf(fact_5453_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_5454_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_5455_diff__zero,axiom,
! [A2: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
= A2 ) ).
% diff_zero
thf(fact_5456_diff__zero,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% diff_zero
thf(fact_5457_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_5458_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_5459_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ A2 )
= zero_z3403309356797280102nteger ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_5460_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: rat] :
( ( minus_minus_rat @ A2 @ A2 )
= zero_zero_rat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_5461_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_5462_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_5463_mult__cancel2,axiom,
! [M4: nat,K3: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ K3 )
= ( times_times_nat @ N2 @ K3 ) )
= ( ( M4 = N2 )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_5464_mult__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ K3 @ M4 )
= ( times_times_nat @ K3 @ N2 ) )
= ( ( M4 = N2 )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_5465_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_5466_mult__is__0,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_5467_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_5468_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_5469_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
= one_one_Code_integer ) ).
% dbl_dec_simps(3)
thf(fact_5470_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_5471_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_5472_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_5473_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_5474_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_5475_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_5476_le__add__same__cancel2,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ A2 ) )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% le_add_same_cancel2
thf(fact_5477_le__add__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_5478_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_5479_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_5480_le__add__same__cancel1,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% le_add_same_cancel1
thf(fact_5481_le__add__same__cancel1,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_5482_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_5483_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_5484_add__le__same__cancel2,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
= ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% add_le_same_cancel2
thf(fact_5485_add__le__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_5486_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_5487_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_5488_add__le__same__cancel1,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A2 ) @ B )
= ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% add_le_same_cancel1
thf(fact_5489_add__le__same__cancel1,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_5490_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_5491_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_5492_diff__ge__0__iff__ge,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
= ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_5493_diff__ge__0__iff__ge,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
= ( ord_less_eq_rat @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_5494_diff__ge__0__iff__ge,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_5495_zero__comp__diff__simps_I1_J,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
= ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% zero_comp_diff_simps(1)
thf(fact_5496_zero__comp__diff__simps_I1_J,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
= ( ord_less_eq_rat @ B @ A2 ) ) ).
% zero_comp_diff_simps(1)
thf(fact_5497_zero__comp__diff__simps_I1_J,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ).
% zero_comp_diff_simps(1)
thf(fact_5498_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_5499_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_5500_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_5501_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_5502_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_5503_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_5504_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_5505_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_5506_less__add__same__cancel2,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ A2 ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% less_add_same_cancel2
thf(fact_5507_less__add__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_5508_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_5509_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_5510_less__add__same__cancel1,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% less_add_same_cancel1
thf(fact_5511_less__add__same__cancel1,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_5512_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_5513_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_5514_add__less__same__cancel2,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
= ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% add_less_same_cancel2
thf(fact_5515_add__less__same__cancel2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_5516_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_5517_add__less__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_5518_add__less__same__cancel1,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A2 ) @ B )
= ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% add_less_same_cancel1
thf(fact_5519_add__less__same__cancel1,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_5520_zero__comp__diff__simps_I2_J,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% zero_comp_diff_simps(2)
thf(fact_5521_zero__comp__diff__simps_I2_J,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
= ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% zero_comp_diff_simps(2)
thf(fact_5522_zero__comp__diff__simps_I2_J,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
= ( ord_less_rat @ B @ A2 ) ) ).
% zero_comp_diff_simps(2)
thf(fact_5523_diff__gt__0__iff__gt,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_5524_diff__gt__0__iff__gt,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
= ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_5525_diff__gt__0__iff__gt,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
= ( ord_less_rat @ B @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_5526_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_5527_mult__cancel__right2,axiom,
! [A2: code_integer,C: code_integer] :
( ( ( times_3573771949741848930nteger @ A2 @ C )
= C )
= ( ( C = zero_z3403309356797280102nteger )
| ( A2 = one_one_Code_integer ) ) ) ).
% mult_cancel_right2
thf(fact_5528_mult__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ( times_times_rat @ A2 @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_5529_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_5530_mult__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( C
= ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_right1
thf(fact_5531_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_5532_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_5533_mult__cancel__left2,axiom,
! [C: code_integer,A2: code_integer] :
( ( ( times_3573771949741848930nteger @ C @ A2 )
= C )
= ( ( C = zero_z3403309356797280102nteger )
| ( A2 = one_one_Code_integer ) ) ) ).
% mult_cancel_left2
thf(fact_5534_mult__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ( times_times_rat @ C @ A2 )
= C )
= ( ( C = zero_zero_rat )
| ( A2 = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_5535_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_5536_mult__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( C
= ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( B = one_one_Code_integer ) ) ) ).
% mult_cancel_left1
thf(fact_5537_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_5538_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_5539_image__add__0,axiom,
! [S: set_rat] :
( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S )
= S ) ).
% image_add_0
thf(fact_5540_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat2 @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_5541_image__add__0,axiom,
! [S: set_int] :
( ( image_int_int2 @ ( plus_plus_int @ zero_zero_int ) @ S )
= S ) ).
% image_add_0
thf(fact_5542_image__add__0,axiom,
! [S: set_Code_integer] :
( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ) @ S )
= S ) ).
% image_add_0
thf(fact_5543_diff__numeral__special_I9_J,axiom,
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(9)
thf(fact_5544_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_5545_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_5546_max__0__1_I2_J,axiom,
( ( ord_max_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
= one_one_Code_natural ) ).
% max_0_1(2)
thf(fact_5547_max__0__1_I2_J,axiom,
( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
= one_one_Code_integer ) ).
% max_0_1(2)
thf(fact_5548_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_5549_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_5550_max__0__1_I1_J,axiom,
( ( ord_max_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
= one_one_Code_natural ) ).
% max_0_1(1)
thf(fact_5551_max__0__1_I1_J,axiom,
( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
= one_one_Code_integer ) ).
% max_0_1(1)
thf(fact_5552_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_5553_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_5554_card_Oempty,axiom,
( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_5555_card_Oempty,axiom,
( ( finite_card_o @ bot_bot_set_o )
= zero_zero_nat ) ).
% card.empty
thf(fact_5556_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_5557_card_Oempty,axiom,
( ( finite_card_int @ bot_bot_set_int )
= zero_zero_nat ) ).
% card.empty
thf(fact_5558_card_Oempty,axiom,
( ( finite1363419556375932405_nat_o @ bot_bo7824918357723582233_nat_o )
= zero_zero_nat ) ).
% card.empty
thf(fact_5559_one__eq__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M4 @ N2 ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_5560_mult__eq__1__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_5561_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M4 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_5562_nat__0__less__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_5563_mult__less__cancel2,axiom,
! [M4: nat,K3: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M4 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_5564_card__0__eq,axiom,
! [A: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( finite4902975817058060853nteger @ A )
= zero_zero_nat )
= ( A = bot_bo3990330152332043303nteger ) ) ) ).
% card_0_eq
thf(fact_5565_card__0__eq,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% card_0_eq
thf(fact_5566_card__0__eq,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( ( finite_card_o @ A )
= zero_zero_nat )
= ( A = bot_bot_set_o ) ) ) ).
% card_0_eq
thf(fact_5567_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_5568_card__0__eq,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( ( finite_card_int @ A )
= zero_zero_nat )
= ( A = bot_bot_set_int ) ) ) ).
% card_0_eq
thf(fact_5569_card__0__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( finite3252695134891459830_nat_o @ A )
=> ( ( ( finite1363419556375932405_nat_o @ A )
= zero_zero_nat )
= ( A = bot_bo7824918357723582233_nat_o ) ) ) ).
% card_0_eq
thf(fact_5570_one__le__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_5571_mult__le__cancel2,axiom,
! [M4: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_5572_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_5573_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_5574_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_5575_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_5576_zero__reorient,axiom,
! [X: code_integer] :
( ( zero_z3403309356797280102nteger = X )
= ( X = zero_z3403309356797280102nteger ) ) ).
% zero_reorient
thf(fact_5577_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_5578_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_5579_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_5580_gr__implies__not__zero,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_5581_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_5582_comm__monoid__add__class_Oadd__0,axiom,
! [A2: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_5583_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_5584_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_5585_comm__monoid__add__class_Oadd__0,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_5586_add_Ocomm__neutral,axiom,
! [A2: rat] :
( ( plus_plus_rat @ A2 @ zero_zero_rat )
= A2 ) ).
% add.comm_neutral
thf(fact_5587_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_5588_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_5589_add_Ocomm__neutral,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
= A2 ) ).
% add.comm_neutral
thf(fact_5590_add_Ogroup__left__neutral,axiom,
! [A2: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_5591_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_5592_add_Ogroup__left__neutral,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_5593_mult__right__cancel,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( ( times_3573771949741848930nteger @ A2 @ C )
= ( times_3573771949741848930nteger @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_5594_mult__right__cancel,axiom,
! [C: rat,A2: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A2 @ C )
= ( times_times_rat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_5595_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_5596_mult__right__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_5597_mult__left__cancel,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( ( times_3573771949741848930nteger @ C @ A2 )
= ( times_3573771949741848930nteger @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_5598_mult__left__cancel,axiom,
! [C: rat,A2: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A2 )
= ( times_times_rat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_5599_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_5600_mult__left__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_5601_no__zero__divisors,axiom,
! [A2: code_integer,B: code_integer] :
( ( A2 != zero_z3403309356797280102nteger )
=> ( ( B != zero_z3403309356797280102nteger )
=> ( ( times_3573771949741848930nteger @ A2 @ B )
!= zero_z3403309356797280102nteger ) ) ) ).
% no_zero_divisors
thf(fact_5602_no__zero__divisors,axiom,
! [A2: rat,B: rat] :
( ( A2 != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A2 @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_5603_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_5604_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_5605_divisors__zero,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A2 @ B )
= zero_z3403309356797280102nteger )
=> ( ( A2 = zero_z3403309356797280102nteger )
| ( B = zero_z3403309356797280102nteger ) ) ) ).
% divisors_zero
thf(fact_5606_divisors__zero,axiom,
! [A2: rat,B: rat] :
( ( ( times_times_rat @ A2 @ B )
= zero_zero_rat )
=> ( ( A2 = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_5607_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_5608_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_5609_mult__not__zero,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A2 @ B )
!= zero_z3403309356797280102nteger )
=> ( ( A2 != zero_z3403309356797280102nteger )
& ( B != zero_z3403309356797280102nteger ) ) ) ).
% mult_not_zero
thf(fact_5610_mult__not__zero,axiom,
! [A2: rat,B: rat] :
( ( ( times_times_rat @ A2 @ B )
!= zero_zero_rat )
=> ( ( A2 != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_5611_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_5612_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_5613_zero__neq__one,axiom,
zero_z2226904508553997617atural != one_one_Code_natural ).
% zero_neq_one
thf(fact_5614_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_5615_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_5616_zero__neq__one,axiom,
zero_z3403309356797280102nteger != one_one_Code_integer ).
% zero_neq_one
thf(fact_5617_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: code_integer,Z4: code_integer] : ( Y4 = Z4 ) )
= ( ^ [A6: code_integer,B6: code_integer] :
( ( minus_8373710615458151222nteger @ A6 @ B6 )
= zero_z3403309356797280102nteger ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_5618_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
= ( ^ [A6: rat,B6: rat] :
( ( minus_minus_rat @ A6 @ B6 )
= zero_zero_rat ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_5619_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
= ( ^ [A6: int,B6: int] :
( ( minus_minus_int @ A6 @ B6 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_5620_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_5621_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ K3 @ M4 )
= ( times_times_nat @ K3 @ N2 ) )
= ( ( K3 = zero_zero_nat )
| ( M4 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_5622_add__scale__eq__noteq,axiom,
! [R2: code_integer,A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( R2 != zero_z3403309356797280102nteger )
=> ( ( ( A2 = B )
& ( C != D2 ) )
=> ( ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ R2 @ C ) )
!= ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_5623_add__scale__eq__noteq,axiom,
! [R2: rat,A2: rat,B: rat,C: rat,D2: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A2 = B )
& ( C != D2 ) )
=> ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_5624_add__scale__eq__noteq,axiom,
! [R2: nat,A2: nat,B: nat,C: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A2 = B )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_5625_add__scale__eq__noteq,axiom,
! [R2: int,A2: int,B: int,C: int,D2: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A2 = B )
& ( C != D2 ) )
=> ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_5626_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).
% Sup_nat_def
thf(fact_5627_add__decreasing,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ C @ B )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_5628_add__decreasing,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_5629_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_5630_add__decreasing,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_5631_add__increasing,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ B @ C )
=> ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_5632_add__increasing,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_5633_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_5634_add__increasing,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_5635_add__decreasing2,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_5636_add__decreasing2,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_5637_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_5638_add__decreasing2,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_5639_add__increasing2,axiom,
! [C: code_integer,B: code_integer,A2: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( ord_le3102999989581377725nteger @ B @ A2 )
=> ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_5640_add__increasing2,axiom,
! [C: rat,B: rat,A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B @ A2 )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_5641_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_5642_add__increasing2,axiom,
! [C: int,B: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_5643_add__nonneg__nonneg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_5644_add__nonneg__nonneg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_5645_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_5646_add__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_5647_add__nonpos__nonpos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_nonpos_nonpos
thf(fact_5648_add__nonpos__nonpos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_5649_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_5650_add__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_5651_add__nonneg__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ( plus_p5714425477246183910nteger @ X @ Y )
= zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_5652_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_5653_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_5654_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_5655_add__nonpos__eq__0__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
=> ( ( ( plus_p5714425477246183910nteger @ X @ Y )
= zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_5656_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_5657_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_5658_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_5659_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5660_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5661_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5662_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5663_zero__le__mult__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
| ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% zero_le_mult_iff
thf(fact_5664_zero__le__mult__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_5665_zero__le__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_5666_mult__nonneg__nonpos2,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_5667_mult__nonneg__nonpos2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A2 ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_5668_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_5669_mult__nonneg__nonpos2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_5670_mult__nonpos__nonneg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_nonpos_nonneg
thf(fact_5671_mult__nonpos__nonneg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_5672_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_5673_mult__nonpos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_5674_mult__nonneg__nonpos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_nonneg_nonpos
thf(fact_5675_mult__nonneg__nonpos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_5676_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_5677_mult__nonneg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_5678_mult__nonneg__nonneg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_5679_mult__nonneg__nonneg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_5680_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_5681_mult__nonneg__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_5682_split__mult__neg__le,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
| ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ).
% split_mult_neg_le
thf(fact_5683_split__mult__neg__le,axiom,
! [A2: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_5684_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_5685_split__mult__neg__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_5686_mult__le__0__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
| ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_5687_mult__le__0__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_5688_mult__le__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_5689_mult__right__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_5690_mult__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_5691_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_5692_mult__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_5693_mult__right__mono__neg,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A2 )
=> ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_5694_mult__right__mono__neg,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_5695_mult__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_5696_mult__left__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_5697_mult__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_5698_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_5699_mult__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_5700_mult__nonpos__nonpos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_5701_mult__nonpos__nonpos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_5702_mult__nonpos__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_5703_mult__left__mono__neg,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ B @ A2 )
=> ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_5704_mult__left__mono__neg,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A2 )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_5705_mult__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_5706_split__mult__pos__le,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
& ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
| ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
& ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_5707_split__mult__pos__le,axiom,
! [A2: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_5708_split__mult__pos__le,axiom,
! [A2: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% split_mult_pos_le
thf(fact_5709_zero__le__square,axiom,
! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_5710_zero__le__square,axiom,
! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_5711_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_5712_mult__mono_H,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_5713_mult__mono_H,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_5714_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_5715_mult__mono_H,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_5716_mult__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_5717_mult__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_5718_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_5719_mult__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_5720_not__one__le__zero,axiom,
~ ( ord_le1926595141338095240atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).
% not_one_le_zero
thf(fact_5721_not__one__le__zero,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% not_one_le_zero
thf(fact_5722_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_5723_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_5724_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_5725_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_5726_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_5727_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_5728_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_5729_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_5730_zero__less__one__class_Ozero__le__one,axiom,
ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% zero_less_one_class.zero_le_one
thf(fact_5731_zero__less__one__class_Ozero__le__one,axiom,
ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% zero_less_one_class.zero_le_one
thf(fact_5732_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_5733_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_5734_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_5735_le__iff__diff__le__0,axiom,
( ord_le3102999989581377725nteger
= ( ^ [A6: code_integer,B6: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ zero_z3403309356797280102nteger ) ) ) ).
% le_iff_diff_le_0
thf(fact_5736_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A6: rat,B6: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A6 @ B6 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_5737_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A6: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_5738_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_5739_pos__add__strict,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_5740_pos__add__strict,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_5741_pos__add__strict,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_5742_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_5743_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_5744_add__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_5745_add__pos__pos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_5746_add__pos__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_5747_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_5748_add__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_5749_add__neg__neg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_neg_neg
thf(fact_5750_add__neg__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_5751_mult__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_5752_mult__neg__neg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_5753_mult__neg__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_5754_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_5755_not__square__less__zero,axiom,
! [A2: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger ) ).
% not_square_less_zero
thf(fact_5756_not__square__less__zero,axiom,
! [A2: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_5757_mult__less__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_5758_mult__less__0__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
& ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
| ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_5759_mult__less__0__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_5760_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_5761_mult__neg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_5762_mult__neg__pos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_neg_pos
thf(fact_5763_mult__neg__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_5764_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_5765_mult__pos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_5766_mult__pos__neg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_pos_neg
thf(fact_5767_mult__pos__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_5768_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_5769_mult__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_5770_mult__pos__pos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_5771_mult__pos__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_5772_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_5773_mult__pos__neg2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_5774_mult__pos__neg2,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).
% mult_pos_neg2
thf(fact_5775_mult__pos__neg2,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B @ A2 ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_5776_zero__less__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_5777_zero__less__mult__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
& ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
| ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% zero_less_mult_iff
thf(fact_5778_zero__less__mult__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A2 @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_5779_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_5780_zero__less__mult__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_5781_zero__less__mult__pos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_5782_zero__less__mult__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_5783_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_5784_zero__less__mult__pos2,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_5785_zero__less__mult__pos2,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A2 ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_5786_zero__less__mult__pos2,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_5787_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_5788_mult__less__cancel__left__neg,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_5789_mult__less__cancel__left__neg,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_5790_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_5791_mult__less__cancel__left__pos,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_5792_mult__less__cancel__left__pos,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_5793_mult__strict__left__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_5794_mult__strict__left__mono__neg,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_5795_mult__strict__left__mono__neg,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_5796_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_5797_mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_5798_mult__strict__left__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_5799_mult__strict__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_5800_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_5801_mult__less__cancel__left__disj,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
& ( ord_le6747313008572928689nteger @ A2 @ B ) )
| ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_5802_mult__less__cancel__left__disj,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A2 @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_5803_mult__strict__right__mono__neg,axiom,
! [B: int,A2: int,C: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_5804_mult__strict__right__mono__neg,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ B @ A2 )
=> ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_5805_mult__strict__right__mono__neg,axiom,
! [B: rat,A2: rat,C: rat] :
( ( ord_less_rat @ B @ A2 )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_5806_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_5807_mult__strict__right__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_5808_mult__strict__right__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_5809_mult__strict__right__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_5810_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_5811_mult__less__cancel__right__disj,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
& ( ord_le6747313008572928689nteger @ A2 @ B ) )
| ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
& ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_5812_mult__less__cancel__right__disj,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A2 @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_5813_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5814_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5815_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5816_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5817_less__numeral__extra_I1_J,axiom,
ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% less_numeral_extra(1)
thf(fact_5818_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_5819_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_5820_less__numeral__extra_I1_J,axiom,
ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% less_numeral_extra(1)
thf(fact_5821_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_5822_zero__less__one,axiom,
ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% zero_less_one
thf(fact_5823_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_5824_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_5825_zero__less__one,axiom,
ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% zero_less_one
thf(fact_5826_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_5827_not__one__less__zero,axiom,
~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).
% not_one_less_zero
thf(fact_5828_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_5829_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_5830_not__one__less__zero,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% not_one_less_zero
thf(fact_5831_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_5832_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A6: int,B6: int] : ( ord_less_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_5833_less__iff__diff__less__0,axiom,
( ord_le6747313008572928689nteger
= ( ^ [A6: code_integer,B6: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ zero_z3403309356797280102nteger ) ) ) ).
% less_iff_diff_less_0
thf(fact_5834_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A6: rat,B6: rat] : ( ord_less_rat @ ( minus_minus_rat @ A6 @ B6 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_5835_inj__on__mult,axiom,
! [A2: code_integer,A: set_Code_integer] :
( ( A2 != zero_z3403309356797280102nteger )
=> ( inj_on1158064722259180589nteger @ ( times_3573771949741848930nteger @ A2 ) @ A ) ) ).
% inj_on_mult
thf(fact_5836_inj__on__mult,axiom,
! [A2: rat,A: set_rat] :
( ( A2 != zero_zero_rat )
=> ( inj_on_rat_rat @ ( times_times_rat @ A2 ) @ A ) ) ).
% inj_on_mult
thf(fact_5837_inj__on__mult,axiom,
! [A2: nat,A: set_nat] :
( ( A2 != zero_zero_nat )
=> ( inj_on_nat_nat @ ( times_times_nat @ A2 ) @ A ) ) ).
% inj_on_mult
thf(fact_5838_inj__on__mult,axiom,
! [A2: int,A: set_int] :
( ( A2 != zero_zero_int )
=> ( inj_on_int_int @ ( times_times_int @ A2 ) @ A ) ) ).
% inj_on_mult
thf(fact_5839_nat__compl__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq_nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_compl_induct
thf(fact_5840_nat__compl__induct_H,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq_nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_compl_induct'
thf(fact_5841_Suc__to__right,axiom,
! [N2: nat,M4: nat] :
( ( ( suc @ N2 )
= M4 )
=> ( N2
= ( minus_minus_nat @ M4 @ ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_to_right
thf(fact_5842_mult__less__mono1,axiom,
! [I2: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_5843_mult__less__mono2,axiom,
! [I2: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I2 ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_5844_nat__mult__eq__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M4 )
= ( times_times_nat @ K3 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_5845_nat__mult__less__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_5846_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ X )
= ( X != zero_zero_nat ) ) ).
% nat_geq_1_eq_neqz
thf(fact_5847_mult__eq__self__implies__10,axiom,
! [M4: nat,N2: nat] :
( ( M4
= ( times_times_nat @ M4 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_5848_add__strict__increasing2,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_5849_add__strict__increasing2,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_5850_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_5851_add__strict__increasing2,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_5852_add__strict__increasing,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ B @ C )
=> ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_5853_add__strict__increasing,axiom,
! [A2: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_5854_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_5855_add__strict__increasing,axiom,
! [A2: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_5856_add__pos__nonneg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_5857_add__pos__nonneg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_5858_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_5859_add__pos__nonneg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_5860_add__nonpos__neg,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_nonpos_neg
thf(fact_5861_add__nonpos__neg,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_5862_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_5863_add__nonpos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_5864_add__nonneg__pos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_5865_add__nonneg__pos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_5866_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_5867_add__nonneg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_5868_add__neg__nonpos,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% add_neg_nonpos
thf(fact_5869_add__neg__nonpos,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_5870_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_5871_add__neg__nonpos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_5872_mult__less__le__imp__less,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le3102999989581377725nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_5873_mult__less__le__imp__less,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_eq_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_5874_mult__less__le__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_5875_mult__less__le__imp__less,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_5876_mult__le__less__imp__less,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_5877_mult__le__less__imp__less,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_5878_mult__le__less__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_5879_mult__le__less__imp__less,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_5880_mult__right__le__imp__le,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_5881_mult__right__le__imp__le,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_5882_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_5883_mult__right__le__imp__le,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_5884_mult__left__le__imp__le,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_5885_mult__left__le__imp__le,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_5886_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_5887_mult__left__le__imp__le,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_5888_mult__le__cancel__left__pos,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_5889_mult__le__cancel__left__pos,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_5890_mult__le__cancel__left__pos,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_5891_mult__le__cancel__left__neg,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_5892_mult__le__cancel__left__neg,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ B @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_5893_mult__le__cancel__left__neg,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A2 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_5894_mult__less__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_5895_mult__less__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_5896_mult__less__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_5897_mult__strict__mono_H,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_5898_mult__strict__mono_H,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_5899_mult__strict__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_5900_mult__strict__mono_H,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_5901_mult__right__less__imp__less,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_5902_mult__right__less__imp__less,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_5903_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_5904_mult__right__less__imp__less,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_5905_mult__less__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_5906_mult__less__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_5907_mult__less__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_5908_mult__strict__mono,axiom,
! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ B )
=> ( ( ord_le6747313008572928689nteger @ C @ D2 )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_5909_mult__strict__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( ord_less_rat @ A2 @ B )
=> ( ( ord_less_rat @ C @ D2 )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_5910_mult__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_5911_mult__strict__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_5912_mult__left__less__imp__less,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_5913_mult__left__less__imp__less,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_5914_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_5915_mult__left__less__imp__less,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_5916_mult__le__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_5917_mult__le__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_5918_mult__le__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_5919_mult__le__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A2 @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_5920_mult__le__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_5921_mult__le__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_5922_sum__squares__ge__zero,axiom,
! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_5923_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_5924_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_5925_mult__left__le,axiom,
! [C: code_natural,A2: code_natural] :
( ( ord_le1926595141338095240atural @ C @ one_one_Code_natural )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
=> ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_5926_mult__left__le,axiom,
! [C: code_integer,A2: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_5927_mult__left__le,axiom,
! [C: rat,A2: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_5928_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_5929_mult__left__le,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_5930_mult__le__one,axiom,
! [A2: code_natural,B: code_natural] :
( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
=> ( ( ord_le1926595141338095240atural @ B @ one_one_Code_natural )
=> ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural ) ) ) ) ).
% mult_le_one
thf(fact_5931_mult__le__one,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer ) ) ) ) ).
% mult_le_one
thf(fact_5932_mult__le__one,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_5933_mult__le__one,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_5934_mult__le__one,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_5935_mult__right__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_5936_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_5937_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_5938_mult__left__le__one__le,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_5939_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
=> ( ( ord_less_eq_rat @ Y @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_5940_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_5941_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_5942_not__sum__squares__lt__zero,axiom,
! [X: code_integer,Y: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).
% not_sum_squares_lt_zero
thf(fact_5943_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_5944_zero__less__two,axiom,
ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural ) ).
% zero_less_two
thf(fact_5945_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_5946_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_5947_zero__less__two,axiom,
ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% zero_less_two
thf(fact_5948_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_5949_card__eq__0__iff,axiom,
! [A: set_Code_integer] :
( ( ( finite4902975817058060853nteger @ A )
= zero_zero_nat )
= ( ( A = bot_bo3990330152332043303nteger )
| ~ ( finite6017078050557962740nteger @ A ) ) ) ).
% card_eq_0_iff
thf(fact_5950_card__eq__0__iff,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( ( A = bot_bo3957492148770167129t_unit )
| ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% card_eq_0_iff
thf(fact_5951_card__eq__0__iff,axiom,
! [A: set_o] :
( ( ( finite_card_o @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_o )
| ~ ( finite_finite_o @ A ) ) ) ).
% card_eq_0_iff
thf(fact_5952_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_5953_card__eq__0__iff,axiom,
! [A: set_int] :
( ( ( finite_card_int @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_int )
| ~ ( finite_finite_int @ A ) ) ) ).
% card_eq_0_iff
thf(fact_5954_card__eq__0__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ( finite1363419556375932405_nat_o @ A )
= zero_zero_nat )
= ( ( A = bot_bo7824918357723582233_nat_o )
| ~ ( finite3252695134891459830_nat_o @ A ) ) ) ).
% card_eq_0_iff
thf(fact_5955_one__less__mult,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_5956_n__less__m__mult__n,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_5957_n__less__n__mult__m,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M4 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_5958_nat__mult__le__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_5959_mult__le__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_5960_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_5961_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_5962_mult__le__cancel__left2,axiom,
! [C: code_integer,A2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_5963_mult__le__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_5964_mult__le__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_5965_mult__le__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_5966_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_5967_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_5968_mult__le__cancel__right2,axiom,
! [A2: code_integer,C: code_integer] :
( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
= ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
& ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_5969_mult__le__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A2 @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_5970_mult__le__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A2 @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_5971_mult__less__cancel__left1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_5972_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_5973_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_5974_mult__less__cancel__left2,axiom,
! [C: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_5975_mult__less__cancel__left2,axiom,
! [C: rat,A2: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_5976_mult__less__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_5977_mult__less__cancel__right1,axiom,
! [C: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_5978_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_5979_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_5980_mult__less__cancel__right2,axiom,
! [A2: code_integer,C: code_integer] :
( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
= ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
& ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_5981_mult__less__cancel__right2,axiom,
! [A2: rat,C: rat] :
( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A2 @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_5982_mult__less__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A2 @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_5983_convex__bound__le,axiom,
! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( ( ord_le3102999989581377725nteger @ X @ A2 )
=> ( ( ord_le3102999989581377725nteger @ Y @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
=> ( ( ( plus_p5714425477246183910nteger @ U @ V )
= one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_5984_convex__bound__le,axiom,
! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A2 )
=> ( ( ord_less_eq_rat @ Y @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_5985_convex__bound__le,axiom,
! [X: int,A2: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A2 )
=> ( ( ord_less_eq_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_5986_card__gt__0__iff,axiom,
! [A: set_Code_integer] :
( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A ) )
= ( ( A != bot_bo3990330152332043303nteger )
& ( finite6017078050557962740nteger @ A ) ) ) ).
% card_gt_0_iff
thf(fact_5987_card__gt__0__iff,axiom,
! [A: set_Product_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
= ( ( A != bot_bo3957492148770167129t_unit )
& ( finite4290736615968046902t_unit @ A ) ) ) ).
% card_gt_0_iff
thf(fact_5988_card__gt__0__iff,axiom,
! [A: set_o] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
= ( ( A != bot_bot_set_o )
& ( finite_finite_o @ A ) ) ) ).
% card_gt_0_iff
thf(fact_5989_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_5990_card__gt__0__iff,axiom,
! [A: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
= ( ( A != bot_bot_set_int )
& ( finite_finite_int @ A ) ) ) ).
% card_gt_0_iff
thf(fact_5991_card__gt__0__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ord_less_nat @ zero_zero_nat @ ( finite1363419556375932405_nat_o @ A ) )
= ( ( A != bot_bo7824918357723582233_nat_o )
& ( finite3252695134891459830_nat_o @ A ) ) ) ).
% card_gt_0_iff
thf(fact_5992_card__1__singleton__iff,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: product_unit] :
( A
= ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% card_1_singleton_iff
thf(fact_5993_card__1__singleton__iff,axiom,
! [A: set_o] :
( ( ( finite_card_o @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: $o] :
( A
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% card_1_singleton_iff
thf(fact_5994_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_5995_card__1__singleton__iff,axiom,
! [A: set_int] :
( ( ( finite_card_int @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: int] :
( A
= ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% card_1_singleton_iff
thf(fact_5996_card__1__singleton__iff,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ( finite1363419556375932405_nat_o @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: produc3658429121746597890et_nat > $o] :
( A
= ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% card_1_singleton_iff
thf(fact_5997_card__eq__SucD,axiom,
! [A: set_Product_unit,K3: nat] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ K3 ) )
=> ? [B2: product_unit,B5: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B2 @ B5 ) )
& ~ ( member_Product_unit @ B2 @ B5 )
& ( ( finite410649719033368117t_unit @ B5 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B5 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% card_eq_SucD
thf(fact_5998_card__eq__SucD,axiom,
! [A: set_o,K3: nat] :
( ( ( finite_card_o @ A )
= ( suc @ K3 ) )
=> ? [B2: $o,B5: set_o] :
( ( A
= ( insert_o @ B2 @ B5 ) )
& ~ ( member_o @ B2 @ B5 )
& ( ( finite_card_o @ B5 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B5 = bot_bot_set_o ) ) ) ) ).
% card_eq_SucD
thf(fact_5999_card__eq__SucD,axiom,
! [A: set_nat,K3: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K3 ) )
=> ? [B2: nat,B5: set_nat] :
( ( A
= ( insert_nat @ B2 @ B5 ) )
& ~ ( member_nat @ B2 @ B5 )
& ( ( finite_card_nat @ B5 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B5 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_6000_card__eq__SucD,axiom,
! [A: set_int,K3: nat] :
( ( ( finite_card_int @ A )
= ( suc @ K3 ) )
=> ? [B2: int,B5: set_int] :
( ( A
= ( insert_int @ B2 @ B5 ) )
& ~ ( member_int @ B2 @ B5 )
& ( ( finite_card_int @ B5 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B5 = bot_bot_set_int ) ) ) ) ).
% card_eq_SucD
thf(fact_6001_card__eq__SucD,axiom,
! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
( ( ( finite1363419556375932405_nat_o @ A )
= ( suc @ K3 ) )
=> ? [B2: produc3658429121746597890et_nat > $o,B5: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ B2 @ B5 ) )
& ~ ( member6576561426505652726_nat_o @ B2 @ B5 )
& ( ( finite1363419556375932405_nat_o @ B5 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B5 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% card_eq_SucD
thf(fact_6002_card__Suc__eq,axiom,
! [A: set_Product_unit,K3: nat] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ K3 ) )
= ( ? [B6: product_unit,B7: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B6 @ B7 ) )
& ~ ( member_Product_unit @ B6 @ B7 )
& ( ( finite410649719033368117t_unit @ B7 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B7 = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% card_Suc_eq
thf(fact_6003_card__Suc__eq,axiom,
! [A: set_o,K3: nat] :
( ( ( finite_card_o @ A )
= ( suc @ K3 ) )
= ( ? [B6: $o,B7: set_o] :
( ( A
= ( insert_o @ B6 @ B7 ) )
& ~ ( member_o @ B6 @ B7 )
& ( ( finite_card_o @ B7 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B7 = bot_bot_set_o ) ) ) ) ) ).
% card_Suc_eq
thf(fact_6004_card__Suc__eq,axiom,
! [A: set_nat,K3: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K3 ) )
= ( ? [B6: nat,B7: set_nat] :
( ( A
= ( insert_nat @ B6 @ B7 ) )
& ~ ( member_nat @ B6 @ B7 )
& ( ( finite_card_nat @ B7 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_6005_card__Suc__eq,axiom,
! [A: set_int,K3: nat] :
( ( ( finite_card_int @ A )
= ( suc @ K3 ) )
= ( ? [B6: int,B7: set_int] :
( ( A
= ( insert_int @ B6 @ B7 ) )
& ~ ( member_int @ B6 @ B7 )
& ( ( finite_card_int @ B7 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B7 = bot_bot_set_int ) ) ) ) ) ).
% card_Suc_eq
thf(fact_6006_card__Suc__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
( ( ( finite1363419556375932405_nat_o @ A )
= ( suc @ K3 ) )
= ( ? [B6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
( ( A
= ( insert5175938949040314269_nat_o @ B6 @ B7 ) )
& ~ ( member6576561426505652726_nat_o @ B6 @ B7 )
& ( ( finite1363419556375932405_nat_o @ B7 )
= K3 )
& ( ( K3 = zero_zero_nat )
=> ( B7 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% card_Suc_eq
thf(fact_6007_nz__le__conv__less,axiom,
! [K3: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ K3 @ M4 )
=> ( ord_less_nat @ ( minus_minus_nat @ K3 @ ( suc @ zero_zero_nat ) ) @ M4 ) ) ) ).
% nz_le_conv_less
thf(fact_6008_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_6009_convex__bound__lt,axiom,
! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
( ( ord_le6747313008572928689nteger @ X @ A2 )
=> ( ( ord_le6747313008572928689nteger @ Y @ A2 )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
=> ( ( ( plus_p5714425477246183910nteger @ U @ V )
= one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_6010_convex__bound__lt,axiom,
! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
( ( ord_less_rat @ X @ A2 )
=> ( ( ord_less_rat @ Y @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_6011_convex__bound__lt,axiom,
! [X: int,A2: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A2 )
=> ( ( ord_less_int @ Y @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_6012_inj__mult__left,axiom,
! [A2: code_integer] :
( ( inj_on1158064722259180589nteger @ ( times_3573771949741848930nteger @ A2 ) @ top_to4645266643341252675nteger )
= ( A2 != zero_z3403309356797280102nteger ) ) ).
% inj_mult_left
thf(fact_6013_inj__mult__left,axiom,
! [A2: rat] :
( ( inj_on_rat_rat @ ( times_times_rat @ A2 ) @ top_top_set_rat )
= ( A2 != zero_zero_rat ) ) ).
% inj_mult_left
thf(fact_6014_inj__mult__left,axiom,
! [A2: int] :
( ( inj_on_int_int @ ( times_times_int @ A2 ) @ top_top_set_int )
= ( A2 != zero_zero_int ) ) ).
% inj_mult_left
thf(fact_6015_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z3: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z3 )
=> ( ( ord_less_rat @ Z3 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq_rat @ X @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_6016_size__prod__simp,axiom,
( basic_5567952622663070111et_nat
= ( ^ [F4: heap_e7401611519738050253t_unit > nat,G3: set_nat > nat,P5: produc3658429121746597890et_nat] : ( plus_plus_nat @ ( plus_plus_nat @ ( F4 @ ( produc1824681642469235216et_nat @ P5 ) ) @ ( G3 @ ( produc8586169260539613262et_nat @ P5 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% size_prod_simp
thf(fact_6017_size__prod__simp,axiom,
( basic_1872990034501187214nt_int
= ( ^ [F4: int > nat,G3: int > nat,P5: product_prod_int_int] : ( plus_plus_nat @ ( plus_plus_nat @ ( F4 @ ( product_fst_int_int @ P5 ) ) @ ( G3 @ ( product_snd_int_int @ P5 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% size_prod_simp
thf(fact_6018_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_6019_sum__squares__gt__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
= ( ( X != zero_z3403309356797280102nteger )
| ( Y != zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_6020_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
= ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_6021_sum__squares__le__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_6022_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_6023_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_6024_mult__le__cancel__iff2,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
= ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_6025_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_6026_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_6027_mult__le__cancel__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
=> ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
= ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_6028_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_eq_rat @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_6029_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_6030_divides__aux__eq,axiom,
! [Q4: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_6031_divides__aux__eq,axiom,
! [Q4: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_6032_divides__aux__eq,axiom,
! [Q4: code_integer,R2: code_integer] :
( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
= ( R2 = zero_z3403309356797280102nteger ) ) ).
% divides_aux_eq
thf(fact_6033_divides__aux__def,axiom,
( unique6322359934112328802ux_nat
= ( ^ [Qr: product_prod_nat_nat] :
( ( product_snd_nat_nat @ Qr )
= zero_zero_nat ) ) ) ).
% divides_aux_def
thf(fact_6034_divides__aux__def,axiom,
( unique6319869463603278526ux_int
= ( ^ [Qr: product_prod_int_int] :
( ( product_snd_int_int @ Qr )
= zero_zero_int ) ) ) ).
% divides_aux_def
thf(fact_6035_divides__aux__def,axiom,
( unique5706413561485394159nteger
= ( ^ [Qr: produc8923325533196201883nteger] :
( ( produc6174133586879617921nteger @ Qr )
= zero_z3403309356797280102nteger ) ) ) ).
% divides_aux_def
thf(fact_6036_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_6037_mult__less__iff1,axiom,
! [Z: code_integer,X: code_integer,Y: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
=> ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
= ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_6038_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
= ( ord_less_rat @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_6039_sum__squares__eq__zero__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
= zero_z3403309356797280102nteger )
= ( ( X = zero_z3403309356797280102nteger )
& ( Y = zero_z3403309356797280102nteger ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_6040_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_6041_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_6042_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_6043_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
= one_one_Code_integer ) ).
% dbl_inc_simps(2)
thf(fact_6044_enumerate__Suc_H,axiom,
! [S: set_Product_unit,N2: nat] :
( ( infini7930543730640340914t_unit @ S @ ( suc @ N2 ) )
= ( infini7930543730640340914t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ ( infini7930543730640340914t_unit @ S @ zero_zero_nat ) @ bot_bo3957492148770167129t_unit ) ) @ N2 ) ) ).
% enumerate_Suc'
thf(fact_6045_enumerate__Suc_H,axiom,
! [S: set_nat,N2: nat] :
( ( infini8530281810654367211te_nat @ S @ ( suc @ N2 ) )
= ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N2 ) ) ).
% enumerate_Suc'
thf(fact_6046_power__decreasing__iff,axiom,
! [B: code_natural,M4: nat,N2: nat] :
( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
=> ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
=> ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ M4 ) @ ( power_7079662738309270450atural @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_6047_power__decreasing__iff,axiom,
! [B: code_integer,M4: nat,N2: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M4 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_6048_power__decreasing__iff,axiom,
! [B: rat,M4: nat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M4 ) @ ( power_power_rat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_6049_power__decreasing__iff,axiom,
! [B: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_6050_power__decreasing__iff,axiom,
! [B: int,M4: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% power_decreasing_iff
thf(fact_6051_dvd__partition,axiom,
! [C5: set_set_Code_integer,K3: nat] :
( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ C5 ) )
=> ( ! [X3: set_Code_integer] :
( ( member1834298987299537450nteger @ X3 @ C5 )
=> ( dvd_dvd_nat @ K3 @ ( finite4902975817058060853nteger @ X3 ) ) )
=> ( ! [X3: set_Code_integer] :
( ( member1834298987299537450nteger @ X3 @ C5 )
=> ! [Xa3: set_Code_integer] :
( ( member1834298987299537450nteger @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_in1364745209274528805nteger @ X3 @ Xa3 )
= bot_bo3990330152332043303nteger ) ) ) )
=> ( dvd_dvd_nat @ K3 @ ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6052_dvd__partition,axiom,
! [C5: set_set_Product_unit,K3: nat] :
( ( finite4290736615968046902t_unit @ ( comple3349607820504768346t_unit @ C5 ) )
=> ( ! [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ C5 )
=> ( dvd_dvd_nat @ K3 @ ( finite410649719033368117t_unit @ X3 ) ) )
=> ( ! [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ C5 )
=> ! [Xa3: set_Product_unit] :
( ( member5877623283571906838t_unit @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_in4660618365625256667t_unit @ X3 @ Xa3 )
= bot_bo3957492148770167129t_unit ) ) ) )
=> ( dvd_dvd_nat @ K3 @ ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6053_dvd__partition,axiom,
! [C5: set_set_o,K3: nat] :
( ( finite_finite_o @ ( comple90263536869209701_set_o @ C5 ) )
=> ( ! [X3: set_o] :
( ( member_set_o @ X3 @ C5 )
=> ( dvd_dvd_nat @ K3 @ ( finite_card_o @ X3 ) ) )
=> ( ! [X3: set_o] :
( ( member_set_o @ X3 @ C5 )
=> ! [Xa3: set_o] :
( ( member_set_o @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_o @ X3 @ Xa3 )
= bot_bot_set_o ) ) ) )
=> ( dvd_dvd_nat @ K3 @ ( finite_card_o @ ( comple90263536869209701_set_o @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6054_dvd__partition,axiom,
! [C5: set_set_nat,K3: nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C5 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ( dvd_dvd_nat @ K3 @ ( finite_card_nat @ X3 ) ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ! [Xa3: set_nat] :
( ( member_set_nat @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_nat @ X3 @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( dvd_dvd_nat @ K3 @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6055_dvd__partition,axiom,
! [C5: set_set_int,K3: nat] :
( ( finite_finite_int @ ( comple3221217463730067765et_int @ C5 ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ C5 )
=> ( dvd_dvd_nat @ K3 @ ( finite_card_int @ X3 ) ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ C5 )
=> ! [Xa3: set_int] :
( ( member_set_int @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_int @ X3 @ Xa3 )
= bot_bot_set_int ) ) ) )
=> ( dvd_dvd_nat @ K3 @ ( finite_card_int @ ( comple3221217463730067765et_int @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6056_dvd__partition,axiom,
! [C5: set_se1353816476878259085_nat_o,K3: nat] :
( ( finite3252695134891459830_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) )
=> ( ! [X3: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X3 @ C5 )
=> ( dvd_dvd_nat @ K3 @ ( finite1363419556375932405_nat_o @ X3 ) ) )
=> ( ! [X3: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ X3 @ C5 )
=> ! [Xa3: set_Pr4532377907799695533_nat_o] :
( ( member1452989533377600726_nat_o @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_in1906310914598751387_nat_o @ X3 @ Xa3 )
= bot_bo7824918357723582233_nat_o ) ) ) )
=> ( dvd_dvd_nat @ K3 @ ( finite1363419556375932405_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6057_cInf__le__cSup,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5488710616941104124bove_o @ A )
=> ( ( condit5413489452508810728elow_o @ A )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6058_cInf__le__cSup,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit4352657709669162033_nat_o @ A )
=> ( ( condit8399017556247434269_nat_o @ A )
=> ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6059_cInf__le__cSup,axiom,
! [A: set_filter_nat] :
( ( A != bot_bo498966703094740906er_nat )
=> ( ( condit4029664723990530684er_nat @ A )
=> ( ( condit3133331395780078992er_nat @ A )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple2454491259644266675er_nat @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6060_cInf__le__cSup,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ A )
=> ( ( condit1735850657277959132ow_int @ A )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Sup_Sup_int @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6061_cInf__le__cSup,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A )
=> ( ( condit68592940725977398et_nat @ A )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6062_cInf__le__cSup,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6063_cInf__le__cSup,axiom,
! [A: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ A )
=> ( ( condit9161856309968323677t_unit @ A )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple4687483117567863418t_unit @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6064_sum__le__card__Max,axiom,
! [A: set_Code_integer,F: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ord_less_eq_nat @ ( groups7237345082560585321er_nat @ F @ A ) @ ( times_times_nat @ ( finite4902975817058060853nteger @ A ) @ ( lattic8265883725875713057ax_nat @ ( image_951025933927791156er_nat @ F @ A ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_6065_sum__le__card__Max,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( times_times_nat @ ( finite_card_nat @ A ) @ ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_6066_gbinomial__reduce__nat,axiom,
! [K3: nat,A2: rat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( gbinomial_rat @ A2 @ K3 )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_6067_power__one,axiom,
! [N2: nat] :
( ( power_power_assn @ one_one_assn @ N2 )
= one_one_assn ) ).
% power_one
thf(fact_6068_power__one,axiom,
! [N2: nat] :
( ( power_8256067586552552935nteger @ one_one_Code_integer @ N2 )
= one_one_Code_integer ) ).
% power_one
thf(fact_6069_power__one,axiom,
! [N2: nat] :
( ( power_7079662738309270450atural @ one_one_Code_natural @ N2 )
= one_one_Code_natural ) ).
% power_one
thf(fact_6070_power__one,axiom,
! [N2: nat] :
( ( power_power_int @ one_one_int @ N2 )
= one_one_int ) ).
% power_one
thf(fact_6071_power__one,axiom,
! [N2: nat] :
( ( power_power_nat @ one_one_nat @ N2 )
= one_one_nat ) ).
% power_one
thf(fact_6072_bdd__below__empty,axiom,
condit9161856309968323677t_unit @ bot_bo3957492148770167129t_unit ).
% bdd_below_empty
thf(fact_6073_bdd__below__empty,axiom,
condit5413489452508810728elow_o @ bot_bot_set_o ).
% bdd_below_empty
thf(fact_6074_bdd__below__empty,axiom,
condit1738341127787009408ow_nat @ bot_bot_set_nat ).
% bdd_below_empty
thf(fact_6075_bdd__below__empty,axiom,
condit1735850657277959132ow_int @ bot_bot_set_int ).
% bdd_below_empty
thf(fact_6076_bdd__below__empty,axiom,
condit8399017556247434269_nat_o @ bot_bo7824918357723582233_nat_o ).
% bdd_below_empty
thf(fact_6077_bdd__below__insert,axiom,
! [A2: nat,A: set_nat] :
( ( condit1738341127787009408ow_nat @ ( insert_nat @ A2 @ A ) )
= ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_below_insert
thf(fact_6078_bdd__below__insert,axiom,
! [A2: int,A: set_int] :
( ( condit1735850657277959132ow_int @ ( insert_int @ A2 @ A ) )
= ( condit1735850657277959132ow_int @ A ) ) ).
% bdd_below_insert
thf(fact_6079_bdd__below__insert,axiom,
! [A2: $o,A: set_o] :
( ( condit5413489452508810728elow_o @ ( insert_o @ A2 @ A ) )
= ( condit5413489452508810728elow_o @ A ) ) ).
% bdd_below_insert
thf(fact_6080_bdd__below__insert,axiom,
! [A2: product_unit,A: set_Product_unit] :
( ( condit9161856309968323677t_unit @ ( insert_Product_unit @ A2 @ A ) )
= ( condit9161856309968323677t_unit @ A ) ) ).
% bdd_below_insert
thf(fact_6081_dvd__times__right__cancel__iff,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( A2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A2 ) @ ( times_3573771949741848930nteger @ C @ A2 ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_6082_dvd__times__right__cancel__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_6083_dvd__times__right__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_6084_dvd__times__left__cancel__iff,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( A2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_6085_dvd__times__left__cancel__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_6086_dvd__times__left__cancel__iff,axiom,
! [A2: int,B: int,C: int] :
( ( A2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_6087_dvd__mult__cancel__right,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_6088_dvd__mult__cancel__right,axiom,
! [A2: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_6089_dvd__mult__cancel__right,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_6090_dvd__mult__cancel__left,axiom,
! [C: code_integer,A2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_6091_dvd__mult__cancel__left,axiom,
! [C: rat,A2: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_6092_dvd__mult__cancel__left,axiom,
! [C: int,A2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A2 @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_6093_dvd__add__times__triv__left__iff,axiom,
! [A2: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ B ) )
= ( dvd_dvd_Code_integer @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_6094_dvd__add__times__triv__left__iff,axiom,
! [A2: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ ( times_times_rat @ C @ A2 ) @ B ) )
= ( dvd_dvd_rat @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_6095_dvd__add__times__triv__left__iff,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_6096_dvd__add__times__triv__left__iff,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_6097_dvd__add__times__triv__right__iff,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A2 ) ) )
= ( dvd_dvd_Code_integer @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_6098_dvd__add__times__triv__right__iff,axiom,
! [A2: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A2 ) ) )
= ( dvd_dvd_rat @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_6099_dvd__add__times__triv__right__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A2 ) ) )
= ( dvd_dvd_nat @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_6100_dvd__add__times__triv__right__iff,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ ( times_times_int @ C @ A2 ) ) )
= ( dvd_dvd_int @ A2 @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_6101_unit__prod,axiom,
! [A2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer ) ) ) ).
% unit_prod
thf(fact_6102_unit__prod,axiom,
! [A2: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural ) ) ) ).
% unit_prod
thf(fact_6103_unit__prod,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_6104_unit__prod,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_6105_power__inject__exp,axiom,
! [A2: code_natural,M4: nat,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ( ( power_7079662738309270450atural @ A2 @ M4 )
= ( power_7079662738309270450atural @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_6106_power__inject__exp,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ( power_power_nat @ A2 @ M4 )
= ( power_power_nat @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_6107_power__inject__exp,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ( power_power_int @ A2 @ M4 )
= ( power_power_int @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_6108_power__inject__exp,axiom,
! [A2: code_integer,M4: nat,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ( ( power_8256067586552552935nteger @ A2 @ M4 )
= ( power_8256067586552552935nteger @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_6109_power__inject__exp,axiom,
! [A2: rat,M4: nat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ( ( power_power_rat @ A2 @ M4 )
= ( power_power_rat @ A2 @ N2 ) )
= ( M4 = N2 ) ) ) ).
% power_inject_exp
thf(fact_6110_nat__mult__dvd__cancel__disj,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( ( K3 = zero_zero_nat )
| ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_6111_gbinomial__0_I1_J,axiom,
! [A2: nat] :
( ( gbinomial_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_6112_gbinomial__0_I1_J,axiom,
! [A2: int] :
( ( gbinomial_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_6113_gbinomial__0_I1_J,axiom,
! [A2: code_integer] :
( ( gbinom8545251970709558553nteger @ A2 @ zero_zero_nat )
= one_one_Code_integer ) ).
% gbinomial_0(1)
thf(fact_6114_gbinomial__0_I1_J,axiom,
! [A2: code_natural] :
( ( gbinom7368847122466276068atural @ A2 @ zero_zero_nat )
= one_one_Code_natural ) ).
% gbinomial_0(1)
thf(fact_6115_power__strict__increasing__iff,axiom,
! [B: code_natural,X: nat,Y: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
=> ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_6116_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_6117_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_6118_power__strict__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
=> ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_6119_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_6120_power__strict__decreasing__iff,axiom,
! [B: code_natural,M4: nat,N2: nat] :
( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
=> ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
=> ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ M4 ) @ ( power_7079662738309270450atural @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_6121_power__strict__decreasing__iff,axiom,
! [B: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_6122_power__strict__decreasing__iff,axiom,
! [B: int,M4: nat,N2: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_6123_power__strict__decreasing__iff,axiom,
! [B: code_integer,M4: nat,N2: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
=> ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M4 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_6124_power__strict__decreasing__iff,axiom,
! [B: rat,M4: nat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M4 ) @ ( power_power_rat @ B @ N2 ) )
= ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_6125_power__increasing__iff,axiom,
! [B: code_natural,X: nat,Y: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
=> ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_6126_power__increasing__iff,axiom,
! [B: code_integer,X: nat,Y: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_6127_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_6128_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_6129_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_6130_is__unit__power__iff,axiom,
! [A2: code_integer,N2: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_6131_is__unit__power__iff,axiom,
! [A2: code_natural,N2: nat] :
( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ one_one_Code_natural )
= ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_6132_is__unit__power__iff,axiom,
! [A2: int,N2: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_6133_is__unit__power__iff,axiom,
! [A2: nat,N2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_6134_power__commuting__commutes,axiom,
! [X: assn,Y: assn,N2: nat] :
( ( ( times_times_assn @ X @ Y )
= ( times_times_assn @ Y @ X ) )
=> ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ Y )
= ( times_times_assn @ Y @ ( power_power_assn @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_6135_power__commuting__commutes,axiom,
! [X: rat,Y: rat,N2: nat] :
( ( ( times_times_rat @ X @ Y )
= ( times_times_rat @ Y @ X ) )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
= ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_6136_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_6137_power__commuting__commutes,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% power_commuting_commutes
thf(fact_6138_power__mult__distrib,axiom,
! [A2: assn,B: assn,N2: nat] :
( ( power_power_assn @ ( times_times_assn @ A2 @ B ) @ N2 )
= ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ ( power_power_assn @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_6139_power__mult__distrib,axiom,
! [A2: rat,B: rat,N2: nat] :
( ( power_power_rat @ ( times_times_rat @ A2 @ B ) @ N2 )
= ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_6140_power__mult__distrib,axiom,
! [A2: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N2 )
= ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_6141_power__mult__distrib,axiom,
! [A2: int,B: int,N2: nat] :
( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N2 )
= ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% power_mult_distrib
thf(fact_6142_power__commutes,axiom,
! [A2: assn,N2: nat] :
( ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ A2 )
= ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% power_commutes
thf(fact_6143_power__commutes,axiom,
! [A2: rat,N2: nat] :
( ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ A2 )
= ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% power_commutes
thf(fact_6144_power__commutes,axiom,
! [A2: nat,N2: nat] :
( ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ A2 )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% power_commutes
thf(fact_6145_power__commutes,axiom,
! [A2: int,N2: nat] :
( ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ A2 )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ).
% power_commutes
thf(fact_6146_dvd__triv__right,axiom,
! [A2: assn,B: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_6147_dvd__triv__right,axiom,
! [A2: rat,B: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_6148_dvd__triv__right,axiom,
! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_6149_dvd__triv__right,axiom,
! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ A2 ) ) ).
% dvd_triv_right
thf(fact_6150_dvd__mult__right,axiom,
! [A2: assn,B: assn,C: assn] :
( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B ) @ C )
=> ( dvd_dvd_assn @ B @ C ) ) ).
% dvd_mult_right
thf(fact_6151_dvd__mult__right,axiom,
! [A2: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B ) @ C )
=> ( dvd_dvd_rat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_6152_dvd__mult__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_6153_dvd__mult__right,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_6154_mult__dvd__mono,axiom,
! [A2: assn,B: assn,C: assn,D2: assn] :
( ( dvd_dvd_assn @ A2 @ B )
=> ( ( dvd_dvd_assn @ C @ D2 )
=> ( dvd_dvd_assn @ ( times_times_assn @ A2 @ C ) @ ( times_times_assn @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_6155_mult__dvd__mono,axiom,
! [A2: rat,B: rat,C: rat,D2: rat] :
( ( dvd_dvd_rat @ A2 @ B )
=> ( ( dvd_dvd_rat @ C @ D2 )
=> ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_6156_mult__dvd__mono,axiom,
! [A2: nat,B: nat,C: nat,D2: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ C @ D2 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_6157_mult__dvd__mono,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ C @ D2 )
=> ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).
% mult_dvd_mono
thf(fact_6158_dvd__triv__left,axiom,
! [A2: assn,B: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_6159_dvd__triv__left,axiom,
! [A2: rat,B: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_6160_dvd__triv__left,axiom,
! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_6161_dvd__triv__left,axiom,
! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B ) ) ).
% dvd_triv_left
thf(fact_6162_dvd__mult__left,axiom,
! [A2: assn,B: assn,C: assn] :
( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B ) @ C )
=> ( dvd_dvd_assn @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_6163_dvd__mult__left,axiom,
! [A2: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B ) @ C )
=> ( dvd_dvd_rat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_6164_dvd__mult__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
=> ( dvd_dvd_nat @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_6165_dvd__mult__left,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
=> ( dvd_dvd_int @ A2 @ C ) ) ).
% dvd_mult_left
thf(fact_6166_dvd__mult2,axiom,
! [A2: assn,B: assn,C: assn] :
( ( dvd_dvd_assn @ A2 @ B )
=> ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_6167_dvd__mult2,axiom,
! [A2: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A2 @ B )
=> ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_6168_dvd__mult2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_6169_dvd__mult2,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_6170_dvd__mult,axiom,
! [A2: assn,C: assn,B: assn] :
( ( dvd_dvd_assn @ A2 @ C )
=> ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% dvd_mult
thf(fact_6171_dvd__mult,axiom,
! [A2: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A2 @ C )
=> ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_6172_dvd__mult,axiom,
! [A2: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ C )
=> ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_6173_dvd__mult,axiom,
! [A2: int,C: int,B: int] :
( ( dvd_dvd_int @ A2 @ C )
=> ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_6174_dvd__def,axiom,
( dvd_dvd_assn
= ( ^ [B6: assn,A6: assn] :
? [K5: assn] :
( A6
= ( times_times_assn @ B6 @ K5 ) ) ) ) ).
% dvd_def
thf(fact_6175_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B6: rat,A6: rat] :
? [K5: rat] :
( A6
= ( times_times_rat @ B6 @ K5 ) ) ) ) ).
% dvd_def
thf(fact_6176_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B6: nat,A6: nat] :
? [K5: nat] :
( A6
= ( times_times_nat @ B6 @ K5 ) ) ) ) ).
% dvd_def
thf(fact_6177_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B6: int,A6: int] :
? [K5: int] :
( A6
= ( times_times_int @ B6 @ K5 ) ) ) ) ).
% dvd_def
thf(fact_6178_dvdI,axiom,
! [A2: assn,B: assn,K3: assn] :
( ( A2
= ( times_times_assn @ B @ K3 ) )
=> ( dvd_dvd_assn @ B @ A2 ) ) ).
% dvdI
thf(fact_6179_dvdI,axiom,
! [A2: rat,B: rat,K3: rat] :
( ( A2
= ( times_times_rat @ B @ K3 ) )
=> ( dvd_dvd_rat @ B @ A2 ) ) ).
% dvdI
thf(fact_6180_dvdI,axiom,
! [A2: nat,B: nat,K3: nat] :
( ( A2
= ( times_times_nat @ B @ K3 ) )
=> ( dvd_dvd_nat @ B @ A2 ) ) ).
% dvdI
thf(fact_6181_dvdI,axiom,
! [A2: int,B: int,K3: int] :
( ( A2
= ( times_times_int @ B @ K3 ) )
=> ( dvd_dvd_int @ B @ A2 ) ) ).
% dvdI
thf(fact_6182_dvdE,axiom,
! [B: assn,A2: assn] :
( ( dvd_dvd_assn @ B @ A2 )
=> ~ ! [K: assn] :
( A2
!= ( times_times_assn @ B @ K ) ) ) ).
% dvdE
thf(fact_6183_dvdE,axiom,
! [B: rat,A2: rat] :
( ( dvd_dvd_rat @ B @ A2 )
=> ~ ! [K: rat] :
( A2
!= ( times_times_rat @ B @ K ) ) ) ).
% dvdE
thf(fact_6184_dvdE,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ A2 )
=> ~ ! [K: nat] :
( A2
!= ( times_times_nat @ B @ K ) ) ) ).
% dvdE
thf(fact_6185_dvdE,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ A2 )
=> ~ ! [K: int] :
( A2
!= ( times_times_int @ B @ K ) ) ) ).
% dvdE
thf(fact_6186_dvd__unit__imp__unit,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_6187_dvd__unit__imp__unit,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_6188_dvd__unit__imp__unit,axiom,
! [A2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ B )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer ) ) ) ).
% dvd_unit_imp_unit
thf(fact_6189_dvd__unit__imp__unit,axiom,
! [A2: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A2 @ B )
=> ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural ) ) ) ).
% dvd_unit_imp_unit
thf(fact_6190_unit__imp__dvd,axiom,
! [B: nat,A2: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A2 ) ) ).
% unit_imp_dvd
thf(fact_6191_unit__imp__dvd,axiom,
! [B: int,A2: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A2 ) ) ).
% unit_imp_dvd
thf(fact_6192_unit__imp__dvd,axiom,
! [B: code_integer,A2: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ B @ A2 ) ) ).
% unit_imp_dvd
thf(fact_6193_unit__imp__dvd,axiom,
! [B: code_natural,A2: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( dvd_dvd_Code_natural @ B @ A2 ) ) ).
% unit_imp_dvd
thf(fact_6194_one__dvd,axiom,
! [A2: assn] : ( dvd_dvd_assn @ one_one_assn @ A2 ) ).
% one_dvd
thf(fact_6195_one__dvd,axiom,
! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% one_dvd
thf(fact_6196_one__dvd,axiom,
! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).
% one_dvd
thf(fact_6197_one__dvd,axiom,
! [A2: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A2 ) ).
% one_dvd
thf(fact_6198_one__dvd,axiom,
! [A2: code_natural] : ( dvd_dvd_Code_natural @ one_one_Code_natural @ A2 ) ).
% one_dvd
thf(fact_6199_power__mult,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( power_power_int @ A2 @ ( times_times_nat @ M4 @ N2 ) )
= ( power_power_int @ ( power_power_int @ A2 @ M4 ) @ N2 ) ) ).
% power_mult
thf(fact_6200_power__mult,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( power_power_nat @ A2 @ ( times_times_nat @ M4 @ N2 ) )
= ( power_power_nat @ ( power_power_nat @ A2 @ M4 ) @ N2 ) ) ).
% power_mult
thf(fact_6201_dvd__power__iff,axiom,
! [X: code_natural,M4: nat,N2: nat] :
( ( X != zero_z2226904508553997617atural )
=> ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ X @ M4 ) @ ( power_7079662738309270450atural @ X @ N2 ) )
= ( ( dvd_dvd_Code_natural @ X @ one_one_Code_natural )
| ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_6202_dvd__power__iff,axiom,
! [X: code_integer,M4: nat,N2: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M4 ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
= ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
| ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_6203_dvd__power__iff,axiom,
! [X: int,M4: nat,N2: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M4 ) @ ( power_power_int @ X @ N2 ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_6204_dvd__power__iff,axiom,
! [X: nat,M4: nat,N2: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M4 ) @ ( power_power_nat @ X @ N2 ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% dvd_power_iff
thf(fact_6205_dvd__power,axiom,
! [N2: nat,X: code_integer] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_Code_integer ) )
=> ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_6206_dvd__power,axiom,
! [N2: nat,X: code_natural] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_Code_natural ) )
=> ( dvd_dvd_Code_natural @ X @ ( power_7079662738309270450atural @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_6207_dvd__power,axiom,
! [N2: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_6208_dvd__power,axiom,
! [N2: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N2 )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).
% dvd_power
thf(fact_6209_gbinomial__Suc__Suc,axiom,
! [A2: rat,K3: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) )
= ( plus_plus_rat @ ( gbinomial_rat @ A2 @ K3 ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_6210_one__le__power,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_6211_one__le__power,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_6212_one__le__power,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_6213_one__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_6214_one__le__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ).
% one_le_power
thf(fact_6215_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural ) ).
% not_is_unit_0
thf(fact_6216_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_6217_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_6218_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% not_is_unit_0
thf(fact_6219_left__right__inverse__power,axiom,
! [X: code_integer,Y: code_integer,N2: nat] :
( ( ( times_3573771949741848930nteger @ X @ Y )
= one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
= one_one_Code_integer ) ) ).
% left_right_inverse_power
thf(fact_6220_left__right__inverse__power,axiom,
! [X: code_natural,Y: code_natural,N2: nat] :
( ( ( times_2397367101498566445atural @ X @ Y )
= one_one_Code_natural )
=> ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ X @ N2 ) @ ( power_7079662738309270450atural @ Y @ N2 ) )
= one_one_Code_natural ) ) ).
% left_right_inverse_power
thf(fact_6221_left__right__inverse__power,axiom,
! [X: assn,Y: assn,N2: nat] :
( ( ( times_times_assn @ X @ Y )
= one_one_assn )
=> ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ ( power_power_assn @ Y @ N2 ) )
= one_one_assn ) ) ).
% left_right_inverse_power
thf(fact_6222_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N2: nat] :
( ( ( times_times_rat @ X @ Y )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_6223_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N2: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_6224_left__right__inverse__power,axiom,
! [X: int,Y: int,N2: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_6225_power__Suc2,axiom,
! [A2: assn,N2: nat] :
( ( power_power_assn @ A2 @ ( suc @ N2 ) )
= ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ A2 ) ) ).
% power_Suc2
thf(fact_6226_power__Suc2,axiom,
! [A2: rat,N2: nat] :
( ( power_power_rat @ A2 @ ( suc @ N2 ) )
= ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ A2 ) ) ).
% power_Suc2
thf(fact_6227_power__Suc2,axiom,
! [A2: nat,N2: nat] :
( ( power_power_nat @ A2 @ ( suc @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ A2 ) ) ).
% power_Suc2
thf(fact_6228_power__Suc2,axiom,
! [A2: int,N2: nat] :
( ( power_power_int @ A2 @ ( suc @ N2 ) )
= ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ A2 ) ) ).
% power_Suc2
thf(fact_6229_power__Suc,axiom,
! [A2: assn,N2: nat] :
( ( power_power_assn @ A2 @ ( suc @ N2 ) )
= ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% power_Suc
thf(fact_6230_power__Suc,axiom,
! [A2: rat,N2: nat] :
( ( power_power_rat @ A2 @ ( suc @ N2 ) )
= ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% power_Suc
thf(fact_6231_power__Suc,axiom,
! [A2: nat,N2: nat] :
( ( power_power_nat @ A2 @ ( suc @ N2 ) )
= ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% power_Suc
thf(fact_6232_power__Suc,axiom,
! [A2: int,N2: nat] :
( ( power_power_int @ A2 @ ( suc @ N2 ) )
= ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ).
% power_Suc
thf(fact_6233_unit__mult__right__cancel,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ B @ A2 )
= ( times_3573771949741848930nteger @ C @ A2 ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_6234_unit__mult__right__cancel,axiom,
! [A2: code_natural,B: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
=> ( ( ( times_2397367101498566445atural @ B @ A2 )
= ( times_2397367101498566445atural @ C @ A2 ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_6235_unit__mult__right__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A2 )
= ( times_times_nat @ C @ A2 ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_6236_unit__mult__right__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ B @ A2 )
= ( times_times_int @ C @ A2 ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_6237_unit__mult__left__cancel,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ A2 @ B )
= ( times_3573771949741848930nteger @ A2 @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_6238_unit__mult__left__cancel,axiom,
! [A2: code_natural,B: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
=> ( ( ( times_2397367101498566445atural @ A2 @ B )
= ( times_2397367101498566445atural @ A2 @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_6239_unit__mult__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( ( times_times_nat @ A2 @ B )
= ( times_times_nat @ A2 @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_6240_unit__mult__left__cancel,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( ( times_times_int @ A2 @ B )
= ( times_times_int @ A2 @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_6241_mult__unit__dvd__iff_H,axiom,
! [A2: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ C )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_6242_mult__unit__dvd__iff_H,axiom,
! [A2: code_natural,B: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ C )
= ( dvd_dvd_Code_natural @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_6243_mult__unit__dvd__iff_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_6244_mult__unit__dvd__iff_H,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_6245_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_6246_dvd__mult__unit__iff_H,axiom,
! [B: code_natural,A2: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ B @ C ) )
= ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_6247_dvd__mult__unit__iff_H,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_6248_dvd__mult__unit__iff_H,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_6249_mult__unit__dvd__iff,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ C )
= ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_6250_mult__unit__dvd__iff,axiom,
! [B: code_natural,A2: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ C )
= ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_6251_mult__unit__dvd__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_6252_mult__unit__dvd__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_6253_dvd__mult__unit__iff,axiom,
! [B: code_integer,A2: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_6254_dvd__mult__unit__iff,axiom,
! [B: code_natural,A2: code_natural,C: code_natural] :
( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
=> ( ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ C @ B ) )
= ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_6255_dvd__mult__unit__iff,axiom,
! [B: nat,A2: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_6256_dvd__mult__unit__iff,axiom,
! [B: int,A2: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A2 @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_6257_is__unit__mult__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
& ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% is_unit_mult_iff
thf(fact_6258_is__unit__mult__iff,axiom,
! [A2: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural )
= ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
& ( dvd_dvd_Code_natural @ B @ one_one_Code_natural ) ) ) ).
% is_unit_mult_iff
thf(fact_6259_is__unit__mult__iff,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A2 @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_6260_is__unit__mult__iff,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A2 @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_6261_power__0,axiom,
! [A2: assn] :
( ( power_power_assn @ A2 @ zero_zero_nat )
= one_one_assn ) ).
% power_0
thf(fact_6262_power__0,axiom,
! [A2: code_integer] :
( ( power_8256067586552552935nteger @ A2 @ zero_zero_nat )
= one_one_Code_integer ) ).
% power_0
thf(fact_6263_power__0,axiom,
! [A2: code_natural] :
( ( power_7079662738309270450atural @ A2 @ zero_zero_nat )
= one_one_Code_natural ) ).
% power_0
thf(fact_6264_power__0,axiom,
! [A2: int] :
( ( power_power_int @ A2 @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_6265_power__0,axiom,
! [A2: nat] :
( ( power_power_nat @ A2 @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_6266_power__add,axiom,
! [A2: assn,M4: nat,N2: nat] :
( ( power_power_assn @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
= ( times_times_assn @ ( power_power_assn @ A2 @ M4 ) @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% power_add
thf(fact_6267_power__add,axiom,
! [A2: rat,M4: nat,N2: nat] :
( ( power_power_rat @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
= ( times_times_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% power_add
thf(fact_6268_power__add,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( power_power_nat @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
= ( times_times_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% power_add
thf(fact_6269_power__add,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( power_power_int @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
= ( times_times_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) ) ) ).
% power_add
thf(fact_6270_gbinomial__addition__formula,axiom,
! [A2: rat,K3: nat] :
( ( gbinomial_rat @ A2 @ ( suc @ K3 ) )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% gbinomial_addition_formula
thf(fact_6271_power__le__one,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
=> ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
=> ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ one_one_Code_natural ) ) ) ).
% power_le_one
thf(fact_6272_power__le__one,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ one_one_Code_integer ) ) ) ).
% power_le_one
thf(fact_6273_power__le__one,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N2 ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_6274_power__le__one,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_6275_power__le__one,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_6276_power__less__power__Suc,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_6277_power__less__power__Suc,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_6278_power__less__power__Suc,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_6279_power__less__power__Suc,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_6280_power__less__power__Suc,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ord_less_rat @ ( power_power_rat @ A2 @ N2 ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% power_less_power_Suc
thf(fact_6281_power__gt1__lemma,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_6282_power__gt1__lemma,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_6283_power__gt1__lemma,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_6284_power__gt1__lemma,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_6285_power__gt1__lemma,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% power_gt1_lemma
thf(fact_6286_unity__coeff__ex,axiom,
! [P: code_integer > $o,L: code_integer] :
( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
= ( ? [X2: code_integer] :
( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_6287_unity__coeff__ex,axiom,
! [P: rat > $o,L: rat] :
( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
= ( ? [X2: rat] :
( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_6288_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
= ( ? [X2: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_6289_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
= ( ? [X2: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_6290_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N2 )
= one_one_Code_natural ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N2 )
= zero_z2226904508553997617atural ) ) ) ).
% power_0_left
thf(fact_6291_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
= one_one_Code_integer ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
= zero_z3403309356797280102nteger ) ) ) ).
% power_0_left
thf(fact_6292_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N2 )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_6293_power__0__left,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_6294_unit__dvdE,axiom,
! [A2: code_natural,B: code_natural] :
( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
=> ~ ( ( A2 != zero_z2226904508553997617atural )
=> ! [C2: code_natural] :
( B
!= ( times_2397367101498566445atural @ A2 @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_6295_unit__dvdE,axiom,
! [A2: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
=> ~ ( ( A2 != zero_z3403309356797280102nteger )
=> ! [C2: code_integer] :
( B
!= ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_6296_unit__dvdE,axiom,
! [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ~ ( ( A2 != zero_zero_nat )
=> ! [C2: nat] :
( B
!= ( times_times_nat @ A2 @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_6297_unit__dvdE,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ~ ( ( A2 != zero_zero_int )
=> ! [C2: int] :
( B
!= ( times_times_int @ A2 @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_6298_power__gt1,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_6299_power__gt1,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_6300_power__gt1,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_6301_power__gt1,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_6302_power__gt1,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) ) ) ).
% power_gt1
thf(fact_6303_power__strict__increasing,axiom,
! [N2: nat,N: nat,A2: code_natural] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).
% power_strict_increasing
thf(fact_6304_power__strict__increasing,axiom,
! [N2: nat,N: nat,A2: nat] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_strict_increasing
thf(fact_6305_power__strict__increasing,axiom,
! [N2: nat,N: nat,A2: int] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_int @ one_one_int @ A2 )
=> ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_strict_increasing
thf(fact_6306_power__strict__increasing,axiom,
! [N2: nat,N: nat,A2: code_integer] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).
% power_strict_increasing
thf(fact_6307_power__strict__increasing,axiom,
! [N2: nat,N: nat,A2: rat] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ord_less_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% power_strict_increasing
thf(fact_6308_power__less__imp__less__exp,axiom,
! [A2: code_natural,M4: nat,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ M4 ) @ ( power_7079662738309270450atural @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_6309_power__less__imp__less__exp,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_6310_power__less__imp__less__exp,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_6311_power__less__imp__less__exp,axiom,
! [A2: code_integer,M4: nat,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ M4 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_6312_power__less__imp__less__exp,axiom,
! [A2: rat,M4: nat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ( ord_less_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) )
=> ( ord_less_nat @ M4 @ N2 ) ) ) ).
% power_less_imp_less_exp
thf(fact_6313_power__increasing,axiom,
! [N2: nat,N: nat,A2: code_integer] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).
% power_increasing
thf(fact_6314_power__increasing,axiom,
! [N2: nat,N: nat,A2: code_natural] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
=> ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).
% power_increasing
thf(fact_6315_power__increasing,axiom,
! [N2: nat,N: nat,A2: rat] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_rat @ one_one_rat @ A2 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% power_increasing
thf(fact_6316_power__increasing,axiom,
! [N2: nat,N: nat,A2: nat] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% power_increasing
thf(fact_6317_power__increasing,axiom,
! [N2: nat,N: nat,A2: int] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% power_increasing
thf(fact_6318_inf__period_I3_J,axiom,
! [D2: code_integer,D3: code_integer,T4: code_integer] :
( ( dvd_dvd_Code_integer @ D2 @ D3 )
=> ! [X6: code_integer,K2: code_integer] :
( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X6 @ T4 ) )
= ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% inf_period(3)
thf(fact_6319_inf__period_I3_J,axiom,
! [D2: rat,D3: rat,T4: rat] :
( ( dvd_dvd_rat @ D2 @ D3 )
=> ! [X6: rat,K2: rat] :
( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T4 ) )
= ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% inf_period(3)
thf(fact_6320_inf__period_I3_J,axiom,
! [D2: int,D3: int,T4: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X6: int,K2: int] :
( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T4 ) )
= ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% inf_period(3)
thf(fact_6321_inf__period_I4_J,axiom,
! [D2: code_integer,D3: code_integer,T4: code_integer] :
( ( dvd_dvd_Code_integer @ D2 @ D3 )
=> ! [X6: code_integer,K2: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X6 @ T4 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% inf_period(4)
thf(fact_6322_inf__period_I4_J,axiom,
! [D2: rat,D3: rat,T4: rat] :
( ( dvd_dvd_rat @ D2 @ D3 )
=> ! [X6: rat,K2: rat] :
( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T4 ) ) )
= ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% inf_period(4)
thf(fact_6323_inf__period_I4_J,axiom,
! [D2: int,D3: int,T4: int] :
( ( dvd_dvd_int @ D2 @ D3 )
=> ! [X6: int,K2: int] :
( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T4 ) ) )
= ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% inf_period(4)
thf(fact_6324_cInf__mono,axiom,
! [B4: set_o,A: set_o] :
( ( B4 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ B4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A )
& ( ord_less_eq_o @ X6 @ B2 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6325_cInf__mono,axiom,
! [B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit8399017556247434269_nat_o @ A )
=> ( ! [B2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ B2 @ B4 )
=> ? [X6: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X6 @ A )
& ( ord_le729326519192465773_nat_o @ X6 @ B2 ) ) )
=> ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6326_cInf__mono,axiom,
! [B4: set_filter_nat,A: set_filter_nat] :
( ( B4 != bot_bo498966703094740906er_nat )
=> ( ( condit3133331395780078992er_nat @ A )
=> ( ! [B2: filter_nat] :
( ( member_filter_nat @ B2 @ B4 )
=> ? [X6: filter_nat] :
( ( member_filter_nat @ X6 @ A )
& ( ord_le2510731241096832064er_nat @ X6 @ B2 ) ) )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple6612786467464916249er_nat @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6327_cInf__mono,axiom,
! [B4: set_int,A: set_int] :
( ( B4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ A )
=> ( ! [B2: int] :
( ( member_int @ B2 @ B4 )
=> ? [X6: int] :
( ( member_int @ X6 @ A )
& ( ord_less_eq_int @ X6 @ B2 ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6328_cInf__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ X6 @ B2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6329_cInf__mono,axiom,
! [B4: set_set_nat,A: set_set_nat] :
( ( B4 != bot_bot_set_set_nat )
=> ( ( condit68592940725977398et_nat @ A )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ B4 )
=> ? [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ X6 @ B2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6330_cInf__mono,axiom,
! [B4: set_Product_unit,A: set_Product_unit] :
( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ A )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ B4 )
=> ? [X6: product_unit] :
( ( member_Product_unit @ X6 @ A )
& ( ord_le3221252021190050221t_unit @ X6 @ B2 ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_6331_le__cInf__iff,axiom,
! [S: set_o,A2: $o] :
( ( S != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ S )
=> ( ( ord_less_eq_o @ A2 @ ( complete_Inf_Inf_o @ S ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ S )
=> ( ord_less_eq_o @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6332_le__cInf__iff,axiom,
! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( S != bot_bo7824918357723582233_nat_o )
=> ( ( condit8399017556247434269_nat_o @ S )
=> ( ( ord_le729326519192465773_nat_o @ A2 @ ( comple90455603224345172_nat_o @ S ) )
= ( ! [X2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X2 @ S )
=> ( ord_le729326519192465773_nat_o @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6333_le__cInf__iff,axiom,
! [S: set_filter_nat,A2: filter_nat] :
( ( S != bot_bo498966703094740906er_nat )
=> ( ( condit3133331395780078992er_nat @ S )
=> ( ( ord_le2510731241096832064er_nat @ A2 @ ( comple6612786467464916249er_nat @ S ) )
= ( ! [X2: filter_nat] :
( ( member_filter_nat @ X2 @ S )
=> ( ord_le2510731241096832064er_nat @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6334_le__cInf__iff,axiom,
! [S: set_int,A2: int] :
( ( S != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ S )
=> ( ( ord_less_eq_int @ A2 @ ( complete_Inf_Inf_int @ S ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ S )
=> ( ord_less_eq_int @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6335_le__cInf__iff,axiom,
! [S: set_nat,A2: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ S )
=> ( ( ord_less_eq_nat @ A2 @ ( complete_Inf_Inf_nat @ S ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ord_less_eq_nat @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6336_le__cInf__iff,axiom,
! [S: set_set_nat,A2: set_nat] :
( ( S != bot_bot_set_set_nat )
=> ( ( condit68592940725977398et_nat @ S )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7806235888213564991et_nat @ S ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ S )
=> ( ord_less_eq_set_nat @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6337_le__cInf__iff,axiom,
! [S: set_Product_unit,A2: product_unit] :
( ( S != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ S )
=> ( ( ord_le3221252021190050221t_unit @ A2 @ ( comple2584293577114468500t_unit @ S ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ S )
=> ( ord_le3221252021190050221t_unit @ A2 @ X2 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6338_cInf__less__iff,axiom,
! [X4: set_int,Y: int] :
( ( X4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ X4 )
=> ( ( ord_less_int @ ( complete_Inf_Inf_int @ X4 ) @ Y )
= ( ? [X2: int] :
( ( member_int @ X2 @ X4 )
& ( ord_less_int @ X2 @ Y ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_6339_cInf__less__iff,axiom,
! [X4: set_nat,Y: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X4 )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X4 ) @ Y )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( ord_less_nat @ X2 @ Y ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_6340_cInf__less__iff,axiom,
! [X4: set_Product_unit,Y: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ X4 )
=> ( ( ord_le361264281704409273t_unit @ ( comple2584293577114468500t_unit @ X4 ) @ Y )
= ( ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ X4 )
& ( ord_le361264281704409273t_unit @ X2 @ Y ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_6341_dvd__mult__cancel,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_6342_nat__mult__dvd__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_6343_power__Suc__less,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
=> ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
=> ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_6344_power__Suc__less,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_6345_power__Suc__less,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_6346_power__Suc__less,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_6347_power__Suc__less,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ A2 @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% power_Suc_less
thf(fact_6348_power__Suc__le__self,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
=> ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
=> ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_6349_power__Suc__le__self,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_6350_power__Suc__le__self,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_6351_power__Suc__le__self,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_6352_power__Suc__le__self,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% power_Suc_le_self
thf(fact_6353_power__Suc__less__one,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
=> ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
=> ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) @ one_one_Code_natural ) ) ) ).
% power_Suc_less_one
thf(fact_6354_power__Suc__less__one,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_6355_power__Suc__less__one,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_6356_power__Suc__less__one,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ) ).
% power_Suc_less_one
thf(fact_6357_power__Suc__less__one,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ A2 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_6358_power__strict__decreasing,axiom,
! [N2: nat,N: nat,A2: code_natural] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
=> ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
=> ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_6359_power__strict__decreasing,axiom,
! [N2: nat,N: nat,A2: nat] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ A2 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_6360_power__strict__decreasing,axiom,
! [N2: nat,N: nat,A2: int] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ A2 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_6361_power__strict__decreasing,axiom,
! [N2: nat,N: nat,A2: code_integer] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
=> ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_6362_power__strict__decreasing,axiom,
! [N2: nat,N: nat,A2: rat] :
( ( ord_less_nat @ N2 @ N )
=> ( ( ord_less_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_rat @ A2 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_6363_power__decreasing,axiom,
! [N2: nat,N: nat,A2: code_natural] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
=> ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
=> ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_6364_power__decreasing,axiom,
! [N2: nat,N: nat,A2: code_integer] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
=> ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_6365_power__decreasing,axiom,
! [N2: nat,N: nat,A2: rat] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
=> ( ( ord_less_eq_rat @ A2 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_6366_power__decreasing,axiom,
! [N2: nat,N: nat,A2: nat] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_6367_power__decreasing,axiom,
! [N2: nat,N: nat,A2: int] :
( ( ord_less_eq_nat @ N2 @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ A2 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% power_decreasing
thf(fact_6368_power__le__imp__le__exp,axiom,
! [A2: code_natural,M4: nat,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ M4 ) @ ( power_7079662738309270450atural @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_6369_power__le__imp__le__exp,axiom,
! [A2: code_integer,M4: nat,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ M4 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_6370_power__le__imp__le__exp,axiom,
! [A2: rat,M4: nat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_6371_power__le__imp__le__exp,axiom,
! [A2: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_6372_power__le__imp__le__exp,axiom,
! [A2: int,M4: nat,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% power_le_imp_le_exp
thf(fact_6373_self__le__power,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_le3102999989581377725nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_6374_self__le__power,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_le1926595141338095240atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_6375_self__le__power,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_6376_self__le__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_6377_self__le__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_eq_int @ one_one_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% self_le_power
thf(fact_6378_one__less__power,axiom,
! [A2: code_natural,N2: nat] :
( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_6379_one__less__power,axiom,
! [A2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_6380_one__less__power,axiom,
! [A2: int,N2: nat] :
( ( ord_less_int @ one_one_int @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_6381_one__less__power,axiom,
! [A2: code_integer,N2: nat] :
( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_6382_one__less__power,axiom,
! [A2: rat,N2: nat] :
( ( ord_less_rat @ one_one_rat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% one_less_power
thf(fact_6383_cINF__mono,axiom,
! [B4: set_Product_unit,F: nat > nat,A: set_nat,G: product_unit > nat] :
( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ! [M2: product_unit] :
( ( member_Product_unit @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6384_cINF__mono,axiom,
! [B4: set_o,F: nat > nat,A: set_nat,G: $o > nat] :
( ( B4 != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ! [M2: $o] :
( ( member_o @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6385_cINF__mono,axiom,
! [B4: set_nat,F: nat > nat,A: set_nat,G: nat > nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ! [M2: nat] :
( ( member_nat @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6386_cINF__mono,axiom,
! [B4: set_int,F: nat > nat,A: set_nat,G: int > nat] :
( ( B4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ! [M2: int] :
( ( member_int @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6387_cINF__mono,axiom,
! [B4: set_Pr4532377907799695533_nat_o,F: nat > nat,A: set_nat,G: ( produc3658429121746597890et_nat > $o ) > nat] :
( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ! [M2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_2174381888560928856_o_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6388_cINF__mono,axiom,
! [B4: set_Product_unit,F: nat > set_nat,A: set_nat,G: product_unit > set_nat] :
( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ! [M2: product_unit] :
( ( member_Product_unit @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6389_cINF__mono,axiom,
! [B4: set_o,F: nat > set_nat,A: set_nat,G: $o > set_nat] :
( ( B4 != bot_bot_set_o )
=> ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ! [M2: $o] :
( ( member_o @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6390_cINF__mono,axiom,
! [B4: set_nat,F: nat > set_nat,A: set_nat,G: nat > set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ! [M2: nat] :
( ( member_nat @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6391_cINF__mono,axiom,
! [B4: set_int,F: nat > set_nat,A: set_nat,G: int > set_nat] :
( ( B4 != bot_bot_set_int )
=> ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ! [M2: int] :
( ( member_int @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6392_cINF__mono,axiom,
! [B4: set_Pr4532377907799695533_nat_o,F: nat > set_nat,A: set_nat,G: ( produc3658429121746597890et_nat > $o ) > set_nat] :
( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ! [M2: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ M2 @ B4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_6888641473154130958et_nat @ G @ B4 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6393_le__cINF__iff,axiom,
! [A: set_Product_unit,F: product_unit > int,U: int] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6394_le__cINF__iff,axiom,
! [A: set_o,F: $o > int,U: int] :
( ( A != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int2 @ F @ A ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6395_le__cINF__iff,axiom,
! [A: set_nat,F: nat > int,U: int] :
( ( A != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int2 @ F @ A ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6396_le__cINF__iff,axiom,
! [A: set_int,F: int > int,U: int] :
( ( A != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int2 @ F @ A ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6397_le__cINF__iff,axiom,
! [A: set_Product_unit,F: product_unit > nat,U: nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6398_le__cINF__iff,axiom,
! [A: set_o,F: $o > nat,U: nat] :
( ( A != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6399_le__cINF__iff,axiom,
! [A: set_nat,F: nat > nat,U: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6400_le__cINF__iff,axiom,
! [A: set_int,F: int > nat,U: nat] :
( ( A != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6401_le__cINF__iff,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6402_le__cINF__iff,axiom,
! [A: set_o,F: $o > product_unit,U: product_unit] :
( ( A != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6403_cInf__superset__mono,axiom,
! [A: set_o,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B4 )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ B4 ) @ ( complete_Inf_Inf_o @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6404_cInf__superset__mono,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit8399017556247434269_nat_o @ B4 )
=> ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
=> ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ B4 ) @ ( comple90455603224345172_nat_o @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6405_cInf__superset__mono,axiom,
! [A: set_filter_nat,B4: set_filter_nat] :
( ( A != bot_bo498966703094740906er_nat )
=> ( ( condit3133331395780078992er_nat @ B4 )
=> ( ( ord_le2426478655948331894er_nat @ A @ B4 )
=> ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ B4 ) @ ( comple6612786467464916249er_nat @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6406_cInf__superset__mono,axiom,
! [A: set_int,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ B4 ) @ ( complete_Inf_Inf_int @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6407_cInf__superset__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ B4 ) @ ( complete_Inf_Inf_nat @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6408_cInf__superset__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit68592940725977398et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ B4 ) @ ( comple7806235888213564991et_nat @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6409_cInf__superset__mono,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ B4 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ B4 ) @ ( comple2584293577114468500t_unit @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6410_cInf__insert,axiom,
! [X4: set_o,A2: $o] :
( ( X4 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ X4 )
=> ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ X4 ) )
= ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ) ).
% cInf_insert
thf(fact_6411_cInf__insert,axiom,
! [X4: set_int,A2: int] :
( ( X4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ X4 )
=> ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
= ( inf_inf_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ) ).
% cInf_insert
thf(fact_6412_cInf__insert,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit8399017556247434269_nat_o @ X4 )
=> ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= ( inf_in1318976480646536635_nat_o @ A2 @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ) ).
% cInf_insert
thf(fact_6413_cInf__insert,axiom,
! [X4: set_nat,A2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X4 )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
= ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ) ).
% cInf_insert
thf(fact_6414_cInf__insert,axiom,
! [X4: set_set_nat,A2: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ( condit68592940725977398et_nat @ X4 )
=> ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
= ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ) ).
% cInf_insert
thf(fact_6415_cInf__insert,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ X4 )
=> ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
= ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ) ).
% cInf_insert
thf(fact_6416_cInf__insert__If,axiom,
! [X4: set_o,A2: $o] :
( ( condit5413489452508810728elow_o @ X4 )
=> ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ X4 ) )
= ( ( ( X4 = bot_bot_set_o )
=> A2 )
& ( ( X4 != bot_bot_set_o )
=> ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6417_cInf__insert__If,axiom,
! [X4: set_int,A2: int] :
( ( condit1735850657277959132ow_int @ X4 )
=> ( ( ( X4 = bot_bot_set_int )
=> ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bot_set_int )
=> ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
= ( inf_inf_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6418_cInf__insert__If,axiom,
! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
( ( condit8399017556247434269_nat_o @ X4 )
=> ( ( ( X4 = bot_bo7824918357723582233_nat_o )
=> ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bo7824918357723582233_nat_o )
=> ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
= ( inf_in1318976480646536635_nat_o @ A2 @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6419_cInf__insert__If,axiom,
! [X4: set_nat,A2: nat] :
( ( condit1738341127787009408ow_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
= ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6420_cInf__insert__If,axiom,
! [X4: set_set_nat,A2: set_nat] :
( ( condit68592940725977398et_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_set_nat )
=> ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bot_set_set_nat )
=> ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
= ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6421_cInf__insert__If,axiom,
! [X4: set_Product_unit,A2: product_unit] :
( ( condit9161856309968323677t_unit @ X4 )
=> ( ( ( X4 = bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
= A2 ) )
& ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
= ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6422_cInf__union__distrib,axiom,
! [A: set_o,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ A )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit5413489452508810728elow_o @ B4 )
=> ( ( complete_Inf_Inf_o @ ( sup_sup_set_o @ A @ B4 ) )
= ( inf_inf_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6423_cInf__union__distrib,axiom,
! [A: set_int,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ A )
=> ( ( B4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ B4 )
=> ( ( complete_Inf_Inf_int @ ( sup_sup_set_int @ A @ B4 ) )
= ( inf_inf_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6424_cInf__union__distrib,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( A != bot_bo7824918357723582233_nat_o )
=> ( ( condit8399017556247434269_nat_o @ A )
=> ( ( B4 != bot_bo7824918357723582233_nat_o )
=> ( ( condit8399017556247434269_nat_o @ B4 )
=> ( ( comple90455603224345172_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
= ( inf_in1318976480646536635_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6425_cInf__union__distrib,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B4 )
=> ( ( complete_Inf_Inf_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6426_cInf__union__distrib,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit68592940725977398et_nat @ A )
=> ( ( B4 != bot_bot_set_set_nat )
=> ( ( condit68592940725977398et_nat @ B4 )
=> ( ( comple7806235888213564991et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6427_cInf__union__distrib,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ A )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ B4 )
=> ( ( comple2584293577114468500t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6428_dbl__inc__def,axiom,
( neg_nu5831290666863070958nteger
= ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).
% dbl_inc_def
thf(fact_6429_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_6430_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_6431_dvd__mult__cancel1,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M4 @ N2 ) @ M4 )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_6432_dvd__mult__cancel2,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M4 ) @ M4 )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_6433_dvd__minus__add,axiom,
! [Q4: nat,N2: nat,R2: nat,M4: nat] :
( ( ord_less_eq_nat @ Q4 @ N2 )
=> ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M4 ) )
=> ( ( dvd_dvd_nat @ M4 @ ( minus_minus_nat @ N2 @ Q4 ) )
= ( dvd_dvd_nat @ M4 @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M4 ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_6434_cINF__superset__mono,axiom,
! [A: set_Product_unit,G: product_unit > int,B4: set_Product_unit,F: product_unit > int] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ G @ B4 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ B4 )
=> ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6435_cINF__superset__mono,axiom,
! [A: set_o,G: $o > int,B4: set_o,F: $o > int] :
( ( A != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_o_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6436_cINF__superset__mono,axiom,
! [A: set_int,G: int > int,B4: set_int,F: int > int] :
( ( A != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B4 )
=> ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6437_cINF__superset__mono,axiom,
! [A: set_nat,G: nat > int,B4: set_nat,F: nat > int] :
( ( A != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6438_cINF__superset__mono,axiom,
! [A: set_Product_unit,G: product_unit > nat,B4: set_Product_unit,F: product_unit > nat] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ G @ B4 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ B4 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6439_cINF__superset__mono,axiom,
! [A: set_o,G: $o > nat,B4: set_o,F: $o > nat] :
( ( A != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6440_cINF__superset__mono,axiom,
! [A: set_int,G: int > nat,B4: set_int,F: int > nat] :
( ( A != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B4 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6441_cINF__superset__mono,axiom,
! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6442_cINF__superset__mono,axiom,
! [A: set_Product_unit,G: product_unit > product_unit,B4: set_Product_unit,F: product_unit > product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6443_cINF__superset__mono,axiom,
! [A: set_o,G: $o > product_unit,B4: set_o,F: $o > product_unit] :
( ( A != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ G @ B4 ) )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ G @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6444_less__eq__cInf__inter,axiom,
! [A: set_o,B4: set_o] :
( ( condit5413489452508810728elow_o @ A )
=> ( ( condit5413489452508810728elow_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
!= bot_bot_set_o )
=> ( ord_less_eq_o @ ( inf_inf_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6445_less__eq__cInf__inter,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( condit8399017556247434269_nat_o @ A )
=> ( ( condit8399017556247434269_nat_o @ B4 )
=> ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
!= bot_bo7824918357723582233_nat_o )
=> ( ord_le729326519192465773_nat_o @ ( inf_in1318976480646536635_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) @ ( comple90455603224345172_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6446_less__eq__cInf__inter,axiom,
! [A: set_filter_nat,B4: set_filter_nat] :
( ( condit3133331395780078992er_nat @ A )
=> ( ( condit3133331395780078992er_nat @ B4 )
=> ( ( ( inf_in7395780696554941096er_nat @ A @ B4 )
!= bot_bo498966703094740906er_nat )
=> ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple6612786467464916249er_nat @ B4 ) ) @ ( comple6612786467464916249er_nat @ ( inf_in7395780696554941096er_nat @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6447_less__eq__cInf__inter,axiom,
! [A: set_int,B4: set_int] :
( ( condit1735850657277959132ow_int @ A )
=> ( ( condit1735850657277959132ow_int @ B4 )
=> ( ( ( inf_inf_set_int @ A @ B4 )
!= bot_bot_set_int )
=> ( ord_less_eq_int @ ( inf_inf_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) @ ( complete_Inf_Inf_int @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6448_less__eq__cInf__inter,axiom,
! [A: set_nat,B4: set_nat] :
( ( condit1738341127787009408ow_nat @ A )
=> ( ( condit1738341127787009408ow_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6449_less__eq__cInf__inter,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( condit68592940725977398et_nat @ A )
=> ( ( condit68592940725977398et_nat @ B4 )
=> ( ( ( inf_inf_set_set_nat @ A @ B4 )
!= bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) @ ( comple7806235888213564991et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6450_less__eq__cInf__inter,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( condit9161856309968323677t_unit @ A )
=> ( ( condit9161856309968323677t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
!= bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6451_cINF__insert,axiom,
! [A: set_Product_unit,F: product_unit > nat,A2: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6452_cINF__insert,axiom,
! [A: set_o,F: $o > nat,A2: $o] :
( ( A != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ ( insert_o @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6453_cINF__insert,axiom,
! [A: set_nat,F: nat > nat,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6454_cINF__insert,axiom,
! [A: set_int,F: int > nat,A2: int] :
( ( A != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
=> ( ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ ( insert_int @ A2 @ A ) ) )
= ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6455_cINF__insert,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,A2: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A2 @ A ) ) )
= ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6456_cINF__insert,axiom,
! [A: set_o,F: $o > product_unit,A2: $o] :
( ( A != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ ( insert_o @ A2 @ A ) ) )
= ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6457_cINF__insert,axiom,
! [A: set_nat,F: nat > product_unit,A2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ A ) ) )
= ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6458_cINF__insert,axiom,
! [A: set_int,F: int > product_unit,A2: int] :
( ( A != bot_bot_set_int )
=> ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ A ) ) )
= ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6459_cINF__insert,axiom,
! [A: set_Product_unit,F: product_unit > set_nat,A2: product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
= ( inf_inf_set_nat @ ( F @ A2 ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6460_cINF__insert,axiom,
! [A: set_o,F: $o > set_nat,A2: $o] :
( ( A != bot_bot_set_o )
=> ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ A ) ) )
= ( inf_inf_set_nat @ ( F @ A2 ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6461_power__eq__if,axiom,
( power_8256067586552552935nteger
= ( ^ [P5: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P5 @ ( power_8256067586552552935nteger @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6462_power__eq__if,axiom,
( power_7079662738309270450atural
= ( ^ [P5: code_natural,M3: nat] : ( if_Code_natural @ ( M3 = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ P5 @ ( power_7079662738309270450atural @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6463_power__eq__if,axiom,
( power_power_assn
= ( ^ [P5: assn,M3: nat] : ( if_assn @ ( M3 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P5 @ ( power_power_assn @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6464_power__eq__if,axiom,
( power_power_rat
= ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6465_power__eq__if,axiom,
( power_power_nat
= ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6466_power__eq__if,axiom,
( power_power_int
= ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_6467_power__minus__mult,axiom,
! [N2: nat,A2: assn] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_assn @ ( power_power_assn @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
= ( power_power_assn @ A2 @ N2 ) ) ) ).
% power_minus_mult
thf(fact_6468_power__minus__mult,axiom,
! [N2: nat,A2: rat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
= ( power_power_rat @ A2 @ N2 ) ) ) ).
% power_minus_mult
thf(fact_6469_power__minus__mult,axiom,
! [N2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
= ( power_power_nat @ A2 @ N2 ) ) ) ).
% power_minus_mult
thf(fact_6470_power__minus__mult,axiom,
! [N2: nat,A2: int] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
= ( power_power_int @ A2 @ N2 ) ) ) ).
% power_minus_mult
thf(fact_6471_cINF__union,axiom,
! [A: set_Product_unit,F: product_unit > nat,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ B4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6472_cINF__union,axiom,
! [A: set_o,F: $o > nat,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ B4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6473_cINF__union,axiom,
! [A: set_nat,F: nat > nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ B4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6474_cINF__union,axiom,
! [A: set_int,F: int > nat,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ B4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6475_cINF__union,axiom,
! [A: set_Product_unit,F: product_unit > product_unit,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6476_cINF__union,axiom,
! [A: set_o,F: $o > product_unit,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ B4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6477_cINF__union,axiom,
! [A: set_nat,F: nat > product_unit,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6478_cINF__union,axiom,
! [A: set_int,F: int > product_unit,B4: set_int] :
( ( A != bot_bot_set_int )
=> ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
=> ( ( B4 != bot_bot_set_int )
=> ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6479_cINF__union,axiom,
! [A: set_Product_unit,F: product_unit > set_nat,B4: set_Product_unit] :
( ( A != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6480_cINF__union,axiom,
! [A: set_o,F: $o > set_nat,B4: set_o] :
( ( A != bot_bot_set_o )
=> ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A ) )
=> ( ( B4 != bot_bot_set_o )
=> ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ B4 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6481_sum_Oinsert,axiom,
! [A: set_int,X: int,G: int > code_integer] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7873554091576472773nteger @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6482_sum_Oinsert,axiom,
! [A: set_o,X: $o,G: $o > code_integer] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6483_sum_Oinsert,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6484_sum_Oinsert,axiom,
! [A: set_nat,X: nat,G: nat > code_integer] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7501900531339628137nteger @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6485_sum_Oinsert,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6486_sum_Oinsert,axiom,
! [A: set_int,X: int,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6487_sum_Oinsert,axiom,
! [A: set_o,X: $o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6488_sum_Oinsert,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6489_sum_Oinsert,axiom,
! [A: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6490_sum_Oinsert,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A ) ) ) ) ) ).
% sum.insert
thf(fact_6491_sum_Oempty,axiom,
! [G: product_unit > nat] :
( ( groups8766797177574957219it_nat @ G @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% sum.empty
thf(fact_6492_sum_Oempty,axiom,
! [G: product_unit > int] :
( ( groups8764306707065906943it_int @ G @ bot_bo3957492148770167129t_unit )
= zero_zero_int ) ).
% sum.empty
thf(fact_6493_sum_Oempty,axiom,
! [G: product_unit > code_integer] :
( ( groups5393439102872367408nteger @ G @ bot_bo3957492148770167129t_unit )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_6494_sum_Oempty,axiom,
! [G: $o > nat] :
( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
= zero_zero_nat ) ).
% sum.empty
thf(fact_6495_sum_Oempty,axiom,
! [G: $o > int] :
( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
= zero_zero_int ) ).
% sum.empty
thf(fact_6496_sum_Oempty,axiom,
! [G: $o > code_integer] :
( ( groups4406642042086082107nteger @ G @ bot_bot_set_o )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_6497_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_6498_sum_Oempty,axiom,
! [G: nat > code_integer] :
( ( groups7501900531339628137nteger @ G @ bot_bot_set_nat )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_6499_sum_Oempty,axiom,
! [G: int > nat] :
( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
= zero_zero_nat ) ).
% sum.empty
thf(fact_6500_sum_Oempty,axiom,
! [G: int > code_integer] :
( ( groups7873554091576472773nteger @ G @ bot_bot_set_int )
= zero_z3403309356797280102nteger ) ).
% sum.empty
thf(fact_6501_member__le__sum,axiom,
! [I2: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
( ( member_Code_integer @ I2 @ A )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
=> ( ( finite6017078050557962740nteger @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups879477027807139574nteger @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6502_member__le__sum,axiom,
! [I2: product_unit,A: set_Product_unit,F: product_unit > code_integer] :
( ( member_Product_unit @ I2 @ A )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups5393439102872367408nteger @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6503_member__le__sum,axiom,
! [I2: $o,A: set_o,F: $o > code_integer] :
( ( member_o @ I2 @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ A @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
=> ( ( finite_finite_o @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups4406642042086082107nteger @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6504_member__le__sum,axiom,
! [I2: int,A: set_int,F: int > code_integer] :
( ( member_int @ I2 @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ A @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
=> ( ( finite_finite_int @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7873554091576472773nteger @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6505_member__le__sum,axiom,
! [I2: nat,A: set_nat,F: nat > code_integer] :
( ( member_nat @ I2 @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
=> ( ( finite_finite_nat @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7501900531339628137nteger @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6506_member__le__sum,axiom,
! [I2: code_integer,A: set_Code_integer,F: code_integer > rat] :
( ( member_Code_integer @ I2 @ A )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( finite6017078050557962740nteger @ A )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups6602215022474089585er_rat @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6507_member__le__sum,axiom,
! [I2: product_unit,A: set_Product_unit,F: product_unit > rat] :
( ( member_Product_unit @ I2 @ A )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups8131667117488461483it_rat @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6508_member__le__sum,axiom,
! [I2: $o,A: set_o,F: $o > rat] :
( ( member_o @ I2 @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ A @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( finite_finite_o @ A )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups7872700643590313910_o_rat @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6509_member__le__sum,axiom,
! [I2: int,A: set_int,F: int > rat] :
( ( member_int @ I2 @ A )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ A @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( finite_finite_int @ A )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups3906332499630173760nt_rat @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6510_member__le__sum,axiom,
! [I2: nat,A: set_nat,F: nat > rat] :
( ( member_nat @ I2 @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups2906978787729119204at_rat @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_6511_sum_Ounion__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( groups879477027807139574nteger @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( plus_p5714425477246183910nteger @ ( groups879477027807139574nteger @ G @ A ) @ ( groups879477027807139574nteger @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6512_sum_Ounion__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A ) @ ( groups6602215022474089585er_rat @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6513_sum_Ounion__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A ) @ ( groups7237345082560585321er_nat @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6514_sum_Ounion__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A ) @ ( groups7234854612051535045er_int @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6515_sum_Ounion__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( groups5393439102872367408nteger @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( plus_p5714425477246183910nteger @ ( groups5393439102872367408nteger @ G @ A ) @ ( groups5393439102872367408nteger @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6516_sum_Ounion__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( groups8131667117488461483it_rat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( plus_plus_rat @ ( groups8131667117488461483it_rat @ G @ A ) @ ( groups8131667117488461483it_rat @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6517_sum_Ounion__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( groups8766797177574957219it_nat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( plus_plus_nat @ ( groups8766797177574957219it_nat @ G @ A ) @ ( groups8766797177574957219it_nat @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6518_sum_Ounion__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > int] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( groups8764306707065906943it_int @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( plus_plus_int @ ( groups8764306707065906943it_int @ G @ A ) @ ( groups8764306707065906943it_int @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6519_sum_Ounion__disjoint,axiom,
! [A: set_o,B4: set_o,G: $o > code_integer] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
=> ( ( groups4406642042086082107nteger @ G @ ( sup_sup_set_o @ A @ B4 ) )
= ( plus_p5714425477246183910nteger @ ( groups4406642042086082107nteger @ G @ A ) @ ( groups4406642042086082107nteger @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6520_sum_Ounion__disjoint,axiom,
! [A: set_o,B4: set_o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
=> ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A @ B4 ) )
= ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A ) @ ( groups7872700643590313910_o_rat @ G @ B4 ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6521_sum__diff1,axiom,
! [A: set_Code_integer,A2: code_integer,F: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ A2 @ A )
=> ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
= ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_Code_integer @ A2 @ A )
=> ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
= ( groups6602215022474089585er_rat @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6522_sum__diff1,axiom,
! [A: set_Product_unit,A2: product_unit,F: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ A2 @ A )
=> ( ( groups8131667117488461483it_rat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ A )
=> ( ( groups8131667117488461483it_rat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( groups8131667117488461483it_rat @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6523_sum__diff1,axiom,
! [A: set_o,A2: $o,F: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ A2 @ A )
=> ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( groups7872700643590313910_o_rat @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6524_sum__diff1,axiom,
! [A: set_int,A2: int,F: int > rat] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ A2 @ A )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_int @ A2 @ A )
=> ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( groups3906332499630173760nt_rat @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6525_sum__diff1,axiom,
! [A: set_Code_integer,A2: code_integer,F: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ A2 @ A )
=> ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
= ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_Code_integer @ A2 @ A )
=> ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
= ( groups7234854612051535045er_int @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6526_sum__diff1,axiom,
! [A: set_Product_unit,A2: product_unit,F: product_unit > int] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ A2 @ A )
=> ( ( groups8764306707065906943it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ A )
=> ( ( groups8764306707065906943it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( groups8764306707065906943it_int @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6527_sum__diff1,axiom,
! [A: set_o,A2: $o,F: $o > int] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ A2 @ A )
=> ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( minus_minus_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( groups8505340233167759370_o_int @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6528_sum__diff1,axiom,
! [A: set_nat,A2: nat,F: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ A2 @ A )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( groups2906978787729119204at_rat @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6529_sum__diff1,axiom,
! [A: set_nat,A2: nat,F: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ A2 @ A )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( groups3539618377306564664at_int @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6530_sum__diff1,axiom,
! [A: set_int,A2: int,F: int > int] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ A2 @ A )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_int @ A2 @ A )
=> ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( groups4538972089207619220nt_int @ F @ A ) ) ) ) ) ).
% sum_diff1
thf(fact_6531_sum_Oremove,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( groups879477027807139574nteger @ G @ A )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6532_sum_Oremove,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( groups6602215022474089585er_rat @ G @ A )
= ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6533_sum_Oremove,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( groups7237345082560585321er_nat @ G @ A )
= ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6534_sum_Oremove,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( groups7234854612051535045er_int @ G @ A )
= ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6535_sum_Oremove,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( groups5393439102872367408nteger @ G @ A )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6536_sum_Oremove,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( groups8131667117488461483it_rat @ G @ A )
= ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6537_sum_Oremove,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( groups8766797177574957219it_nat @ G @ A )
= ( plus_plus_nat @ ( G @ X ) @ ( groups8766797177574957219it_nat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6538_sum_Oremove,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > int] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( groups8764306707065906943it_int @ G @ A )
= ( plus_plus_int @ ( G @ X ) @ ( groups8764306707065906943it_int @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6539_sum_Oremove,axiom,
! [A: set_o,X: $o,G: $o > code_integer] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( groups4406642042086082107nteger @ G @ A )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6540_sum_Oremove,axiom,
! [A: set_o,X: $o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( groups7872700643590313910_o_rat @ G @ A )
= ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_6541_sum_Oinsert__remove,axiom,
! [A: set_Code_integer,G: code_integer > code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6542_sum_Oinsert__remove,axiom,
! [A: set_Code_integer,G: code_integer > rat,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6543_sum_Oinsert__remove,axiom,
! [A: set_Code_integer,G: code_integer > nat,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6544_sum_Oinsert__remove,axiom,
! [A: set_Code_integer,G: code_integer > int,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6545_sum_Oinsert__remove,axiom,
! [A: set_Product_unit,G: product_unit > code_integer,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6546_sum_Oinsert__remove,axiom,
! [A: set_Product_unit,G: product_unit > rat,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6547_sum_Oinsert__remove,axiom,
! [A: set_Product_unit,G: product_unit > nat,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( groups8766797177574957219it_nat @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_plus_nat @ ( G @ X ) @ ( groups8766797177574957219it_nat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6548_sum_Oinsert__remove,axiom,
! [A: set_Product_unit,G: product_unit > int,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( groups8764306707065906943it_int @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_plus_int @ ( G @ X ) @ ( groups8764306707065906943it_int @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6549_sum_Oinsert__remove,axiom,
! [A: set_o,G: $o > code_integer,X: $o] :
( ( finite_finite_o @ A )
=> ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6550_sum_Oinsert__remove,axiom,
! [A: set_o,G: $o > rat,X: $o] :
( ( finite_finite_o @ A )
=> ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_6551_sum__strict__mono,axiom,
! [A: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A ) @ ( groups7237345082560585321er_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6552_sum__strict__mono,axiom,
! [A: set_Product_unit,F: product_unit > nat,G: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( groups8766797177574957219it_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6553_sum__strict__mono,axiom,
! [A: set_o,F: $o > nat,G: $o > nat] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( groups8507830703676809646_o_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6554_sum__strict__mono,axiom,
! [A: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6555_sum__strict__mono,axiom,
! [A: set_Code_integer,F: code_integer > int,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A ) @ ( groups7234854612051535045er_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6556_sum__strict__mono,axiom,
! [A: set_Product_unit,F: product_unit > int,G: product_unit > int] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( groups8764306707065906943it_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6557_sum__strict__mono,axiom,
! [A: set_o,F: $o > int,G: $o > int] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( groups8505340233167759370_o_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6558_sum__strict__mono,axiom,
! [A: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6559_sum__strict__mono,axiom,
! [A: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le6747313008572928689nteger @ ( groups879477027807139574nteger @ F @ A ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6560_sum__strict__mono,axiom,
! [A: set_Product_unit,F: product_unit > code_integer,G: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_le6747313008572928689nteger @ ( groups5393439102872367408nteger @ F @ A ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_6561_sum_Oinsert__if,axiom,
! [A: set_int,X: int,G: int > code_integer] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X @ A )
=> ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
= ( groups7873554091576472773nteger @ G @ A ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7873554091576472773nteger @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6562_sum_Oinsert__if,axiom,
! [A: set_o,X: $o,G: $o > code_integer] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ X @ A )
=> ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
= ( groups4406642042086082107nteger @ G @ A ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6563_sum_Oinsert__if,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
= ( groups5393439102872367408nteger @ G @ A ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6564_sum_Oinsert__if,axiom,
! [A: set_nat,X: nat,G: nat > code_integer] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
= ( groups7501900531339628137nteger @ G @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7501900531339628137nteger @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6565_sum_Oinsert__if,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ X @ A )
=> ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
= ( groups879477027807139574nteger @ G @ A ) ) )
& ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6566_sum_Oinsert__if,axiom,
! [A: set_int,X: int,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X @ A )
=> ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
= ( groups3906332499630173760nt_rat @ G @ A ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6567_sum_Oinsert__if,axiom,
! [A: set_o,X: $o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ X @ A )
=> ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
= ( groups7872700643590313910_o_rat @ G @ A ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6568_sum_Oinsert__if,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( groups8131667117488461483it_rat @ G @ A ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6569_sum_Oinsert__if,axiom,
! [A: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
= ( groups2906978787729119204at_rat @ G @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6570_sum_Oinsert__if,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ X @ A )
=> ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( groups6602215022474089585er_rat @ G @ A ) ) )
& ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_6571_sum__pos,axiom,
! [I: set_Code_integer,F: code_integer > nat] :
( ( finite6017078050557962740nteger @ I )
=> ( ( I != bot_bo3990330152332043303nteger )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups7237345082560585321er_nat @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6572_sum__pos,axiom,
! [I: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups8766797177574957219it_nat @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6573_sum__pos,axiom,
! [I: set_o,F: $o > nat] :
( ( finite_finite_o @ I )
=> ( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6574_sum__pos,axiom,
! [I: set_int,F: int > nat] :
( ( finite_finite_int @ I )
=> ( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6575_sum__pos,axiom,
! [I: set_Code_integer,F: code_integer > int] :
( ( finite6017078050557962740nteger @ I )
=> ( ( I != bot_bo3990330152332043303nteger )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups7234854612051535045er_int @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6576_sum__pos,axiom,
! [I: set_Product_unit,F: product_unit > int] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups8764306707065906943it_int @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6577_sum__pos,axiom,
! [I: set_o,F: $o > int] :
( ( finite_finite_o @ I )
=> ( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6578_sum__pos,axiom,
! [I: set_nat,F: nat > int] :
( ( finite_finite_nat @ I )
=> ( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6579_sum__pos,axiom,
! [I: set_Code_integer,F: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ I )
=> ( ( I != bot_bo3990330152332043303nteger )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups879477027807139574nteger @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6580_sum__pos,axiom,
! [I: set_Product_unit,F: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups5393439102872367408nteger @ F @ I ) ) ) ) ) ).
% sum_pos
thf(fact_6581_sum_OUnion__disjoint,axiom,
! [C5: set_set_nat,G: nat > nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ( finite_finite_nat @ X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ! [Xa3: set_nat] :
( ( member_set_nat @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_nat @ X3 @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( comple7399068483239264473et_nat @ C5 ) )
= ( comp_s5456809700045850692at_nat @ groups8294997508430121362at_nat @ groups3542108847815614940at_nat @ G @ C5 ) ) ) ) ).
% sum.Union_disjoint
thf(fact_6582_sum_OUnion__disjoint,axiom,
! [C5: set_set_int,G: int > int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ C5 )
=> ( finite_finite_int @ X3 ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ C5 )
=> ! [Xa3: set_int] :
( ( member_set_int @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_int @ X3 @ Xa3 )
= bot_bot_set_int ) ) ) )
=> ( ( groups4538972089207619220nt_int @ G @ ( comple3221217463730067765et_int @ C5 ) )
= ( comp_s9098231955081793132nt_int @ groups1256056575759317066nt_int @ groups4538972089207619220nt_int @ G @ C5 ) ) ) ) ).
% sum.Union_disjoint
thf(fact_6583_sum__diff1__nat,axiom,
! [A2: product_unit,A: set_Product_unit,F: product_unit > nat] :
( ( ( member_Product_unit @ A2 @ A )
=> ( ( groups8766797177574957219it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( minus_minus_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_Product_unit @ A2 @ A )
=> ( ( groups8766797177574957219it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
= ( groups8766797177574957219it_nat @ F @ A ) ) ) ) ).
% sum_diff1_nat
thf(fact_6584_sum__diff1__nat,axiom,
! [A2: $o,A: set_o,F: $o > nat] :
( ( ( member_o @ A2 @ A )
=> ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( groups8507830703676809646_o_nat @ F @ A ) ) ) ) ).
% sum_diff1_nat
thf(fact_6585_sum__diff1__nat,axiom,
! [A2: int,A: set_int,F: int > nat] :
( ( ( member_int @ A2 @ A )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_int @ A2 @ A )
=> ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= ( groups4541462559716669496nt_nat @ F @ A ) ) ) ) ).
% sum_diff1_nat
thf(fact_6586_sum__diff1__nat,axiom,
! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > nat] :
( ( ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( groups6742167275697360483_o_nat @ F @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
= ( minus_minus_nat @ ( groups6742167275697360483_o_nat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
=> ( ( groups6742167275697360483_o_nat @ F @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
= ( groups6742167275697360483_o_nat @ F @ A ) ) ) ) ).
% sum_diff1_nat
thf(fact_6587_sum__diff1__nat,axiom,
! [A2: nat,A: set_nat,F: nat > nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( F @ A2 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( groups3542108847815614940at_nat @ F @ A ) ) ) ) ).
% sum_diff1_nat
thf(fact_6588_bezout__add__strong__nat,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ? [D: nat,X3: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ B )
& ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_6589_bezout1__nat,axiom,
! [A2: nat,B: nat] :
? [D: nat,X3: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
= D )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y2 ) )
= D ) ) ) ).
% bezout1_nat
thf(fact_6590_bezout__add__nat,axiom,
! [A2: nat,B: nat] :
? [D: nat,X3: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A2 )
& ( dvd_dvd_nat @ D @ B )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D ) ) ) ) ).
% bezout_add_nat
thf(fact_6591_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
=> ( ( dvd_dvd_nat @ D2 @ B )
=> ( ( ( ( times_times_nat @ A2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D2 ) ) )
=> ? [X3: nat,Y2: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A2 @ B ) )
& ( ( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y2 ) @ D2 ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_6592_dvd__productE,axiom,
! [P3: nat,A2: nat,B: nat] :
( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A2 @ B ) )
=> ~ ! [X3: nat,Y2: nat] :
( ( P3
= ( times_times_nat @ X3 @ Y2 ) )
=> ( ( dvd_dvd_nat @ X3 @ A2 )
=> ~ ( dvd_dvd_nat @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_6593_dvd__productE,axiom,
! [P3: int,A2: int,B: int] :
( ( dvd_dvd_int @ P3 @ ( times_times_int @ A2 @ B ) )
=> ~ ! [X3: int,Y2: int] :
( ( P3
= ( times_times_int @ X3 @ Y2 ) )
=> ( ( dvd_dvd_int @ X3 @ A2 )
=> ~ ( dvd_dvd_int @ Y2 @ B ) ) ) ) ).
% dvd_productE
thf(fact_6594_division__decomp,axiom,
! [A2: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
=> ? [B10: nat,C8: nat] :
( ( A2
= ( times_times_nat @ B10 @ C8 ) )
& ( dvd_dvd_nat @ B10 @ B )
& ( dvd_dvd_nat @ C8 @ C ) ) ) ).
% division_decomp
thf(fact_6595_division__decomp,axiom,
! [A2: int,B: int,C: int] :
( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
=> ? [B10: int,C8: int] :
( ( A2
= ( times_times_int @ B10 @ C8 ) )
& ( dvd_dvd_int @ B10 @ B )
& ( dvd_dvd_int @ C8 @ C ) ) ) ).
% division_decomp
thf(fact_6596_card__Min__le__sum,axiom,
! [A: set_Code_integer,F: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite4902975817058060853nteger @ A ) @ ( lattic8721135487736765967in_nat @ ( image_951025933927791156er_nat @ F @ A ) ) ) @ ( groups7237345082560585321er_nat @ F @ A ) ) ) ).
% card_Min_le_sum
thf(fact_6597_card__Min__le__sum,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_nat @ A ) @ ( lattic8721135487736765967in_nat @ ( image_nat_nat2 @ F @ A ) ) ) @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% card_Min_le_sum
thf(fact_6598_Min__singleton,axiom,
! [X: product_unit] :
( ( lattic8554435388794137742t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% Min_singleton
thf(fact_6599_Min__singleton,axiom,
! [X: $o] :
( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% Min_singleton
thf(fact_6600_Min__singleton,axiom,
! [X: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% Min_singleton
thf(fact_6601_Min__singleton,axiom,
! [X: int] :
( ( lattic8718645017227715691in_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% Min_singleton
thf(fact_6602_Min_Obounded__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
= ( ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6603_Min_Obounded__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6604_Min_Obounded__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6605_Min_Obounded__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
= ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6606_Min_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6607_Min_Obounded__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6608_Min__gr__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
= ( ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le361264281704409273t_unit @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6609_Min__gr__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6610_Min__gr__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6611_Min__gr__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6612_Min__gr__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le6747313008572928689nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
= ( ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6613_Min__gr__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
= ( ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_rat @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6614_Min__in,axiom,
! [A: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( member_Code_integer @ ( lattic1063845414844153500nteger @ A ) @ A ) ) ) ).
% Min_in
thf(fact_6615_Min__in,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( member_Product_unit @ ( lattic8554435388794137742t_unit @ A ) @ A ) ) ) ).
% Min_in
thf(fact_6616_Min__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( member_o @ ( lattic1973801136483472281_Min_o @ A ) @ A ) ) ) ).
% Min_in
thf(fact_6617_Min__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% Min_in
thf(fact_6618_Min__in,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( member_int @ ( lattic8718645017227715691in_int @ A ) @ A ) ) ) ).
% Min_in
thf(fact_6619_Min__eq__iff,axiom,
! [A: set_Code_integer,M4: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ( lattic1063845414844153500nteger @ A )
= M4 )
= ( ( member_Code_integer @ M4 @ A )
& ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le3102999989581377725nteger @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6620_Min__eq__iff,axiom,
! [A: set_Product_unit,M4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ( lattic8554435388794137742t_unit @ A )
= M4 )
= ( ( member_Product_unit @ M4 @ A )
& ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6621_Min__eq__iff,axiom,
! [A: set_o,M4: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ( lattic1973801136483472281_Min_o @ A )
= M4 )
= ( ( member_o @ M4 @ A )
& ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6622_Min__eq__iff,axiom,
! [A: set_rat,M4: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ( lattic8086005427650270231in_rat @ A )
= M4 )
= ( ( member_rat @ M4 @ A )
& ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6623_Min__eq__iff,axiom,
! [A: set_nat,M4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M4 )
= ( ( member_nat @ M4 @ A )
& ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6624_Min__eq__iff,axiom,
! [A: set_int,M4: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ( lattic8718645017227715691in_int @ A )
= M4 )
= ( ( member_int @ M4 @ A )
& ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ M4 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6625_Min__le__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ A ) @ X )
= ( ? [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
& ( ord_le3102999989581377725nteger @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6626_Min__le__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ A ) @ X )
= ( ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
& ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6627_Min__le__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ A ) @ X )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6628_Min__le__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ A ) @ X )
= ( ? [X2: rat] :
( ( member_rat @ X2 @ A )
& ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6629_Min__le__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6630_Min__le__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A ) @ X )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6631_eq__Min__iff,axiom,
! [A: set_Code_integer,M4: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( M4
= ( lattic1063845414844153500nteger @ A ) )
= ( ( member_Code_integer @ M4 @ A )
& ! [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
=> ( ord_le3102999989581377725nteger @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6632_eq__Min__iff,axiom,
! [A: set_Product_unit,M4: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( M4
= ( lattic8554435388794137742t_unit @ A ) )
= ( ( member_Product_unit @ M4 @ A )
& ! [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
=> ( ord_le3221252021190050221t_unit @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6633_eq__Min__iff,axiom,
! [A: set_o,M4: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( M4
= ( lattic1973801136483472281_Min_o @ A ) )
= ( ( member_o @ M4 @ A )
& ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6634_eq__Min__iff,axiom,
! [A: set_rat,M4: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( M4
= ( lattic8086005427650270231in_rat @ A ) )
= ( ( member_rat @ M4 @ A )
& ! [X2: rat] :
( ( member_rat @ X2 @ A )
=> ( ord_less_eq_rat @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6635_eq__Min__iff,axiom,
! [A: set_nat,M4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M4
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat @ M4 @ A )
& ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6636_eq__Min__iff,axiom,
! [A: set_int,M4: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( M4
= ( lattic8718645017227715691in_int @ A ) )
= ( ( member_int @ M4 @ A )
& ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ord_less_eq_int @ M4 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6637_Min_OboundedE,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
=> ! [A10: code_integer] :
( ( member_Code_integer @ A10 @ A )
=> ( ord_le3102999989581377725nteger @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_6638_Min_OboundedE,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
=> ! [A10: product_unit] :
( ( member_Product_unit @ A10 @ A )
=> ( ord_le3221252021190050221t_unit @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_6639_Min_OboundedE,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
=> ! [A10: $o] :
( ( member_o @ A10 @ A )
=> ( ord_less_eq_o @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_6640_Min_OboundedE,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
=> ! [A10: rat] :
( ( member_rat @ A10 @ A )
=> ( ord_less_eq_rat @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_6641_Min_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A )
=> ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_6642_Min_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
=> ! [A10: int] :
( ( member_int @ A10 @ A )
=> ( ord_less_eq_int @ X @ A10 ) ) ) ) ) ).
% Min.boundedE
thf(fact_6643_Min_OboundedI,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ A )
=> ( ord_le3102999989581377725nteger @ X @ A4 ) )
=> ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_6644_Min_OboundedI,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ord_le3221252021190050221t_unit @ X @ A4 ) )
=> ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_6645_Min_OboundedI,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_o @ X @ A4 ) )
=> ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_6646_Min_OboundedI,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ! [A4: rat] :
( ( member_rat @ A4 @ A )
=> ( ord_less_eq_rat @ X @ A4 ) )
=> ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_6647_Min_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_6648_Min_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ X @ A4 ) )
=> ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_6649_Min__less__iff,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( ord_le361264281704409273t_unit @ ( lattic8554435388794137742t_unit @ A ) @ X )
= ( ? [X2: product_unit] :
( ( member_Product_unit @ X2 @ A )
& ( ord_le361264281704409273t_unit @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6650_Min__less__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_o @ ( lattic1973801136483472281_Min_o @ A ) @ X )
= ( ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_o @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6651_Min__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6652_Min__less__iff,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A ) @ X )
= ( ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6653_Min__less__iff,axiom,
! [A: set_Code_integer,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( ord_le6747313008572928689nteger @ ( lattic1063845414844153500nteger @ A ) @ X )
= ( ? [X2: code_integer] :
( ( member_Code_integer @ X2 @ A )
& ( ord_le6747313008572928689nteger @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6654_Min__less__iff,axiom,
! [A: set_rat,X: rat] :
( ( finite_finite_rat @ A )
=> ( ( A != bot_bot_set_rat )
=> ( ( ord_less_rat @ ( lattic8086005427650270231in_rat @ A ) @ X )
= ( ? [X2: rat] :
( ( member_rat @ X2 @ A )
& ( ord_less_rat @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6655_Min__insert2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ A )
=> ( ord_less_eq_o @ A2 @ B2 ) )
=> ( ( lattic1973801136483472281_Min_o @ ( insert_o @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_6656_Min__insert2,axiom,
! [A: set_Product_unit,A2: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ A )
=> ( ord_le3221252021190050221t_unit @ A2 @ B2 ) )
=> ( ( lattic8554435388794137742t_unit @ ( insert_Product_unit @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_6657_Min__insert2,axiom,
! [A: set_Code_integer,A2: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ A )
=> ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
=> ( ( lattic1063845414844153500nteger @ ( insert_Code_integer @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_6658_Min__insert2,axiom,
! [A: set_rat,A2: rat] :
( ( finite_finite_rat @ A )
=> ( ! [B2: rat] :
( ( member_rat @ B2 @ A )
=> ( ord_less_eq_rat @ A2 @ B2 ) )
=> ( ( lattic8086005427650270231in_rat @ ( insert_rat @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_6659_Min__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ A )
=> ( ord_less_eq_nat @ A2 @ B2 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_6660_Min__insert2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ! [B2: int] :
( ( member_int @ B2 @ A )
=> ( ord_less_eq_int @ A2 @ B2 ) )
=> ( ( lattic8718645017227715691in_int @ ( insert_int @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_6661_cInf__eq__Min,axiom,
! [X4: set_int] :
( ( finite_finite_int @ X4 )
=> ( ( X4 != bot_bot_set_int )
=> ( ( complete_Inf_Inf_int @ X4 )
= ( lattic8718645017227715691in_int @ X4 ) ) ) ) ).
% cInf_eq_Min
thf(fact_6662_cInf__eq__Min,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ X4 )
= ( lattic8721135487736765967in_nat @ X4 ) ) ) ) ).
% cInf_eq_Min
thf(fact_6663_cInf__eq__Min,axiom,
! [X4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ X4 )
=> ( ( X4 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit @ X4 )
= ( lattic8554435388794137742t_unit @ X4 ) ) ) ) ).
% cInf_eq_Min
thf(fact_6664_Min__Inf,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( lattic8554435388794137742t_unit @ A )
= ( comple2584293577114468500t_unit @ A ) ) ) ) ).
% Min_Inf
thf(fact_6665_Min__antimono,axiom,
! [M: set_Code_integer,N: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ M @ N )
=> ( ( M != bot_bo3990330152332043303nteger )
=> ( ( finite6017078050557962740nteger @ N )
=> ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ N ) @ ( lattic1063845414844153500nteger @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_6666_Min__antimono,axiom,
! [M: set_Product_unit,N: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ M @ N )
=> ( ( M != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ N )
=> ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ N ) @ ( lattic8554435388794137742t_unit @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_6667_Min__antimono,axiom,
! [M: set_o,N: set_o] :
( ( ord_less_eq_set_o @ M @ N )
=> ( ( M != bot_bot_set_o )
=> ( ( finite_finite_o @ N )
=> ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ N ) @ ( lattic1973801136483472281_Min_o @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_6668_Min__antimono,axiom,
! [M: set_rat,N: set_rat] :
( ( ord_less_eq_set_rat @ M @ N )
=> ( ( M != bot_bot_set_rat )
=> ( ( finite_finite_rat @ N )
=> ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ N ) @ ( lattic8086005427650270231in_rat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_6669_Min__antimono,axiom,
! [M: set_int,N: set_int] :
( ( ord_less_eq_set_int @ M @ N )
=> ( ( M != bot_bot_set_int )
=> ( ( finite_finite_int @ N )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N ) @ ( lattic8718645017227715691in_int @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_6670_Min__antimono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_6671_Min_Osubset__imp,axiom,
! [A: set_Code_integer,B4: set_Code_integer] :
( ( ord_le7084787975880047091nteger @ A @ B4 )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ B4 ) @ ( lattic1063845414844153500nteger @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6672_Min_Osubset__imp,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ B4 ) @ ( lattic8554435388794137742t_unit @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6673_Min_Osubset__imp,axiom,
! [A: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A @ B4 )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B4 )
=> ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ B4 ) @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6674_Min_Osubset__imp,axiom,
! [A: set_rat,B4: set_rat] :
( ( ord_less_eq_set_rat @ A @ B4 )
=> ( ( A != bot_bot_set_rat )
=> ( ( finite_finite_rat @ B4 )
=> ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ B4 ) @ ( lattic8086005427650270231in_rat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6675_Min_Osubset__imp,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( A != bot_bot_set_int )
=> ( ( finite_finite_int @ B4 )
=> ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B4 ) @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6676_Min_Osubset__imp,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6677_minus__Min__eq__Max,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( uminus_uminus_int @ ( lattic8718645017227715691in_int @ S ) )
= ( lattic8263393255366662781ax_int @ ( image_int_int2 @ uminus_uminus_int @ S ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_6678_minus__Min__eq__Max,axiom,
! [S: set_Code_integer] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ( ( uminus1351360451143612070nteger @ ( lattic1063845414844153500nteger @ S ) )
= ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_6679_minus__Max__eq__Min,axiom,
! [S: set_int] :
( ( finite_finite_int @ S )
=> ( ( S != bot_bot_set_int )
=> ( ( uminus_uminus_int @ ( lattic8263393255366662781ax_int @ S ) )
= ( lattic8718645017227715691in_int @ ( image_int_int2 @ uminus_uminus_int @ S ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_6680_minus__Max__eq__Min,axiom,
! [S: set_Code_integer] :
( ( finite6017078050557962740nteger @ S )
=> ( ( S != bot_bo3990330152332043303nteger )
=> ( ( uminus1351360451143612070nteger @ ( lattic4901227151466704046nteger @ S ) )
= ( lattic1063845414844153500nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_6681_Gcd__0__iff,axiom,
! [A: set_int] :
( ( ( gcd_Gcd_int @ A )
= zero_zero_int )
= ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% Gcd_0_iff
thf(fact_6682_Gcd__0__iff,axiom,
! [A: set_nat] :
( ( ( gcd_Gcd_nat @ A )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_6683_Gcd__fin__0__iff,axiom,
! [A: set_int] :
( ( ( semiri4256215615220890538in_int @ A )
= zero_zero_int )
= ( ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
& ( finite_finite_int @ A ) ) ) ).
% Gcd_fin_0_iff
thf(fact_6684_Gcd__fin__0__iff,axiom,
! [A: set_nat] :
( ( ( semiri4258706085729940814in_nat @ A )
= zero_zero_nat )
= ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
& ( finite_finite_nat @ A ) ) ) ).
% Gcd_fin_0_iff
thf(fact_6685_sum__bounded__above__strict,axiom,
! [A: set_nat,F: nat > rat,K4: rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6686_sum__bounded__above__strict,axiom,
! [A: set_int,F: int > rat,K4: rat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6687_sum__bounded__above__strict,axiom,
! [A: set_o,F: $o > rat,K4: rat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
=> ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6688_sum__bounded__above__strict,axiom,
! [A: set_Product_unit,F: product_unit > rat,K4: rat] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
=> ( ord_less_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6689_sum__bounded__above__strict,axiom,
! [A: set_nat,F: nat > int,K4: int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_int @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6690_sum__bounded__above__strict,axiom,
! [A: set_o,F: $o > int,K4: int] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_int @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
=> ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6691_sum__bounded__above__strict,axiom,
! [A: set_Product_unit,F: product_unit > int,K4: int] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_less_int @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
=> ( ord_less_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6692_sum__bounded__above__strict,axiom,
! [A: set_int,F: int > nat,K4: nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6693_sum__bounded__above__strict,axiom,
! [A: set_o,F: $o > nat,K4: nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
=> ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6694_sum__bounded__above__strict,axiom,
! [A: set_Product_unit,F: product_unit > nat,K4: nat] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
=> ( ord_less_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_6695_prod__le__power,axiom,
! [A: set_nat,F: nat > code_natural,N2: code_natural,K3: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
& ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
=> ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
=> ( ord_le1926595141338095240atural @ ( groups2279045934846249631atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6696_prod__le__power,axiom,
! [A: set_int,F: int > code_natural,N2: code_natural,K3: nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
& ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
=> ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
=> ( ord_le1926595141338095240atural @ ( groups2650699495083094267atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6697_prod__le__power,axiom,
! [A: set_o,F: $o > code_natural,N2: code_natural,K3: nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
& ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ K3 )
=> ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
=> ( ord_le1926595141338095240atural @ ( groups6518289543945209051atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6698_prod__le__power,axiom,
! [A: set_Product_unit,F: product_unit > code_natural,N2: code_natural,K3: nat] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
& ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ K3 )
=> ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
=> ( ord_le1926595141338095240atural @ ( groups7008954963700796496atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6699_prod__le__power,axiom,
! [A: set_nat,F: nat > code_integer,N2: code_integer,K3: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
=> ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
=> ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6700_prod__le__power,axiom,
! [A: set_int,F: int > code_integer,N2: code_integer,K3: nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
=> ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
=> ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6701_prod__le__power,axiom,
! [A: set_o,F: $o > code_integer,N2: code_integer,K3: nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ K3 )
=> ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
=> ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6702_prod__le__power,axiom,
! [A: set_Product_unit,F: product_unit > code_integer,N2: code_integer,K3: nat] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ K3 )
=> ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
=> ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6703_prod__le__power,axiom,
! [A: set_nat,F: nat > rat,N2: rat,K3: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
=> ( ( ord_less_eq_rat @ one_one_rat @ N2 )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A ) @ ( power_power_rat @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6704_prod__le__power,axiom,
! [A: set_int,F: int > rat,N2: rat,K3: nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
=> ( ( ord_less_eq_rat @ one_one_rat @ N2 )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( power_power_rat @ N2 @ K3 ) ) ) ) ) ).
% prod_le_power
thf(fact_6705_prod__mono2,axiom,
! [B4: set_int,A: set_int,F: int > code_integer] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( groups3827104343326376752nteger @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6706_prod__mono2,axiom,
! [B4: set_o,A: set_o,F: $o > code_integer] :
( ( finite_finite_o @ B4 )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( groups7694694392188491536nteger @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6707_prod__mono2,axiom,
! [B4: set_Product_unit,A: set_Product_unit,F: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( groups8185359811944078981nteger @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6708_prod__mono2,axiom,
! [B4: set_Code_integer,A: set_Code_integer,F: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ B4 )
=> ( ( ord_le7084787975880047091nteger @ A @ B4 )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B4 @ A ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ A )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3674199335183972705nteger @ F @ A ) @ ( groups3674199335183972705nteger @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6709_prod__mono2,axiom,
! [B4: set_int,A: set_int,F: int > rat] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( groups1072433553688619179nt_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6710_prod__mono2,axiom,
! [B4: set_o,A: set_o,F: $o > rat] :
( ( finite_finite_o @ B4 )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
=> ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A ) @ ( groups2869687844427037835_o_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6711_prod__mono2,axiom,
! [B4: set_Product_unit,A: set_Product_unit,F: product_unit > rat] :
( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B4 )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
=> ( ord_less_eq_rat @ ( groups8984663225794451456it_rat @ F @ A ) @ ( groups8984663225794451456it_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6712_prod__mono2,axiom,
! [B4: set_Code_integer,A: set_Code_integer,F: code_integer > rat] :
( ( finite6017078050557962740nteger @ B4 )
=> ( ( ord_le7084787975880047091nteger @ A @ B4 )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B4 @ A ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
=> ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A ) @ ( groups2555765274223993564er_rat @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6713_prod__mono2,axiom,
! [B4: set_int,A: set_int,F: int > int] :
( ( finite_finite_int @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
=> ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F @ A ) @ ( groups1705073143266064639nt_int @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6714_prod__mono2,axiom,
! [B4: set_o,A: set_o,F: $o > int] :
( ( finite_finite_o @ B4 )
=> ( ( ord_less_eq_set_o @ A @ B4 )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
=> ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A ) @ ( groups3502327434004483295_o_int @ F @ B4 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_6715_neg__equal__iff__equal,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_6716_neg__equal__iff__equal,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A2 )
= ( uminus1351360451143612070nteger @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_6717_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_6718_add_Oinverse__inverse,axiom,
! [A2: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_6719_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) )
= ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_6720_Compl__anti__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B4 ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_6721_neg__le__iff__le,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) )
= ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_6722_neg__le__iff__le,axiom,
! [B: rat,A2: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_eq_rat @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_6723_neg__le__iff__le,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_6724_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_6725_neg__equal__zero,axiom,
! [A2: code_integer] :
( ( ( uminus1351360451143612070nteger @ A2 )
= A2 )
= ( A2 = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
thf(fact_6726_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_6727_equal__neg__zero,axiom,
! [A2: code_integer] :
( ( A2
= ( uminus1351360451143612070nteger @ A2 ) )
= ( A2 = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
thf(fact_6728_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_6729_neg__equal__0__iff__equal,axiom,
! [A2: code_integer] :
( ( ( uminus1351360451143612070nteger @ A2 )
= zero_z3403309356797280102nteger )
= ( A2 = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
thf(fact_6730_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_6731_neg__0__equal__iff__equal,axiom,
! [A2: code_integer] :
( ( zero_z3403309356797280102nteger
= ( uminus1351360451143612070nteger @ A2 ) )
= ( zero_z3403309356797280102nteger = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_6732_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_6733_add_Oinverse__neutral,axiom,
( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% add.inverse_neutral
thf(fact_6734_neg__less__iff__less,axiom,
! [B: rat,A2: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_rat @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_6735_neg__less__iff__less,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_6736_neg__less__iff__less,axiom,
! [B: code_integer,A2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) )
= ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% neg_less_iff_less
thf(fact_6737_minus__add__distrib,axiom,
! [A2: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_add_distrib
thf(fact_6738_minus__add__distrib,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_6739_minus__add__distrib,axiom,
! [A2: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_add_distrib
thf(fact_6740_minus__add__cancel,axiom,
! [A2: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_6741_minus__add__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_6742_minus__add__cancel,axiom,
! [A2: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_6743_add__minus__cancel,axiom,
! [A2: rat,B: rat] :
( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_6744_add__minus__cancel,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_6745_add__minus__cancel,axiom,
! [A2: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_6746_mult__minus__left,axiom,
! [A2: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_6747_mult__minus__left,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_6748_mult__minus__left,axiom,
! [A2: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% mult_minus_left
thf(fact_6749_minus__mult__minus,axiom,
! [A2: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B ) )
= ( times_times_rat @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_6750_minus__mult__minus,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_6751_minus__mult__minus,axiom,
! [A2: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B ) )
= ( times_3573771949741848930nteger @ A2 @ B ) ) ).
% minus_mult_minus
thf(fact_6752_mult__minus__right,axiom,
! [A2: rat,B: rat] :
( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_6753_mult__minus__right,axiom,
! [A2: int,B: int] :
( ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_6754_mult__minus__right,axiom,
! [A2: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% mult_minus_right
thf(fact_6755_minus__diff__eq,axiom,
! [A2: rat,B: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B ) )
= ( minus_minus_rat @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_6756_minus__diff__eq,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
= ( minus_minus_int @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_6757_minus__diff__eq,axiom,
! [A2: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
= ( minus_8373710615458151222nteger @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_6758_Compl__disjoint,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
= bot_bo3957492148770167129t_unit ) ).
% Compl_disjoint
thf(fact_6759_Compl__disjoint,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ ( uminus_uminus_set_o @ A ) )
= bot_bot_set_o ) ).
% Compl_disjoint
thf(fact_6760_Compl__disjoint,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_6761_Compl__disjoint,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
= bot_bot_set_int ) ).
% Compl_disjoint
thf(fact_6762_Compl__disjoint,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ A @ ( uminus5254974436814262692_nat_o @ A ) )
= bot_bo7824918357723582233_nat_o ) ).
% Compl_disjoint
thf(fact_6763_Compl__disjoint2,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ A ) @ A )
= bot_bo3957492148770167129t_unit ) ).
% Compl_disjoint2
thf(fact_6764_Compl__disjoint2,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A ) @ A )
= bot_bot_set_o ) ).
% Compl_disjoint2
thf(fact_6765_Compl__disjoint2,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_6766_Compl__disjoint2,axiom,
! [A: set_int] :
( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
= bot_bot_set_int ) ).
% Compl_disjoint2
thf(fact_6767_Compl__disjoint2,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ A ) @ A )
= bot_bo7824918357723582233_nat_o ) ).
% Compl_disjoint2
thf(fact_6768_inter__compl__diff__conv,axiom,
! [A: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) )
= ( minus_minus_set_nat @ A @ B4 ) ) ).
% inter_compl_diff_conv
thf(fact_6769_Diff__Compl,axiom,
! [A: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ).
% Diff_Compl
thf(fact_6770_Compl__Diff__eq,axiom,
! [A: set_nat,B4: set_nat] :
( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ B4 ) ) ).
% Compl_Diff_eq
thf(fact_6771_neg__0__le__iff__le,axiom,
! [A2: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
= ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% neg_0_le_iff_le
thf(fact_6772_neg__0__le__iff__le,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_6773_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_6774_neg__le__0__iff__le,axiom,
! [A2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_6775_neg__le__0__iff__le,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_6776_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_6777_less__eq__neg__nonpos,axiom,
! [A2: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
= ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_6778_less__eq__neg__nonpos,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_6779_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_6780_neg__less__eq__nonneg,axiom,
! [A2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_6781_neg__less__eq__nonneg,axiom,
! [A2: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
= ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_6782_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_6783_less__neg__neg,axiom,
! [A2: rat] :
( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_6784_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_6785_less__neg__neg,axiom,
! [A2: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
= ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_6786_neg__less__pos,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% neg_less_pos
thf(fact_6787_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_6788_neg__less__pos,axiom,
! [A2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% neg_less_pos
thf(fact_6789_neg__0__less__iff__less,axiom,
! [A2: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
= ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_6790_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_6791_neg__0__less__iff__less,axiom,
! [A2: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
= ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% neg_0_less_iff_less
thf(fact_6792_neg__less__0__iff__less,axiom,
! [A2: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_6793_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_6794_neg__less__0__iff__less,axiom,
! [A2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_6795_ab__left__minus,axiom,
! [A2: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_6796_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_6797_ab__left__minus,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_6798_add_Oright__inverse,axiom,
! [A2: rat] :
( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_6799_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_6800_add_Oright__inverse,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_6801_diff__0,axiom,
! [A2: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A2 )
= ( uminus_uminus_rat @ A2 ) ) ).
% diff_0
thf(fact_6802_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_6803_diff__0,axiom,
! [A2: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
= ( uminus1351360451143612070nteger @ A2 ) ) ).
% diff_0
thf(fact_6804_diff__minus__eq__add,axiom,
! [A2: rat,B: rat] :
( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B ) )
= ( plus_plus_rat @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_6805_diff__minus__eq__add,axiom,
! [A2: int,B: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_6806_diff__minus__eq__add,axiom,
! [A2: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
= ( plus_p5714425477246183910nteger @ A2 @ B ) ) ).
% diff_minus_eq_add
thf(fact_6807_uminus__add__conv__diff,axiom,
! [A2: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( minus_minus_rat @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_6808_uminus__add__conv__diff,axiom,
! [A2: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( minus_minus_int @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_6809_uminus__add__conv__diff,axiom,
! [A2: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
= ( minus_8373710615458151222nteger @ B @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_6810_mult__minus1,axiom,
! [Z: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1
thf(fact_6811_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_6812_mult__minus1,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1
thf(fact_6813_mult__minus1__right,axiom,
! [Z: rat] :
( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ Z ) ) ).
% mult_minus1_right
thf(fact_6814_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_6815_mult__minus1__right,axiom,
! [Z: code_integer] :
( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z ) ) ).
% mult_minus1_right
thf(fact_6816_boolean__algebra_Ocompl__one,axiom,
( ( uminus5254974436814262692_nat_o @ top_to4869887016220417533_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% boolean_algebra.compl_one
thf(fact_6817_boolean__algebra_Ocompl__one,axiom,
( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.compl_one
thf(fact_6818_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.compl_one
thf(fact_6819_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_6820_boolean__algebra_Ocompl__one,axiom,
( ( uminus1532241313380277803et_int @ top_top_set_int )
= bot_bot_set_int ) ).
% boolean_algebra.compl_one
thf(fact_6821_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_assn @ top_top_assn )
= bot_bot_assn ) ).
% boolean_algebra.compl_one
thf(fact_6822_boolean__algebra_Ocompl__one,axiom,
( ( uminus2952777764628376836t_unit @ top_top_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.compl_one
thf(fact_6823_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5254974436814262692_nat_o @ bot_bo7824918357723582233_nat_o )
= top_to4869887016220417533_nat_o ) ).
% boolean_algebra.compl_zero
thf(fact_6824_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.compl_zero
thf(fact_6825_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% boolean_algebra.compl_zero
thf(fact_6826_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_6827_boolean__algebra_Ocompl__zero,axiom,
( ( uminus1532241313380277803et_int @ bot_bot_set_int )
= top_top_set_int ) ).
% boolean_algebra.compl_zero
thf(fact_6828_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_assn @ bot_bot_assn )
= top_top_assn ) ).
% boolean_algebra.compl_zero
thf(fact_6829_boolean__algebra_Ocompl__zero,axiom,
( ( uminus2952777764628376836t_unit @ bot_bot_Product_unit )
= top_top_Product_unit ) ).
% boolean_algebra.compl_zero
thf(fact_6830_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ ( uminus5944136376168626660t_unit @ X ) )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6831_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6832_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6833_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
= bot_bot_set_int ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6834_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ X @ ( uminus5254974436814262692_nat_o @ X ) )
= bot_bo7824918357723582233_nat_o ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6835_boolean__algebra_Oconj__cancel__right,axiom,
! [X: assn] :
( ( inf_inf_assn @ X @ ( uminus_uminus_assn @ X ) )
= bot_bot_assn ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6836_boolean__algebra_Oconj__cancel__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ X ) )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_cancel_right
thf(fact_6837_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ X )
= bot_bo3957492148770167129t_unit ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6838_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6839_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6840_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_int] :
( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
= bot_bot_set_int ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6841_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ X )
= bot_bo7824918357723582233_nat_o ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6842_boolean__algebra_Oconj__cancel__left,axiom,
! [X: assn] :
( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ X )
= bot_bot_assn ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6843_boolean__algebra_Oconj__cancel__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ X )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_cancel_left
thf(fact_6844_inf__compl__bot__right,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ ( inf_in4660618365625256667t_unit @ Y @ ( uminus5944136376168626660t_unit @ X ) ) )
= bot_bo3957492148770167129t_unit ) ).
% inf_compl_bot_right
thf(fact_6845_inf__compl__bot__right,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
= bot_bot_set_o ) ).
% inf_compl_bot_right
thf(fact_6846_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_6847_inf__compl__bot__right,axiom,
! [X: set_int,Y: set_int] :
( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
= bot_bot_set_int ) ).
% inf_compl_bot_right
thf(fact_6848_inf__compl__bot__right,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ Y @ ( uminus5254974436814262692_nat_o @ X ) ) )
= bot_bo7824918357723582233_nat_o ) ).
% inf_compl_bot_right
thf(fact_6849_inf__compl__bot__right,axiom,
! [X: assn,Y: assn] :
( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ ( uminus_uminus_assn @ X ) ) )
= bot_bot_assn ) ).
% inf_compl_bot_right
thf(fact_6850_inf__compl__bot__right,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) ) )
= bot_bot_Product_unit ) ).
% inf_compl_bot_right
thf(fact_6851_inf__compl__bot__left2,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ Y ) )
= bot_bo3957492148770167129t_unit ) ).
% inf_compl_bot_left2
thf(fact_6852_inf__compl__bot__left2,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left2
thf(fact_6853_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_6854_inf__compl__bot__left2,axiom,
! [X: set_int,Y: set_int] :
( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
= bot_bot_set_int ) ).
% inf_compl_bot_left2
thf(fact_6855_inf__compl__bot__left2,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ Y ) )
= bot_bo7824918357723582233_nat_o ) ).
% inf_compl_bot_left2
thf(fact_6856_inf__compl__bot__left2,axiom,
! [X: assn,Y: assn] :
( ( inf_inf_assn @ X @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
= bot_bot_assn ) ).
% inf_compl_bot_left2
thf(fact_6857_inf__compl__bot__left2,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) )
= bot_bot_Product_unit ) ).
% inf_compl_bot_left2
thf(fact_6858_inf__compl__bot__left1,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ ( inf_in4660618365625256667t_unit @ X @ Y ) )
= bot_bo3957492148770167129t_unit ) ).
% inf_compl_bot_left1
thf(fact_6859_inf__compl__bot__left1,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
= bot_bot_set_o ) ).
% inf_compl_bot_left1
thf(fact_6860_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_6861_inf__compl__bot__left1,axiom,
! [X: set_int,Y: set_int] :
( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
= bot_bot_set_int ) ).
% inf_compl_bot_left1
thf(fact_6862_inf__compl__bot__left1,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ ( inf_in1906310914598751387_nat_o @ X @ Y ) )
= bot_bo7824918357723582233_nat_o ) ).
% inf_compl_bot_left1
thf(fact_6863_inf__compl__bot__left1,axiom,
! [X: assn,Y: assn] :
( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( inf_inf_assn @ X @ Y ) )
= bot_bot_assn ) ).
% inf_compl_bot_left1
thf(fact_6864_inf__compl__bot__left1,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( inf_inf_Product_unit @ X @ Y ) )
= bot_bot_Product_unit ) ).
% inf_compl_bot_left1
thf(fact_6865_subset__Compl__singleton,axiom,
! [A: set_Product_unit,B: product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
= ( ~ ( member_Product_unit @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_6866_subset__Compl__singleton,axiom,
! [A: set_o,B: $o] :
( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_6867_subset__Compl__singleton,axiom,
! [A: set_int,B: int] :
( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_6868_subset__Compl__singleton,axiom,
! [A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
= ( ~ ( member6576561426505652726_nat_o @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_6869_subset__Compl__singleton,axiom,
! [A: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_6870_prod_Oempty,axiom,
! [G: product_unit > assn] :
( ( groups5360233303959380816t_assn @ G @ bot_bo3957492148770167129t_unit )
= one_one_assn ) ).
% prod.empty
thf(fact_6871_prod_Oempty,axiom,
! [G: product_unit > nat] :
( ( groups396421249026171384it_nat @ G @ bot_bo3957492148770167129t_unit )
= one_one_nat ) ).
% prod.empty
thf(fact_6872_prod_Oempty,axiom,
! [G: product_unit > int] :
( ( groups393930778517121108it_int @ G @ bot_bo3957492148770167129t_unit )
= one_one_int ) ).
% prod.empty
thf(fact_6873_prod_Oempty,axiom,
! [G: product_unit > code_integer] :
( ( groups8185359811944078981nteger @ G @ bot_bo3957492148770167129t_unit )
= one_one_Code_integer ) ).
% prod.empty
thf(fact_6874_prod_Oempty,axiom,
! [G: product_unit > code_natural] :
( ( groups7008954963700796496atural @ G @ bot_bo3957492148770167129t_unit )
= one_one_Code_natural ) ).
% prod.empty
thf(fact_6875_prod_Oempty,axiom,
! [G: $o > assn] :
( ( groups5301882518646026715o_assn @ G @ bot_bot_set_o )
= one_one_assn ) ).
% prod.empty
thf(fact_6876_prod_Oempty,axiom,
! [G: $o > nat] :
( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
= one_one_nat ) ).
% prod.empty
thf(fact_6877_prod_Oempty,axiom,
! [G: $o > int] :
( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
= one_one_int ) ).
% prod.empty
thf(fact_6878_prod_Oempty,axiom,
! [G: $o > code_integer] :
( ( groups7694694392188491536nteger @ G @ bot_bot_set_o )
= one_one_Code_integer ) ).
% prod.empty
thf(fact_6879_prod_Oempty,axiom,
! [G: $o > code_natural] :
( ( groups6518289543945209051atural @ G @ bot_bot_set_o )
= one_one_Code_natural ) ).
% prod.empty
thf(fact_6880_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > assn] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups6906906614972039071t_assn @ G @ A )
= one_one_assn ) ) ).
% prod.infinite
thf(fact_6881_prod_Oinfinite,axiom,
! [A: set_Code_integer,G: code_integer > assn] :
( ~ ( finite6017078050557962740nteger @ A )
=> ( ( groups1304777262505850412r_assn @ G @ A )
= one_one_assn ) ) ).
% prod.infinite
thf(fact_6882_prod_Oinfinite,axiom,
! [A: set_Code_integer,G: code_integer > nat] :
( ~ ( finite6017078050557962740nteger @ A )
=> ( ( groups3190895334310489300er_nat @ G @ A )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_6883_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups705719431365010083at_int @ G @ A )
= one_one_int ) ) ).
% prod.infinite
thf(fact_6884_prod_Oinfinite,axiom,
! [A: set_Code_integer,G: code_integer > int] :
( ~ ( finite6017078050557962740nteger @ A )
=> ( ( groups3188404863801439024er_int @ G @ A )
= one_one_int ) ) ).
% prod.infinite
thf(fact_6885_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > code_integer] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3455450783089532116nteger @ G @ A )
= one_one_Code_integer ) ) ).
% prod.infinite
thf(fact_6886_prod_Oinfinite,axiom,
! [A: set_Code_integer,G: code_integer > code_integer] :
( ~ ( finite6017078050557962740nteger @ A )
=> ( ( groups3674199335183972705nteger @ G @ A )
= one_one_Code_integer ) ) ).
% prod.infinite
thf(fact_6887_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > code_natural] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups2279045934846249631atural @ G @ A )
= one_one_Code_natural ) ) ).
% prod.infinite
thf(fact_6888_prod_Oinfinite,axiom,
! [A: set_Code_integer,G: code_integer > code_natural] :
( ~ ( finite6017078050557962740nteger @ A )
=> ( ( groups2497794486940690220atural @ G @ A )
= one_one_Code_natural ) ) ).
% prod.infinite
thf(fact_6889_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G @ A )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_6890_of__nat__mult,axiom,
! [M4: nat,N2: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M4 @ N2 ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M4 ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_6891_of__nat__mult,axiom,
! [M4: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M4 @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_6892_of__nat__mult,axiom,
! [M4: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M4 @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_6893_of__nat__mult,axiom,
! [M4: nat,N2: nat] :
( ( semiri4939895301339042750nteger @ ( times_times_nat @ M4 @ N2 ) )
= ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M4 ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).
% of_nat_mult
thf(fact_6894_of__nat__1,axiom,
( ( semiri3763490453095760265atural @ one_one_nat )
= one_one_Code_natural ) ).
% of_nat_1
thf(fact_6895_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_6896_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_6897_of__nat__1,axiom,
( ( semiri4939895301339042750nteger @ one_one_nat )
= one_one_Code_integer ) ).
% of_nat_1
thf(fact_6898_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_Code_natural
= ( semiri3763490453095760265atural @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6899_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6900_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6901_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_Code_integer
= ( semiri4939895301339042750nteger @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6902_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri3763490453095760265atural @ N2 )
= one_one_Code_natural )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6903_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6904_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6905_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri4939895301339042750nteger @ N2 )
= one_one_Code_integer )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6906_Gcd__empty,axiom,
( ( gcd_Gcd_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_empty
thf(fact_6907_Gcd__empty,axiom,
( ( gcd_Gcd_int @ bot_bot_set_int )
= zero_zero_int ) ).
% Gcd_empty
thf(fact_6908_Gcd__UNIV,axiom,
( ( gcd_Gcd_nat @ top_top_set_nat )
= one_one_nat ) ).
% Gcd_UNIV
thf(fact_6909_Gcd__UNIV,axiom,
( ( gcd_Gcd_int @ top_top_set_int )
= one_one_int ) ).
% Gcd_UNIV
thf(fact_6910_Gcd__fin_Oempty,axiom,
( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_fin.empty
thf(fact_6911_Gcd__fin_Oempty,axiom,
( ( semiri4256215615220890538in_int @ bot_bot_set_int )
= zero_zero_int ) ).
% Gcd_fin.empty
thf(fact_6912_Gcd__fin_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( semiri4258706085729940814in_nat @ A )
= one_one_nat ) ) ).
% Gcd_fin.infinite
thf(fact_6913_Gcd__fin_Oinfinite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ A )
=> ( ( semiri4256215615220890538in_int @ A )
= one_one_int ) ) ).
% Gcd_fin.infinite
thf(fact_6914_is__unit__Gcd__fin__iff,axiom,
! [A: set_nat] :
( ( dvd_dvd_nat @ ( semiri4258706085729940814in_nat @ A ) @ one_one_nat )
= ( ( semiri4258706085729940814in_nat @ A )
= one_one_nat ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_6915_is__unit__Gcd__fin__iff,axiom,
! [A: set_int] :
( ( dvd_dvd_int @ ( semiri4256215615220890538in_int @ A ) @ one_one_int )
= ( ( semiri4256215615220890538in_int @ A )
= one_one_int ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_6916_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_6917_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_6918_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% add_neg_numeral_special(7)
thf(fact_6919_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_6920_add__neg__numeral__special_I7_J,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(7)
thf(fact_6921_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= zero_zero_rat ) ).
% add_neg_numeral_special(8)
thf(fact_6922_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_6923_add__neg__numeral__special_I8_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(8)
thf(fact_6924_diff__numeral__special_I12_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% diff_numeral_special(12)
thf(fact_6925_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_6926_diff__numeral__special_I12_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(12)
thf(fact_6927_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
= one_one_rat ) ).
% minus_one_mult_self
thf(fact_6928_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_6929_minus__one__mult__self,axiom,
! [N2: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
= one_one_Code_integer ) ).
% minus_one_mult_self
thf(fact_6930_left__minus__one__mult__self,axiom,
! [N2: nat,A2: rat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_6931_left__minus__one__mult__self,axiom,
! [N2: nat,A2: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_6932_left__minus__one__mult__self,axiom,
! [N2: nat,A2: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A2 ) )
= A2 ) ).
% left_minus_one_mult_self
thf(fact_6933_of__nat__Suc,axiom,
! [M4: nat] :
( ( semiri3763490453095760265atural @ ( suc @ M4 ) )
= ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( semiri3763490453095760265atural @ M4 ) ) ) ).
% of_nat_Suc
thf(fact_6934_of__nat__Suc,axiom,
! [M4: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M4 ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M4 ) ) ) ).
% of_nat_Suc
thf(fact_6935_of__nat__Suc,axiom,
! [M4: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M4 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M4 ) ) ) ).
% of_nat_Suc
thf(fact_6936_of__nat__Suc,axiom,
! [M4: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M4 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M4 ) ) ) ).
% of_nat_Suc
thf(fact_6937_of__nat__Suc,axiom,
! [M4: nat] :
( ( semiri4939895301339042750nteger @ ( suc @ M4 ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M4 ) ) ) ).
% of_nat_Suc
thf(fact_6938_prod_Oinsert,axiom,
! [A: set_int,X: int,G: int > assn] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6939_prod_Oinsert,axiom,
! [A: set_o,X: $o,G: $o > assn] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6940_prod_Oinsert,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6941_prod_Oinsert,axiom,
! [A: set_nat,X: nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6942_prod_Oinsert,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6943_prod_Oinsert,axiom,
! [A: set_int,X: int,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6944_prod_Oinsert,axiom,
! [A: set_o,X: $o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6945_prod_Oinsert,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6946_prod_Oinsert,axiom,
! [A: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6947_prod_Oinsert,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ).
% prod.insert
thf(fact_6948_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_6949_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_6950_minus__equation__iff,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( uminus_uminus_int @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_6951_minus__equation__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A2 )
= B )
= ( ( uminus1351360451143612070nteger @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_6952_equation__minus__iff,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_6953_equation__minus__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( A2
= ( uminus1351360451143612070nteger @ B ) )
= ( B
= ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_6954_prod_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ A )
= one_one_nat ) ) ).
% prod.neutral
thf(fact_6955_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > assn,A: set_nat] :
( ( ( groups6906906614972039071t_assn @ G @ A )
!= one_one_assn )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= one_one_assn ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6956_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > assn,A: set_int] :
( ( ( groups7882442080178216443t_assn @ G @ A )
!= one_one_assn )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( G @ A4 )
= one_one_assn ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6957_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > assn,A: set_o] :
( ( ( groups5301882518646026715o_assn @ G @ A )
!= one_one_assn )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= one_one_assn ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6958_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: product_unit > assn,A: set_Product_unit] :
( ( ( groups5360233303959380816t_assn @ G @ A )
!= one_one_assn )
=> ~ ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ( G @ A4 )
= one_one_assn ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6959_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > nat,A: set_int] :
( ( ( groups1707563613775114915nt_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( G @ A4 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6960_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > nat,A: set_o] :
( ( ( groups3504817904513533571_o_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6961_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: product_unit > nat,A: set_Product_unit] :
( ( ( groups396421249026171384it_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ A )
=> ( ( G @ A4 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6962_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A: set_nat] :
( ( ( groups705719431365010083at_int @ G @ A )
!= one_one_int )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( G @ A4 )
= one_one_int ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6963_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > int,A: set_int] :
( ( ( groups1705073143266064639nt_int @ G @ A )
!= one_one_int )
=> ~ ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( G @ A4 )
= one_one_int ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6964_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: $o > int,A: set_o] :
( ( ( groups3502327434004483295_o_int @ G @ A )
!= one_one_int )
=> ~ ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( G @ A4 )
= one_one_int ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_6965_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
= ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6966_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6967_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6968_mult__of__nat__commute,axiom,
! [X: nat,Y: code_integer] :
( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
= ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6969_le__imp__neg__le,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_6970_le__imp__neg__le,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_6971_le__imp__neg__le,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_6972_minus__le__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
= ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_6973_minus__le__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_6974_minus__le__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_6975_le__minus__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% le_minus_iff
thf(fact_6976_le__minus__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% le_minus_iff
thf(fact_6977_le__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_6978_minus__less__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_6979_minus__less__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_6980_minus__less__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
= ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 ) ) ).
% minus_less_iff
thf(fact_6981_less__minus__iff,axiom,
! [A2: rat,B: rat] :
( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% less_minus_iff
thf(fact_6982_less__minus__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% less_minus_iff
thf(fact_6983_less__minus__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% less_minus_iff
thf(fact_6984_add_Oinverse__distrib__swap,axiom,
! [A2: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_6985_add_Oinverse__distrib__swap,axiom,
! [A2: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_6986_add_Oinverse__distrib__swap,axiom,
! [A2: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_6987_group__cancel_Oneg1,axiom,
! [A: rat,K3: rat,A2: rat] :
( ( A
= ( plus_plus_rat @ K3 @ A2 ) )
=> ( ( uminus_uminus_rat @ A )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_6988_group__cancel_Oneg1,axiom,
! [A: int,K3: int,A2: int] :
( ( A
= ( plus_plus_int @ K3 @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K3 ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_6989_group__cancel_Oneg1,axiom,
! [A: code_integer,K3: code_integer,A2: code_integer] :
( ( A
= ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
=> ( ( uminus1351360451143612070nteger @ A )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_6990_square__eq__iff,axiom,
! [A2: rat,B: rat] :
( ( ( times_times_rat @ A2 @ A2 )
= ( times_times_rat @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_rat @ B ) ) ) ) ).
% square_eq_iff
thf(fact_6991_square__eq__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ A2 )
= ( times_times_int @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_6992_square__eq__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A2 @ A2 )
= ( times_3573771949741848930nteger @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% square_eq_iff
thf(fact_6993_minus__mult__commute,axiom,
! [A2: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B )
= ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_mult_commute
thf(fact_6994_minus__mult__commute,axiom,
! [A2: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
= ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_6995_minus__mult__commute,axiom,
! [A2: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
= ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_mult_commute
thf(fact_6996_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_6997_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_6998_minus__diff__commute,axiom,
! [B: rat,A2: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A2 )
= ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_6999_minus__diff__commute,axiom,
! [B: int,A2: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
= ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_7000_minus__diff__commute,axiom,
! [B: code_integer,A2: code_integer] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 )
= ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_7001_Gcd__1,axiom,
! [A: set_nat] :
( ( member_nat @ one_one_nat @ A )
=> ( ( gcd_Gcd_nat @ A )
= one_one_nat ) ) ).
% Gcd_1
thf(fact_7002_Gcd__1,axiom,
! [A: set_int] :
( ( member_int @ one_one_int @ A )
=> ( ( gcd_Gcd_int @ A )
= one_one_int ) ) ).
% Gcd_1
thf(fact_7003_minus__assn__def,axiom,
( minus_minus_assn
= ( ^ [A6: assn,B6: assn] : ( inf_inf_assn @ A6 @ ( uminus_uminus_assn @ B6 ) ) ) ) ).
% minus_assn_def
thf(fact_7004_prod__ge__1,axiom,
! [A: set_nat,F: nat > code_integer] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7005_prod__ge__1,axiom,
! [A: set_int,F: int > code_integer] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7006_prod__ge__1,axiom,
! [A: set_o,F: $o > code_integer] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7007_prod__ge__1,axiom,
! [A: set_Product_unit,F: product_unit > code_integer] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7008_prod__ge__1,axiom,
! [A: set_nat,F: nat > code_natural] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups2279045934846249631atural @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7009_prod__ge__1,axiom,
! [A: set_int,F: int > code_natural] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups2650699495083094267atural @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7010_prod__ge__1,axiom,
! [A: set_o,F: $o > code_natural] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups6518289543945209051atural @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7011_prod__ge__1,axiom,
! [A: set_Product_unit,F: product_unit > code_natural] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
=> ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups7008954963700796496atural @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7012_prod__ge__1,axiom,
! [A: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7013_prod__ge__1,axiom,
! [A: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_7014_prod_OUnion__comp,axiom,
! [B4: set_set_int,G: int > assn] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ B4 )
=> ( finite_finite_int @ X3 ) )
=> ( ! [A12: set_int] :
( ( member_set_int @ A12 @ B4 )
=> ! [A23: set_int] :
( ( member_set_int @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: int] :
( ( member_int @ X3 @ A12 )
=> ( ( member_int @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_assn ) ) ) ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ ( comple3221217463730067765et_int @ B4 ) )
= ( comp_s4595925057662634336t_assn @ groups5950967880310095793t_assn @ groups7882442080178216443t_assn @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7015_prod_OUnion__comp,axiom,
! [B4: set_set_o,G: $o > assn] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ B4 )
=> ( finite_finite_o @ X3 ) )
=> ( ! [A12: set_o] :
( ( member_set_o @ A12 @ B4 )
=> ! [A23: set_o] :
( ( member_set_o @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: $o] :
( ( member_o @ X3 @ A12 )
=> ( ( member_o @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_assn ) ) ) ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ ( comple90263536869209701_set_o @ B4 ) )
= ( comp_s518380907012905326o_assn @ groups6963733262650904763o_assn @ groups5301882518646026715o_assn @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7016_prod_OUnion__comp,axiom,
! [B4: set_set_Product_unit,G: product_unit > assn] :
( ! [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ B4 )
=> ( finite4290736615968046902t_unit @ X3 ) )
=> ( ! [A12: set_Product_unit] :
( ( member5877623283571906838t_unit @ A12 @ B4 )
=> ! [A23: set_Product_unit] :
( ( member5877623283571906838t_unit @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A12 )
=> ( ( member_Product_unit @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_assn ) ) ) ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ ( comple3349607820504768346t_unit @ B4 ) )
= ( comp_s8247530669214396195t_assn @ groups2392624107559085104t_assn @ groups5360233303959380816t_assn @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7017_prod_OUnion__comp,axiom,
! [B4: set_set_nat,G: nat > assn] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B4 )
=> ( finite_finite_nat @ X3 ) )
=> ( ! [A12: set_nat] :
( ( member_set_nat @ A12 @ B4 )
=> ! [A23: set_nat] :
( ( member_set_nat @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A12 )
=> ( ( member_nat @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_assn ) ) ) ) ) )
=> ( ( groups6906906614972039071t_assn @ G @ ( comple7399068483239264473et_nat @ B4 ) )
= ( comp_s2451231738238893644t_assn @ groups1535173586962943829t_assn @ groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7018_prod_OUnion__comp,axiom,
! [B4: set_set_Code_integer,G: code_integer > assn] :
( ! [X3: set_Code_integer] :
( ( member1834298987299537450nteger @ X3 @ B4 )
=> ( finite6017078050557962740nteger @ X3 ) )
=> ( ! [A12: set_Code_integer] :
( ( member1834298987299537450nteger @ A12 @ B4 )
=> ! [A23: set_Code_integer] :
( ( member1834298987299537450nteger @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A12 )
=> ( ( member_Code_integer @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_assn ) ) ) ) ) )
=> ( ( groups1304777262505850412r_assn @ G @ ( comple739944243200306918nteger @ B4 ) )
= ( comp_s3531641609786562035r_assn @ groups4245288774754739042r_assn @ groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7019_prod_OUnion__comp,axiom,
! [B4: set_set_int,G: int > nat] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ B4 )
=> ( finite_finite_int @ X3 ) )
=> ( ! [A12: set_int] :
( ( member_set_int @ A12 @ B4 )
=> ! [A23: set_int] :
( ( member_set_int @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: int] :
( ( member_int @ X3 @ A12 )
=> ( ( member_int @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_nat ) ) ) ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ ( comple3221217463730067765et_int @ B4 ) )
= ( comp_s3237642481825588056nt_nat @ groups6435469334873047129nt_nat @ groups1707563613775114915nt_nat @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7020_prod_OUnion__comp,axiom,
! [B4: set_set_o,G: $o > nat] :
( ! [X3: set_o] :
( ( member_set_o @ X3 @ B4 )
=> ( finite_finite_o @ X3 ) )
=> ( ! [A12: set_o] :
( ( member_set_o @ A12 @ B4 )
=> ! [A23: set_o] :
( ( member_set_o @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: $o] :
( ( member_o @ X3 @ A12 )
=> ( ( member_o @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_nat ) ) ) ) ) )
=> ( ( groups3504817904513533571_o_nat @ G @ ( comple90263536869209701_set_o @ B4 ) )
= ( comp_s6084864194472624150_o_nat @ groups4868773677410035043_o_nat @ groups3504817904513533571_o_nat @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7021_prod_OUnion__comp,axiom,
! [B4: set_set_Product_unit,G: product_unit > nat] :
( ! [X3: set_Product_unit] :
( ( member5877623283571906838t_unit @ X3 @ B4 )
=> ( finite4290736615968046902t_unit @ X3 ) )
=> ( ! [A12: set_Product_unit] :
( ( member5877623283571906838t_unit @ A12 @ B4 )
=> ! [A23: set_Product_unit] :
( ( member5877623283571906838t_unit @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A12 )
=> ( ( member_Product_unit @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_nat ) ) ) ) ) )
=> ( ( groups396421249026171384it_nat @ G @ ( comple3349607820504768346t_unit @ B4 ) )
= ( comp_s8928436226432208331it_nat @ groups8454329047868613336it_nat @ groups396421249026171384it_nat @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7022_prod_OUnion__comp,axiom,
! [B4: set_set_Code_integer,G: code_integer > nat] :
( ! [X3: set_Code_integer] :
( ( member1834298987299537450nteger @ X3 @ B4 )
=> ( finite6017078050557962740nteger @ X3 ) )
=> ( ! [A12: set_Code_integer] :
( ( member1834298987299537450nteger @ A12 @ B4 )
=> ! [A23: set_Code_integer] :
( ( member1834298987299537450nteger @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ A12 )
=> ( ( member_Code_integer @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_nat ) ) ) ) ) )
=> ( ( groups3190895334310489300er_nat @ G @ ( comple739944243200306918nteger @ B4 ) )
= ( comp_s8759123648413916139er_nat @ groups6044384099095135754er_nat @ groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7023_prod_OUnion__comp,axiom,
! [B4: set_set_int,G: int > int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ B4 )
=> ( finite_finite_int @ X3 ) )
=> ( ! [A12: set_int] :
( ( member_set_int @ A12 @ B4 )
=> ! [A23: set_int] :
( ( member_set_int @ A23 @ B4 )
=> ( ( A12 != A23 )
=> ! [X3: int] :
( ( member_int @ X3 @ A12 )
=> ( ( member_int @ X3 @ A23 )
=> ( ( G @ X3 )
= one_one_int ) ) ) ) ) )
=> ( ( groups1705073143266064639nt_int @ G @ ( comple3221217463730067765et_int @ B4 ) )
= ( comp_s9098231955081793132nt_int @ groups6432978864363996853nt_int @ groups1705073143266064639nt_int @ G @ B4 ) ) ) ) ).
% prod.Union_comp
thf(fact_7024_neg__eq__iff__add__eq__0,axiom,
! [A2: rat,B: rat] :
( ( ( uminus_uminus_rat @ A2 )
= B )
= ( ( plus_plus_rat @ A2 @ B )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_7025_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B: int] :
( ( ( uminus_uminus_int @ A2 )
= B )
= ( ( plus_plus_int @ A2 @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_7026_neg__eq__iff__add__eq__0,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A2 )
= B )
= ( ( plus_p5714425477246183910nteger @ A2 @ B )
= zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_7027_eq__neg__iff__add__eq__0,axiom,
! [A2: rat,B: rat] :
( ( A2
= ( uminus_uminus_rat @ B ) )
= ( ( plus_plus_rat @ A2 @ B )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_7028_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B: int] :
( ( A2
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A2 @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_7029_eq__neg__iff__add__eq__0,axiom,
! [A2: code_integer,B: code_integer] :
( ( A2
= ( uminus1351360451143612070nteger @ B ) )
= ( ( plus_p5714425477246183910nteger @ A2 @ B )
= zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_7030_add_Oinverse__unique,axiom,
! [A2: rat,B: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_7031_add_Oinverse__unique,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_7032_add_Oinverse__unique,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A2 @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A2 )
= B ) ) ).
% add.inverse_unique
thf(fact_7033_ab__group__add__class_Oab__left__minus,axiom,
! [A2: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_7034_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_7035_ab__group__add__class_Oab__left__minus,axiom,
! [A2: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
= zero_z3403309356797280102nteger ) ).
% ab_group_add_class.ab_left_minus
thf(fact_7036_add__eq__0__iff,axiom,
! [A2: rat,B: rat] :
( ( ( plus_plus_rat @ A2 @ B )
= zero_zero_rat )
= ( B
= ( uminus_uminus_rat @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_7037_add__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_7038_add__eq__0__iff,axiom,
! [A2: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A2 @ B )
= zero_z3403309356797280102nteger )
= ( B
= ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_7039_le__minus__one__simps_I4_J,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(4)
thf(fact_7040_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_7041_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_7042_le__minus__one__simps_I2_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% le_minus_one_simps(2)
thf(fact_7043_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_7044_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_7045_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_7046_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_7047_less__minus__one__simps_I2_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% less_minus_one_simps(2)
thf(fact_7048_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_7049_less__minus__one__simps_I2_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% less_minus_one_simps(2)
thf(fact_7050_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(4)
thf(fact_7051_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_7052_less__minus__one__simps_I4_J,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(4)
thf(fact_7053_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B6 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7054_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7055_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B6 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7056_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B6 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_7057_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_7058_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B6 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_7059_group__cancel_Osub2,axiom,
! [B4: rat,K3: rat,B: rat,A2: rat] :
( ( B4
= ( plus_plus_rat @ K3 @ B ) )
=> ( ( minus_minus_rat @ A2 @ B4 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K3 ) @ ( minus_minus_rat @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_7060_group__cancel_Osub2,axiom,
! [B4: int,K3: int,B: int,A2: int] :
( ( B4
= ( plus_plus_int @ K3 @ B ) )
=> ( ( minus_minus_int @ A2 @ B4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K3 ) @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_7061_group__cancel_Osub2,axiom,
! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
( ( B4
= ( plus_p5714425477246183910nteger @ K3 @ B ) )
=> ( ( minus_8373710615458151222nteger @ A2 @ B4 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K3 ) @ ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_7062_square__eq__1__iff,axiom,
! [X: rat] :
( ( ( times_times_rat @ X @ X )
= one_one_rat )
= ( ( X = one_one_rat )
| ( X
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% square_eq_1_iff
thf(fact_7063_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_7064_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( ( times_3573771949741848930nteger @ X @ X )
= one_one_Code_integer )
= ( ( X = one_one_Code_integer )
| ( X
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% square_eq_1_iff
thf(fact_7065_inf__cancel__left2,axiom,
! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ A2 ) @ ( inf_in4660618365625256667t_unit @ X @ B ) )
= bot_bo3957492148770167129t_unit ) ).
% inf_cancel_left2
thf(fact_7066_inf__cancel__left2,axiom,
! [X: set_o,A2: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A2 ) @ ( inf_inf_set_o @ X @ B ) )
= bot_bot_set_o ) ).
% inf_cancel_left2
thf(fact_7067_inf__cancel__left2,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( inf_inf_set_nat @ X @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_7068_inf__cancel__left2,axiom,
! [X: set_int,A2: set_int,B: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A2 ) @ ( inf_inf_set_int @ X @ B ) )
= bot_bot_set_int ) ).
% inf_cancel_left2
thf(fact_7069_inf__cancel__left2,axiom,
! [X: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ A2 ) @ ( inf_in1906310914598751387_nat_o @ X @ B ) )
= bot_bo7824918357723582233_nat_o ) ).
% inf_cancel_left2
thf(fact_7070_inf__cancel__left2,axiom,
! [X: assn,A2: assn,B: assn] :
( ( inf_inf_assn @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ A2 ) @ ( inf_inf_assn @ X @ B ) )
= bot_bot_assn ) ).
% inf_cancel_left2
thf(fact_7071_inf__cancel__left2,axiom,
! [X: product_unit,A2: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ A2 ) @ ( inf_inf_Product_unit @ X @ B ) )
= bot_bot_Product_unit ) ).
% inf_cancel_left2
thf(fact_7072_inf__cancel__left1,axiom,
! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ X @ A2 ) @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ B ) )
= bot_bo3957492148770167129t_unit ) ).
% inf_cancel_left1
thf(fact_7073_inf__cancel__left1,axiom,
! [X: set_o,A2: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
= bot_bot_set_o ) ).
% inf_cancel_left1
thf(fact_7074_inf__cancel__left1,axiom,
! [X: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_7075_inf__cancel__left1,axiom,
! [X: set_int,A2: set_int,B: set_int] :
( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
= bot_bot_set_int ) ).
% inf_cancel_left1
thf(fact_7076_inf__cancel__left1,axiom,
! [X: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ X @ A2 ) @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ B ) )
= bot_bo7824918357723582233_nat_o ) ).
% inf_cancel_left1
thf(fact_7077_inf__cancel__left1,axiom,
! [X: assn,A2: assn,B: assn] :
( ( inf_inf_assn @ ( inf_inf_assn @ X @ A2 ) @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ B ) )
= bot_bot_assn ) ).
% inf_cancel_left1
thf(fact_7078_inf__cancel__left1,axiom,
! [X: product_unit,A2: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ A2 ) @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ B ) )
= bot_bot_Product_unit ) ).
% inf_cancel_left1
thf(fact_7079_subset__Compl__self__eq,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% subset_Compl_self_eq
thf(fact_7080_subset__Compl__self__eq,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ A ) )
= ( A = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_7081_subset__Compl__self__eq,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
= ( A = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_7082_subset__Compl__self__eq,axiom,
! [A: set_Pr4532377907799695533_nat_o] :
( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ A ) )
= ( A = bot_bo7824918357723582233_nat_o ) ) ).
% subset_Compl_self_eq
thf(fact_7083_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_7084_Compl__UNIV__eq,axiom,
( ( uminus5254974436814262692_nat_o @ top_to4869887016220417533_nat_o )
= bot_bo7824918357723582233_nat_o ) ).
% Compl_UNIV_eq
thf(fact_7085_Compl__UNIV__eq,axiom,
( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Compl_UNIV_eq
thf(fact_7086_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_o @ top_top_set_o )
= bot_bot_set_o ) ).
% Compl_UNIV_eq
thf(fact_7087_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_7088_Compl__UNIV__eq,axiom,
( ( uminus1532241313380277803et_int @ top_top_set_int )
= bot_bot_set_int ) ).
% Compl_UNIV_eq
thf(fact_7089_Compl__empty__eq,axiom,
( ( uminus5254974436814262692_nat_o @ bot_bo7824918357723582233_nat_o )
= top_to4869887016220417533_nat_o ) ).
% Compl_empty_eq
thf(fact_7090_Compl__empty__eq,axiom,
( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
= top_to1996260823553986621t_unit ) ).
% Compl_empty_eq
thf(fact_7091_Compl__empty__eq,axiom,
( ( uminus_uminus_set_o @ bot_bot_set_o )
= top_top_set_o ) ).
% Compl_empty_eq
thf(fact_7092_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_7093_Compl__empty__eq,axiom,
( ( uminus1532241313380277803et_int @ bot_bot_set_int )
= top_top_set_int ) ).
% Compl_empty_eq
thf(fact_7094_Compl__partition2,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ A ) @ A )
= top_to1996260823553986621t_unit ) ).
% Compl_partition2
thf(fact_7095_Compl__partition2,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A ) @ A )
= top_top_set_o ) ).
% Compl_partition2
thf(fact_7096_Compl__partition2,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_7097_Compl__partition2,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
= top_top_set_int ) ).
% Compl_partition2
thf(fact_7098_Compl__partition,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
= top_to1996260823553986621t_unit ) ).
% Compl_partition
thf(fact_7099_Compl__partition,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ ( uminus_uminus_set_o @ A ) )
= top_top_set_o ) ).
% Compl_partition
thf(fact_7100_Compl__partition,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_7101_Compl__partition,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
= top_top_set_int ) ).
% Compl_partition
thf(fact_7102_Compl__eq__Diff__UNIV,axiom,
( uminus5944136376168626660t_unit
= ( minus_6452836326544984404t_unit @ top_to1996260823553986621t_unit ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_7103_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_o
= ( minus_minus_set_o @ top_top_set_o ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_7104_Compl__eq__Diff__UNIV,axiom,
( uminus1532241313380277803et_int
= ( minus_minus_set_int @ top_top_set_int ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_7105_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_7106_Compl__Int,axiom,
! [A: set_nat,B4: set_nat] :
( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% Compl_Int
thf(fact_7107_Compl__Un,axiom,
! [A: set_nat,B4: set_nat] :
( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% Compl_Un
thf(fact_7108_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A6: rat,K5: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K5 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K5 ) @ A6 ) @ one_one_rat ) @ K5 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_7109_gbinomial__index__swap,axiom,
! [K3: nat,N2: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K3 ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ N2 ) ) ) ).
% gbinomial_index_swap
thf(fact_7110_prod_OUnion__disjoint,axiom,
! [C5: set_set_nat,G: nat > nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ( finite_finite_nat @ X3 ) )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C5 )
=> ! [Xa3: set_nat] :
( ( member_set_nat @ Xa3 @ C5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf_set_nat @ X3 @ Xa3 )
= bot_bot_set_nat ) ) ) )
=> ( ( groups708209901874060359at_nat @ G @ ( comple7399068483239264473et_nat @ C5 ) )
= ( comp_s5456809700045850692at_nat @ groups4248547760180025341at_nat @ groups708209901874060359at_nat @ G @ C5 ) ) ) ) ).
% prod.Union_disjoint
thf(fact_7111_Gcd__remove0__nat,axiom,
! [M: set_nat] :
( ( finite_finite_nat @ M )
=> ( ( gcd_Gcd_nat @ M )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_7112_Gcd__eq__1__I,axiom,
! [A2: nat,A: set_nat] :
( ( dvd_dvd_nat @ A2 @ one_one_nat )
=> ( ( member_nat @ A2 @ A )
=> ( ( gcd_Gcd_nat @ A )
= one_one_nat ) ) ) ).
% Gcd_eq_1_I
thf(fact_7113_Gcd__eq__1__I,axiom,
! [A2: int,A: set_int] :
( ( dvd_dvd_int @ A2 @ one_one_int )
=> ( ( member_int @ A2 @ A )
=> ( ( gcd_Gcd_int @ A )
= one_one_int ) ) ) ).
% Gcd_eq_1_I
thf(fact_7114_prod__le__1,axiom,
! [A: set_nat,F: nat > code_natural] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
& ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
=> ( ord_le1926595141338095240atural @ ( groups2279045934846249631atural @ F @ A ) @ one_one_Code_natural ) ) ).
% prod_le_1
thf(fact_7115_prod__le__1,axiom,
! [A: set_int,F: int > code_natural] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
& ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
=> ( ord_le1926595141338095240atural @ ( groups2650699495083094267atural @ F @ A ) @ one_one_Code_natural ) ) ).
% prod_le_1
thf(fact_7116_prod__le__1,axiom,
! [A: set_o,F: $o > code_natural] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
& ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
=> ( ord_le1926595141338095240atural @ ( groups6518289543945209051atural @ F @ A ) @ one_one_Code_natural ) ) ).
% prod_le_1
thf(fact_7117_prod__le__1,axiom,
! [A: set_Product_unit,F: product_unit > code_natural] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
& ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
=> ( ord_le1926595141338095240atural @ ( groups7008954963700796496atural @ F @ A ) @ one_one_Code_natural ) ) ).
% prod_le_1
thf(fact_7118_prod__le__1,axiom,
! [A: set_nat,F: nat > code_integer] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7119_prod__le__1,axiom,
! [A: set_int,F: int > code_integer] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7120_prod__le__1,axiom,
! [A: set_o,F: $o > code_integer] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7121_prod__le__1,axiom,
! [A: set_Product_unit,F: product_unit > code_integer] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
& ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
=> ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% prod_le_1
thf(fact_7122_prod__le__1,axiom,
! [A: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7123_prod__le__1,axiom,
! [A: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_7124_prod_Orelated,axiom,
! [R3: code_integer > code_integer > $o,S: set_nat,H2: nat > code_integer,G: nat > code_integer] :
( ( R3 @ one_one_Code_integer @ one_one_Code_integer )
=> ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups3455450783089532116nteger @ H2 @ S ) @ ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7125_prod_Orelated,axiom,
! [R3: code_integer > code_integer > $o,S: set_Code_integer,H2: code_integer > code_integer,G: code_integer > code_integer] :
( ( R3 @ one_one_Code_integer @ one_one_Code_integer )
=> ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups3674199335183972705nteger @ H2 @ S ) @ ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7126_prod_Orelated,axiom,
! [R3: code_natural > code_natural > $o,S: set_nat,H2: nat > code_natural,G: nat > code_natural] :
( ( R3 @ one_one_Code_natural @ one_one_Code_natural )
=> ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_2397367101498566445atural @ X12 @ Y12 ) @ ( times_2397367101498566445atural @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups2279045934846249631atural @ H2 @ S ) @ ( groups2279045934846249631atural @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7127_prod_Orelated,axiom,
! [R3: code_natural > code_natural > $o,S: set_Code_integer,H2: code_integer > code_natural,G: code_integer > code_natural] :
( ( R3 @ one_one_Code_natural @ one_one_Code_natural )
=> ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_2397367101498566445atural @ X12 @ Y12 ) @ ( times_2397367101498566445atural @ X23 @ Y23 ) ) )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups2497794486940690220atural @ H2 @ S ) @ ( groups2497794486940690220atural @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7128_prod_Orelated,axiom,
! [R3: assn > assn > $o,S: set_nat,H2: nat > assn,G: nat > assn] :
( ( R3 @ one_one_assn @ one_one_assn )
=> ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups6906906614972039071t_assn @ H2 @ S ) @ ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7129_prod_Orelated,axiom,
! [R3: assn > assn > $o,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
( ( R3 @ one_one_assn @ one_one_assn )
=> ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X23 @ Y23 ) ) )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups1304777262505850412r_assn @ H2 @ S ) @ ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7130_prod_Orelated,axiom,
! [R3: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
( ( R3 @ one_one_rat @ one_one_rat )
=> ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups73079841787564623at_rat @ H2 @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7131_prod_Orelated,axiom,
! [R3: rat > rat > $o,S: set_Code_integer,H2: code_integer > rat,G: code_integer > rat] :
( ( R3 @ one_one_rat @ one_one_rat )
=> ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups2555765274223993564er_rat @ H2 @ S ) @ ( groups2555765274223993564er_rat @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7132_prod_Orelated,axiom,
! [R3: nat > nat > $o,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
( ( R3 @ one_one_nat @ one_one_nat )
=> ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_times_nat @ X12 @ Y12 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups3190895334310489300er_nat @ H2 @ S ) @ ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7133_prod_Orelated,axiom,
! [R3: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
( ( R3 @ one_one_int @ one_one_int )
=> ( ! [X12: int,Y12: int,X23: int,Y23: int] :
( ( ( R3 @ X12 @ X23 )
& ( R3 @ Y12 @ Y23 ) )
=> ( R3 @ ( times_times_int @ X12 @ Y12 ) @ ( times_times_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups705719431365010083at_int @ H2 @ S ) @ ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ) ).
% prod.related
thf(fact_7134_prod_Oinsert__if,axiom,
! [A: set_int,X: int,G: int > assn] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X @ A )
=> ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
= ( groups7882442080178216443t_assn @ G @ A ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7135_prod_Oinsert__if,axiom,
! [A: set_o,X: $o,G: $o > assn] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ X @ A )
=> ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
= ( groups5301882518646026715o_assn @ G @ A ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7136_prod_Oinsert__if,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
= ( groups5360233303959380816t_assn @ G @ A ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7137_prod_Oinsert__if,axiom,
! [A: set_nat,X: nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
= ( groups6906906614972039071t_assn @ G @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7138_prod_Oinsert__if,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ X @ A )
=> ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
= ( groups1304777262505850412r_assn @ G @ A ) ) )
& ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7139_prod_Oinsert__if,axiom,
! [A: set_int,X: int,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X @ A )
=> ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
= ( groups1072433553688619179nt_rat @ G @ A ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7140_prod_Oinsert__if,axiom,
! [A: set_o,X: $o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( ( member_o @ X @ A )
=> ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
= ( groups2869687844427037835_o_rat @ G @ A ) ) )
& ( ~ ( member_o @ X @ A )
=> ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7141_prod_Oinsert__if,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( groups8984663225794451456it_rat @ G @ A ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7142_prod_Oinsert__if,axiom,
! [A: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
= ( groups73079841787564623at_rat @ G @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7143_prod_Oinsert__if,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( ( member_Code_integer @ X @ A )
=> ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( groups2555765274223993564er_rat @ G @ A ) ) )
& ( ~ ( member_Code_integer @ X @ A )
=> ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_7144_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T5: set_int,S: set_int,I2: int > int,J: int > int,T2: set_int,G: int > assn,H2: int > assn] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ S )
= ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7145_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T5: set_o,S: set_int,I2: $o > int,J: int > $o,T2: set_o,G: int > assn,H2: $o > assn] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_o @ T5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ S )
= ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7146_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T5: set_Product_unit,S: set_int,I2: product_unit > int,J: int > product_unit,T2: set_Product_unit,G: int > assn,H2: product_unit > assn] :
( ( finite_finite_int @ S4 )
=> ( ( finite4290736615968046902t_unit @ T5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ S )
= ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7147_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_o,T5: set_int,S: set_o,I2: int > $o,J: $o > int,T2: set_int,G: $o > assn,H2: int > assn] :
( ( finite_finite_o @ S4 )
=> ( ( finite_finite_int @ T5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
=> ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
=> ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ S )
= ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7148_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_o,T5: set_o,S: set_o,I2: $o > $o,J: $o > $o,T2: set_o,G: $o > assn,H2: $o > assn] :
( ( finite_finite_o @ S4 )
=> ( ( finite_finite_o @ T5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
=> ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
=> ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ S )
= ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7149_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_o,T5: set_Product_unit,S: set_o,I2: product_unit > $o,J: $o > product_unit,T2: set_Product_unit,G: $o > assn,H2: product_unit > assn] :
( ( finite_finite_o @ S4 )
=> ( ( finite4290736615968046902t_unit @ T5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
=> ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
=> ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ S )
= ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7150_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Product_unit,T5: set_int,S: set_Product_unit,I2: int > product_unit,J: product_unit > int,T2: set_int,G: product_unit > assn,H2: int > assn] :
( ( finite4290736615968046902t_unit @ S4 )
=> ( ( finite_finite_int @ T5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
=> ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
=> ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ S )
= ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7151_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Product_unit,T5: set_o,S: set_Product_unit,I2: $o > product_unit,J: product_unit > $o,T2: set_o,G: product_unit > assn,H2: $o > assn] :
( ( finite4290736615968046902t_unit @ S4 )
=> ( ( finite_finite_o @ T5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
=> ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
=> ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ S )
= ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7152_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_Product_unit,T5: set_Product_unit,S: set_Product_unit,I2: product_unit > product_unit,J: product_unit > product_unit,T2: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
( ( finite4290736615968046902t_unit @ S4 )
=> ( ( finite4290736615968046902t_unit @ T5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
=> ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
=> ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ S )
= ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7153_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T5: set_Code_integer,S: set_int,I2: code_integer > int,J: int > code_integer,T2: set_Code_integer,G: int > assn,H2: code_integer > assn] :
( ( finite_finite_int @ S4 )
=> ( ( finite6017078050557962740nteger @ T5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
=> ( member_Code_integer @ ( J @ A4 ) @ ( minus_2355218937544613996nteger @ T2 @ T5 ) ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T2 @ T5 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T2 @ T5 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S4 )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ T5 )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ! [A4: int] :
( ( member_int @ A4 @ S )
=> ( ( H2 @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ S )
= ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_7154_gbinomial__minus,axiom,
! [A2: rat,K3: nat] :
( ( gbinomial_rat @ ( uminus_uminus_rat @ A2 ) @ K3 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) ) ) ).
% gbinomial_minus
thf(fact_7155_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_7156_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(3)
thf(fact_7157_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_7158_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_7159_le__minus__one__simps_I1_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% le_minus_one_simps(1)
thf(fact_7160_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_7161_less__minus__one__simps_I1_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% less_minus_one_simps(1)
thf(fact_7162_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_7163_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_7164_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(3)
thf(fact_7165_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_7166_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_7167_inf__shunt,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X @ Y )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ X @ ( uminus5944136376168626660t_unit @ Y ) ) ) ).
% inf_shunt
thf(fact_7168_inf__shunt,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).
% inf_shunt
thf(fact_7169_inf__shunt,axiom,
! [X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% inf_shunt
thf(fact_7170_inf__shunt,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ X @ Y )
= bot_bo7824918357723582233_nat_o )
= ( ord_le2965882846123202637_nat_o @ X @ ( uminus5254974436814262692_nat_o @ Y ) ) ) ).
% inf_shunt
thf(fact_7171_inf__shunt,axiom,
! [X: assn,Y: assn] :
( ( ( inf_inf_assn @ X @ Y )
= bot_bot_assn )
= ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).
% inf_shunt
thf(fact_7172_inf__shunt,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( inf_inf_Product_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% inf_shunt
thf(fact_7173_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_7174_power__minus,axiom,
! [A2: rat,N2: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N2 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% power_minus
thf(fact_7175_power__minus,axiom,
! [A2: int,N2: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N2 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A2 @ N2 ) ) ) ).
% power_minus
thf(fact_7176_power__minus,axiom,
! [A2: code_integer,N2: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N2 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ).
% power_minus
thf(fact_7177_disjoint__eq__subset__Compl,axiom,
! [A: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_7178_disjoint__eq__subset__Compl,axiom,
! [A: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_7179_disjoint__eq__subset__Compl,axiom,
! [A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_7180_disjoint__eq__subset__Compl,axiom,
! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
= bot_bo7824918357723582233_nat_o )
= ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_7181_disjoint__eq__subset__Compl,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_7182_Compl__insert,axiom,
! [X: product_unit,A: set_Product_unit] :
( ( uminus5944136376168626660t_unit @ ( insert_Product_unit @ X @ A ) )
= ( minus_6452836326544984404t_unit @ ( uminus5944136376168626660t_unit @ A ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ).
% Compl_insert
thf(fact_7183_Compl__insert,axiom,
! [X: $o,A: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X @ A ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_7184_Compl__insert,axiom,
! [X: int,A: set_int] :
( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A ) )
= ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% Compl_insert
thf(fact_7185_Compl__insert,axiom,
! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
( ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
= ( minus_1801376950450012436_nat_o @ ( uminus5254974436814262692_nat_o @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ).
% Compl_insert
thf(fact_7186_Compl__insert,axiom,
! [X: nat,A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_7187_Image__subset__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A ) @ B4 )
= ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ ( converse_nat_nat @ R2 ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_7188_less__1__prod2,axiom,
! [I: set_int,I2: int,F: int > code_integer] :
( ( finite_finite_int @ I )
=> ( ( member_int @ I2 @ I )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7189_less__1__prod2,axiom,
! [I: set_o,I2: $o,F: $o > code_integer] :
( ( finite_finite_o @ I )
=> ( ( member_o @ I2 @ I )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7190_less__1__prod2,axiom,
! [I: set_Product_unit,I2: product_unit,F: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( member_Product_unit @ I2 @ I )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7191_less__1__prod2,axiom,
! [I: set_nat,I2: nat,F: nat > code_integer] :
( ( finite_finite_nat @ I )
=> ( ( member_nat @ I2 @ I )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7192_less__1__prod2,axiom,
! [I: set_Code_integer,I2: code_integer,F: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ I )
=> ( ( member_Code_integer @ I2 @ I )
=> ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7193_less__1__prod2,axiom,
! [I: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I )
=> ( ( member_int @ I2 @ I )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7194_less__1__prod2,axiom,
! [I: set_o,I2: $o,F: $o > rat] :
( ( finite_finite_o @ I )
=> ( ( member_o @ I2 @ I )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7195_less__1__prod2,axiom,
! [I: set_Product_unit,I2: product_unit,F: product_unit > rat] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( member_Product_unit @ I2 @ I )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups8984663225794451456it_rat @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7196_less__1__prod2,axiom,
! [I: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I )
=> ( ( member_nat @ I2 @ I )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7197_less__1__prod2,axiom,
! [I: set_Code_integer,I2: code_integer,F: code_integer > rat] :
( ( finite6017078050557962740nteger @ I )
=> ( ( member_Code_integer @ I2 @ I )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_7198_less__1__prod,axiom,
! [I: set_Code_integer,F: code_integer > int] :
( ( finite6017078050557962740nteger @ I )
=> ( ( I != bot_bo3990330152332043303nteger )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7199_less__1__prod,axiom,
! [I: set_Product_unit,F: product_unit > int] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups393930778517121108it_int @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7200_less__1__prod,axiom,
! [I: set_o,F: $o > int] :
( ( finite_finite_o @ I )
=> ( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7201_less__1__prod,axiom,
! [I: set_nat,F: nat > int] :
( ( finite_finite_nat @ I )
=> ( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7202_less__1__prod,axiom,
! [I: set_int,F: int > int] :
( ( finite_finite_int @ I )
=> ( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7203_less__1__prod,axiom,
! [I: set_Code_integer,F: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ I )
=> ( ( I != bot_bo3990330152332043303nteger )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7204_less__1__prod,axiom,
! [I: set_Product_unit,F: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ I )
=> ( ( I != bot_bo3957492148770167129t_unit )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7205_less__1__prod,axiom,
! [I: set_o,F: $o > code_integer] :
( ( finite_finite_o @ I )
=> ( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7206_less__1__prod,axiom,
! [I: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ I )
=> ( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7207_less__1__prod,axiom,
! [I: set_int,F: int > code_integer] :
( ( finite_finite_int @ I )
=> ( ( I != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
=> ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I ) ) ) ) ) ).
% less_1_prod
thf(fact_7208_prod_Oreindex__nontrivial,axiom,
! [A: set_nat,H2: nat > nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_assn ) ) ) ) )
=> ( ( groups6906906614972039071t_assn @ G @ ( image_nat_nat2 @ H2 @ A ) )
= ( groups6906906614972039071t_assn @ ( comp_nat_assn_nat @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7209_prod_Oreindex__nontrivial,axiom,
! [A: set_nat,H2: nat > code_integer,G: code_integer > assn] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_assn ) ) ) ) )
=> ( ( groups1304777262505850412r_assn @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
= ( groups6906906614972039071t_assn @ ( comp_C8462447471145125178sn_nat @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7210_prod_Oreindex__nontrivial,axiom,
! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ( member_Code_integer @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_assn ) ) ) ) )
=> ( ( groups1304777262505850412r_assn @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
= ( groups1304777262505850412r_assn @ ( comp_C3026223900822616519nteger @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7211_prod_Oreindex__nontrivial,axiom,
! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ( member_Code_integer @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_nat ) ) ) ) )
=> ( ( groups3190895334310489300er_nat @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
= ( groups3190895334310489300er_nat @ ( comp_C7238605732685392495nteger @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7212_prod_Oreindex__nontrivial,axiom,
! [A: set_nat,H2: nat > nat,G: nat > int] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_int ) ) ) ) )
=> ( ( groups705719431365010083at_int @ G @ ( image_nat_nat2 @ H2 @ A ) )
= ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7213_prod_Oreindex__nontrivial,axiom,
! [A: set_nat,H2: nat > code_integer,G: code_integer > int] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_int ) ) ) ) )
=> ( ( groups3188404863801439024er_int @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
= ( groups705719431365010083at_int @ ( comp_C5048910307564104766nt_nat @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7214_prod_Oreindex__nontrivial,axiom,
! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ( member_Code_integer @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_int ) ) ) ) )
=> ( ( groups3188404863801439024er_int @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
= ( groups3188404863801439024er_int @ ( comp_C7610259292922237131nteger @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7215_prod_Oreindex__nontrivial,axiom,
! [A: set_nat,H2: nat > nat,G: nat > code_integer] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_Code_integer ) ) ) ) )
=> ( ( groups3455450783089532116nteger @ G @ ( image_nat_nat2 @ H2 @ A ) )
= ( groups3455450783089532116nteger @ ( comp_n3898172953802868194er_nat @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7216_prod_Oreindex__nontrivial,axiom,
! [A: set_nat,H2: nat > code_integer,G: code_integer > code_integer] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_Code_integer ) ) ) ) )
=> ( ( groups3674199335183972705nteger @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
= ( groups3455450783089532116nteger @ ( comp_C6974050283906349679er_nat @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7217_prod_Oreindex__nontrivial,axiom,
! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [X3: code_integer,Y2: code_integer] :
( ( member_Code_integer @ X3 @ A )
=> ( ( member_Code_integer @ Y2 @ A )
=> ( ( X3 != Y2 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y2 ) )
=> ( ( G @ ( H2 @ X3 ) )
= one_one_Code_integer ) ) ) ) )
=> ( ( groups3674199335183972705nteger @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
= ( groups3674199335183972705nteger @ ( comp_C7449957260575251196nteger @ G @ H2 ) @ A ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_7218_prod_Osubset__diff,axiom,
! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > assn] :
( ( ord_le7084787975880047091nteger @ B4 @ A )
=> ( ( finite6017078050557962740nteger @ A )
=> ( ( groups1304777262505850412r_assn @ G @ A )
= ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7219_prod_Osubset__diff,axiom,
! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > rat] :
( ( ord_le7084787975880047091nteger @ B4 @ A )
=> ( ( finite6017078050557962740nteger @ A )
=> ( ( groups2555765274223993564er_rat @ G @ A )
= ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7220_prod_Osubset__diff,axiom,
! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > nat] :
( ( ord_le7084787975880047091nteger @ B4 @ A )
=> ( ( finite6017078050557962740nteger @ A )
=> ( ( groups3190895334310489300er_nat @ G @ A )
= ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7221_prod_Osubset__diff,axiom,
! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > int] :
( ( ord_le7084787975880047091nteger @ B4 @ A )
=> ( ( finite6017078050557962740nteger @ A )
=> ( ( groups3188404863801439024er_int @ G @ A )
= ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7222_prod_Osubset__diff,axiom,
! [B4: set_nat,A: set_nat,G: nat > assn] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups6906906614972039071t_assn @ G @ A )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7223_prod_Osubset__diff,axiom,
! [B4: set_nat,A: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups73079841787564623at_rat @ G @ A )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7224_prod_Osubset__diff,axiom,
! [B4: set_nat,A: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups705719431365010083at_int @ G @ A )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7225_prod_Osubset__diff,axiom,
! [B4: set_nat,A: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G @ A )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_7226_prod_Osame__carrier,axiom,
! [C5: set_int,A: set_int,B4: set_int,G: int > assn,H2: int > assn] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A @ C5 )
=> ( ( ord_less_eq_set_int @ B4 @ C5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups7882442080178216443t_assn @ G @ A )
= ( groups7882442080178216443t_assn @ H2 @ B4 ) )
= ( ( groups7882442080178216443t_assn @ G @ C5 )
= ( groups7882442080178216443t_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7227_prod_Osame__carrier,axiom,
! [C5: set_o,A: set_o,B4: set_o,G: $o > assn,H2: $o > assn] :
( ( finite_finite_o @ C5 )
=> ( ( ord_less_eq_set_o @ A @ C5 )
=> ( ( ord_less_eq_set_o @ B4 @ C5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups5301882518646026715o_assn @ G @ A )
= ( groups5301882518646026715o_assn @ H2 @ B4 ) )
= ( ( groups5301882518646026715o_assn @ G @ C5 )
= ( groups5301882518646026715o_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7228_prod_Osame__carrier,axiom,
! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
( ( finite4290736615968046902t_unit @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ A @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups5360233303959380816t_assn @ G @ A )
= ( groups5360233303959380816t_assn @ H2 @ B4 ) )
= ( ( groups5360233303959380816t_assn @ G @ C5 )
= ( groups5360233303959380816t_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7229_prod_Osame__carrier,axiom,
! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
( ( finite6017078050557962740nteger @ C5 )
=> ( ( ord_le7084787975880047091nteger @ A @ C5 )
=> ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups1304777262505850412r_assn @ G @ A )
= ( groups1304777262505850412r_assn @ H2 @ B4 ) )
= ( ( groups1304777262505850412r_assn @ G @ C5 )
= ( groups1304777262505850412r_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7230_prod_Osame__carrier,axiom,
! [C5: set_int,A: set_int,B4: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A @ C5 )
=> ( ( ord_less_eq_set_int @ B4 @ C5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups1707563613775114915nt_nat @ G @ A )
= ( groups1707563613775114915nt_nat @ H2 @ B4 ) )
= ( ( groups1707563613775114915nt_nat @ G @ C5 )
= ( groups1707563613775114915nt_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7231_prod_Osame__carrier,axiom,
! [C5: set_o,A: set_o,B4: set_o,G: $o > nat,H2: $o > nat] :
( ( finite_finite_o @ C5 )
=> ( ( ord_less_eq_set_o @ A @ C5 )
=> ( ( ord_less_eq_set_o @ B4 @ C5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups3504817904513533571_o_nat @ G @ A )
= ( groups3504817904513533571_o_nat @ H2 @ B4 ) )
= ( ( groups3504817904513533571_o_nat @ G @ C5 )
= ( groups3504817904513533571_o_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7232_prod_Osame__carrier,axiom,
! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
( ( finite4290736615968046902t_unit @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ A @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups396421249026171384it_nat @ G @ A )
= ( groups396421249026171384it_nat @ H2 @ B4 ) )
= ( ( groups396421249026171384it_nat @ G @ C5 )
= ( groups396421249026171384it_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7233_prod_Osame__carrier,axiom,
! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
( ( finite6017078050557962740nteger @ C5 )
=> ( ( ord_le7084787975880047091nteger @ A @ C5 )
=> ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups3190895334310489300er_nat @ G @ A )
= ( groups3190895334310489300er_nat @ H2 @ B4 ) )
= ( ( groups3190895334310489300er_nat @ G @ C5 )
= ( groups3190895334310489300er_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7234_prod_Osame__carrier,axiom,
! [C5: set_int,A: set_int,B4: set_int,G: int > int,H2: int > int] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A @ C5 )
=> ( ( ord_less_eq_set_int @ B4 @ C5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_int ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups1705073143266064639nt_int @ G @ A )
= ( groups1705073143266064639nt_int @ H2 @ B4 ) )
= ( ( groups1705073143266064639nt_int @ G @ C5 )
= ( groups1705073143266064639nt_int @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7235_prod_Osame__carrier,axiom,
! [C5: set_o,A: set_o,B4: set_o,G: $o > int,H2: $o > int] :
( ( finite_finite_o @ C5 )
=> ( ( ord_less_eq_set_o @ A @ C5 )
=> ( ( ord_less_eq_set_o @ B4 @ C5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_int ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups3502327434004483295_o_int @ G @ A )
= ( groups3502327434004483295_o_int @ H2 @ B4 ) )
= ( ( groups3502327434004483295_o_int @ G @ C5 )
= ( groups3502327434004483295_o_int @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_7236_prod_Osame__carrierI,axiom,
! [C5: set_int,A: set_int,B4: set_int,G: int > assn,H2: int > assn] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A @ C5 )
=> ( ( ord_less_eq_set_int @ B4 @ C5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups7882442080178216443t_assn @ G @ C5 )
= ( groups7882442080178216443t_assn @ H2 @ C5 ) )
=> ( ( groups7882442080178216443t_assn @ G @ A )
= ( groups7882442080178216443t_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7237_prod_Osame__carrierI,axiom,
! [C5: set_o,A: set_o,B4: set_o,G: $o > assn,H2: $o > assn] :
( ( finite_finite_o @ C5 )
=> ( ( ord_less_eq_set_o @ A @ C5 )
=> ( ( ord_less_eq_set_o @ B4 @ C5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups5301882518646026715o_assn @ G @ C5 )
= ( groups5301882518646026715o_assn @ H2 @ C5 ) )
=> ( ( groups5301882518646026715o_assn @ G @ A )
= ( groups5301882518646026715o_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7238_prod_Osame__carrierI,axiom,
! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
( ( finite4290736615968046902t_unit @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ A @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups5360233303959380816t_assn @ G @ C5 )
= ( groups5360233303959380816t_assn @ H2 @ C5 ) )
=> ( ( groups5360233303959380816t_assn @ G @ A )
= ( groups5360233303959380816t_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7239_prod_Osame__carrierI,axiom,
! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
( ( finite6017078050557962740nteger @ C5 )
=> ( ( ord_le7084787975880047091nteger @ A @ C5 )
=> ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_assn ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_assn ) )
=> ( ( ( groups1304777262505850412r_assn @ G @ C5 )
= ( groups1304777262505850412r_assn @ H2 @ C5 ) )
=> ( ( groups1304777262505850412r_assn @ G @ A )
= ( groups1304777262505850412r_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7240_prod_Osame__carrierI,axiom,
! [C5: set_int,A: set_int,B4: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A @ C5 )
=> ( ( ord_less_eq_set_int @ B4 @ C5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups1707563613775114915nt_nat @ G @ C5 )
= ( groups1707563613775114915nt_nat @ H2 @ C5 ) )
=> ( ( groups1707563613775114915nt_nat @ G @ A )
= ( groups1707563613775114915nt_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7241_prod_Osame__carrierI,axiom,
! [C5: set_o,A: set_o,B4: set_o,G: $o > nat,H2: $o > nat] :
( ( finite_finite_o @ C5 )
=> ( ( ord_less_eq_set_o @ A @ C5 )
=> ( ( ord_less_eq_set_o @ B4 @ C5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups3504817904513533571_o_nat @ G @ C5 )
= ( groups3504817904513533571_o_nat @ H2 @ C5 ) )
=> ( ( groups3504817904513533571_o_nat @ G @ A )
= ( groups3504817904513533571_o_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7242_prod_Osame__carrierI,axiom,
! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
( ( finite4290736615968046902t_unit @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ A @ C5 )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
=> ( ! [A4: product_unit] :
( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: product_unit] :
( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups396421249026171384it_nat @ G @ C5 )
= ( groups396421249026171384it_nat @ H2 @ C5 ) )
=> ( ( groups396421249026171384it_nat @ G @ A )
= ( groups396421249026171384it_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7243_prod_Osame__carrierI,axiom,
! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
( ( finite6017078050557962740nteger @ C5 )
=> ( ( ord_le7084787975880047091nteger @ A @ C5 )
=> ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
=> ( ! [A4: code_integer] :
( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_nat ) )
=> ( ! [B2: code_integer] :
( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups3190895334310489300er_nat @ G @ C5 )
= ( groups3190895334310489300er_nat @ H2 @ C5 ) )
=> ( ( groups3190895334310489300er_nat @ G @ A )
= ( groups3190895334310489300er_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7244_prod_Osame__carrierI,axiom,
! [C5: set_int,A: set_int,B4: set_int,G: int > int,H2: int > int] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A @ C5 )
=> ( ( ord_less_eq_set_int @ B4 @ C5 )
=> ( ! [A4: int] :
( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_int ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups1705073143266064639nt_int @ G @ C5 )
= ( groups1705073143266064639nt_int @ H2 @ C5 ) )
=> ( ( groups1705073143266064639nt_int @ G @ A )
= ( groups1705073143266064639nt_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7245_prod_Osame__carrierI,axiom,
! [C5: set_o,A: set_o,B4: set_o,G: $o > int,H2: $o > int] :
( ( finite_finite_o @ C5 )
=> ( ( ord_less_eq_set_o @ A @ C5 )
=> ( ( ord_less_eq_set_o @ B4 @ C5 )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
=> ( ( G @ A4 )
= one_one_int ) )
=> ( ! [B2: $o] :
( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
=> ( ( H2 @ B2 )
= one_one_int ) )
=> ( ( ( groups3502327434004483295_o_int @ G @ C5 )
= ( groups3502327434004483295_o_int @ H2 @ C5 ) )
=> ( ( groups3502327434004483295_o_int @ G @ A )
= ( groups3502327434004483295_o_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_7246_prod_Omono__neutral__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ( groups1304777262505850412r_assn @ G @ S )
= ( groups1304777262505850412r_assn @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7247_prod_Omono__neutral__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups3190895334310489300er_nat @ G @ S )
= ( groups3190895334310489300er_nat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7248_prod_Omono__neutral__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups3188404863801439024er_int @ G @ S )
= ( groups3188404863801439024er_int @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7249_prod_Omono__neutral__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_integer ) )
=> ( ( groups3674199335183972705nteger @ G @ S )
= ( groups3674199335183972705nteger @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7250_prod_Omono__neutral__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_natural] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_natural ) )
=> ( ( groups2497794486940690220atural @ G @ S )
= ( groups2497794486940690220atural @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7251_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S: set_nat,G: nat > assn] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ( groups6906906614972039071t_assn @ G @ S )
= ( groups6906906614972039071t_assn @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7252_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S: set_nat,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ S )
= ( groups705719431365010083at_int @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7253_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S: set_nat,G: nat > code_integer] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_integer ) )
=> ( ( groups3455450783089532116nteger @ G @ S )
= ( groups3455450783089532116nteger @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7254_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S: set_nat,G: nat > code_natural] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_natural ) )
=> ( ( groups2279045934846249631atural @ G @ S )
= ( groups2279045934846249631atural @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7255_prod_Omono__neutral__left,axiom,
! [T2: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ G @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_7256_prod_Omono__neutral__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ( groups1304777262505850412r_assn @ G @ T2 )
= ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7257_prod_Omono__neutral__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups3190895334310489300er_nat @ G @ T2 )
= ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7258_prod_Omono__neutral__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups3188404863801439024er_int @ G @ T2 )
= ( groups3188404863801439024er_int @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7259_prod_Omono__neutral__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_integer ) )
=> ( ( groups3674199335183972705nteger @ G @ T2 )
= ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7260_prod_Omono__neutral__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_natural] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_natural ) )
=> ( ( groups2497794486940690220atural @ G @ T2 )
= ( groups2497794486940690220atural @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7261_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S: set_nat,G: nat > assn] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ( groups6906906614972039071t_assn @ G @ T2 )
= ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7262_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S: set_nat,G: nat > int] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ T2 )
= ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7263_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S: set_nat,G: nat > code_integer] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_integer ) )
=> ( ( groups3455450783089532116nteger @ G @ T2 )
= ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7264_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S: set_nat,G: nat > code_natural] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_Code_natural ) )
=> ( ( groups2279045934846249631atural @ G @ T2 )
= ( groups2279045934846249631atural @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7265_prod_Omono__neutral__right,axiom,
! [T2: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ T2 )
= ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_7266_prod_Omono__neutral__cong__left,axiom,
! [T2: set_int,S: set_int,H2: int > assn,G: int > assn] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_assn ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ S )
= ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7267_prod_Omono__neutral__cong__left,axiom,
! [T2: set_o,S: set_o,H2: $o > assn,G: $o > assn] :
( ( finite_finite_o @ T2 )
=> ( ( ord_less_eq_set_o @ S @ T2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_assn ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ S )
= ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7268_prod_Omono__neutral__cong__left,axiom,
! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > assn,G: product_unit > assn] :
( ( finite4290736615968046902t_unit @ T2 )
=> ( ( ord_le3507040750410214029t_unit @ S @ T2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_assn ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ S )
= ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7269_prod_Omono__neutral__cong__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_assn ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1304777262505850412r_assn @ G @ S )
= ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7270_prod_Omono__neutral__cong__left,axiom,
! [T2: set_int,S: set_int,H2: int > nat,G: int > nat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S )
= ( groups1707563613775114915nt_nat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7271_prod_Omono__neutral__cong__left,axiom,
! [T2: set_o,S: set_o,H2: $o > nat,G: $o > nat] :
( ( finite_finite_o @ T2 )
=> ( ( ord_less_eq_set_o @ S @ T2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3504817904513533571_o_nat @ G @ S )
= ( groups3504817904513533571_o_nat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7272_prod_Omono__neutral__cong__left,axiom,
! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > nat,G: product_unit > nat] :
( ( finite4290736615968046902t_unit @ T2 )
=> ( ( ord_le3507040750410214029t_unit @ S @ T2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups396421249026171384it_nat @ G @ S )
= ( groups396421249026171384it_nat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7273_prod_Omono__neutral__cong__left,axiom,
! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3190895334310489300er_nat @ G @ S )
= ( groups3190895334310489300er_nat @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7274_prod_Omono__neutral__cong__left,axiom,
! [T2: set_int,S: set_int,H2: int > int,G: int > int] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_int ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1705073143266064639nt_int @ G @ S )
= ( groups1705073143266064639nt_int @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7275_prod_Omono__neutral__cong__left,axiom,
! [T2: set_o,S: set_o,H2: $o > int,G: $o > int] :
( ( finite_finite_o @ T2 )
=> ( ( ord_less_eq_set_o @ S @ T2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( H2 @ X3 )
= one_one_int ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3502327434004483295_o_int @ G @ S )
= ( groups3502327434004483295_o_int @ H2 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_7276_prod_Omono__neutral__cong__right,axiom,
! [T2: set_int,S: set_int,G: int > assn,H2: int > assn] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ T2 )
= ( groups7882442080178216443t_assn @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7277_prod_Omono__neutral__cong__right,axiom,
! [T2: set_o,S: set_o,G: $o > assn,H2: $o > assn] :
( ( finite_finite_o @ T2 )
=> ( ( ord_less_eq_set_o @ S @ T2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ T2 )
= ( groups5301882518646026715o_assn @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7278_prod_Omono__neutral__cong__right,axiom,
! [T2: set_Product_unit,S: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
( ( finite4290736615968046902t_unit @ T2 )
=> ( ( ord_le3507040750410214029t_unit @ S @ T2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ T2 )
= ( groups5360233303959380816t_assn @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7279_prod_Omono__neutral__cong__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1304777262505850412r_assn @ G @ T2 )
= ( groups1304777262505850412r_assn @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7280_prod_Omono__neutral__cong__right,axiom,
! [T2: set_int,S: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ T2 )
= ( groups1707563613775114915nt_nat @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7281_prod_Omono__neutral__cong__right,axiom,
! [T2: set_o,S: set_o,G: $o > nat,H2: $o > nat] :
( ( finite_finite_o @ T2 )
=> ( ( ord_less_eq_set_o @ S @ T2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3504817904513533571_o_nat @ G @ T2 )
= ( groups3504817904513533571_o_nat @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7282_prod_Omono__neutral__cong__right,axiom,
! [T2: set_Product_unit,S: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
( ( finite4290736615968046902t_unit @ T2 )
=> ( ( ord_le3507040750410214029t_unit @ S @ T2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups396421249026171384it_nat @ G @ T2 )
= ( groups396421249026171384it_nat @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7283_prod_Omono__neutral__cong__right,axiom,
! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( ord_le7084787975880047091nteger @ S @ T2 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3190895334310489300er_nat @ G @ T2 )
= ( groups3190895334310489300er_nat @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7284_prod_Omono__neutral__cong__right,axiom,
! [T2: set_int,S: set_int,G: int > int,H2: int > int] :
( ( finite_finite_int @ T2 )
=> ( ( ord_less_eq_set_int @ S @ T2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1705073143266064639nt_int @ G @ T2 )
= ( groups1705073143266064639nt_int @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7285_prod_Omono__neutral__cong__right,axiom,
! [T2: set_o,S: set_o,G: $o > int,H2: $o > int] :
( ( finite_finite_o @ T2 )
=> ( ( ord_less_eq_set_o @ S @ T2 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ S )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3502327434004483295_o_int @ G @ T2 )
= ( groups3502327434004483295_o_int @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_7286_prod_Ounion__inter,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
= ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7287_prod_Ounion__inter,axiom,
! [A: set_nat,B4: set_nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7288_prod_Ounion__inter,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
= ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7289_prod_Ounion__inter,axiom,
! [A: set_nat,B4: set_nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7290_prod_Ounion__inter,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
= ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7291_prod_Ounion__inter,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
= ( times_times_int @ ( groups3188404863801439024er_int @ G @ A ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7292_prod_Ounion__inter,axiom,
! [A: set_nat,B4: set_nat,G: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ A ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7293_prod_Ounion__inter,axiom,
! [A: set_nat,B4: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% prod.union_inter
thf(fact_7294_prod_OInt__Diff,axiom,
! [A: set_Code_integer,G: code_integer > assn,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups1304777262505850412r_assn @ G @ A )
= ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7295_prod_OInt__Diff,axiom,
! [A: set_nat,G: nat > assn,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( groups6906906614972039071t_assn @ G @ A )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7296_prod_OInt__Diff,axiom,
! [A: set_Code_integer,G: code_integer > rat,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups2555765274223993564er_rat @ G @ A )
= ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7297_prod_OInt__Diff,axiom,
! [A: set_nat,G: nat > rat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( groups73079841787564623at_rat @ G @ A )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7298_prod_OInt__Diff,axiom,
! [A: set_Code_integer,G: code_integer > nat,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups3190895334310489300er_nat @ G @ A )
= ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7299_prod_OInt__Diff,axiom,
! [A: set_Code_integer,G: code_integer > int,B4: set_Code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups3188404863801439024er_int @ G @ A )
= ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7300_prod_OInt__Diff,axiom,
! [A: set_nat,G: nat > int,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( groups705719431365010083at_int @ G @ A )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7301_prod_OInt__Diff,axiom,
! [A: set_nat,G: nat > nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G @ A )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_7302_prod_Omono__neutral__cong,axiom,
! [T2: set_int,S: set_int,H2: int > assn,G: int > assn] :
( ( finite_finite_int @ T2 )
=> ( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_assn ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_assn ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7882442080178216443t_assn @ G @ S )
= ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7303_prod_Omono__neutral__cong,axiom,
! [T2: set_o,S: set_o,H2: $o > assn,G: $o > assn] :
( ( finite_finite_o @ T2 )
=> ( ( finite_finite_o @ S )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_assn ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_assn ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5301882518646026715o_assn @ G @ S )
= ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7304_prod_Omono__neutral__cong,axiom,
! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > assn,G: product_unit > assn] :
( ( finite4290736615968046902t_unit @ T2 )
=> ( ( finite4290736615968046902t_unit @ S )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_assn ) )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_assn ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( inf_in4660618365625256667t_unit @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups5360233303959380816t_assn @ G @ S )
= ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7305_prod_Omono__neutral__cong,axiom,
! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_assn ) )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_assn ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1304777262505850412r_assn @ G @ S )
= ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7306_prod_Omono__neutral__cong,axiom,
! [T2: set_int,S: set_int,H2: int > nat,G: int > nat] :
( ( finite_finite_int @ T2 )
=> ( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_nat ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_nat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S )
= ( groups1707563613775114915nt_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7307_prod_Omono__neutral__cong,axiom,
! [T2: set_o,S: set_o,H2: $o > nat,G: $o > nat] :
( ( finite_finite_o @ T2 )
=> ( ( finite_finite_o @ S )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_nat ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3504817904513533571_o_nat @ G @ S )
= ( groups3504817904513533571_o_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7308_prod_Omono__neutral__cong,axiom,
! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > nat,G: product_unit > nat] :
( ( finite4290736615968046902t_unit @ T2 )
=> ( ( finite4290736615968046902t_unit @ S )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_nat ) )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_nat ) )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ ( inf_in4660618365625256667t_unit @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups396421249026171384it_nat @ G @ S )
= ( groups396421249026171384it_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7309_prod_Omono__neutral__cong,axiom,
! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ T2 )
=> ( ( finite6017078050557962740nteger @ S )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_nat ) )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_nat ) )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3190895334310489300er_nat @ G @ S )
= ( groups3190895334310489300er_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7310_prod_Omono__neutral__cong,axiom,
! [T2: set_int,S: set_int,H2: int > int,G: int > int] :
( ( finite_finite_int @ T2 )
=> ( ( finite_finite_int @ S )
=> ( ! [I3: int] :
( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_int ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_int ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1705073143266064639nt_int @ G @ S )
= ( groups1705073143266064639nt_int @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7311_prod_Omono__neutral__cong,axiom,
! [T2: set_o,S: set_o,H2: $o > int,G: $o > int] :
( ( finite_finite_o @ T2 )
=> ( ( finite_finite_o @ S )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
=> ( ( H2 @ I3 )
= one_one_int ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
=> ( ( G @ I3 )
= one_one_int ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3502327434004483295_o_int @ G @ S )
= ( groups3502327434004483295_o_int @ H2 @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_7312_prod_Oeq__fold,axiom,
( groups708209901874060359at_nat
= ( ^ [G3: nat > nat] : ( finite_fold_nat_nat @ ( comp_nat_nat_nat_nat @ times_times_nat @ G3 ) @ one_one_nat ) ) ) ).
% prod.eq_fold
thf(fact_7313_sum__bounded__above,axiom,
! [A: set_nat,F: nat > code_integer,K4: code_integer] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
=> ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7314_sum__bounded__above,axiom,
! [A: set_int,F: int > code_integer,K4: code_integer] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
=> ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7315_sum__bounded__above,axiom,
! [A: set_o,F: $o > code_integer,K4: code_integer] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
=> ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7316_sum__bounded__above,axiom,
! [A: set_Product_unit,F: product_unit > code_integer,K4: code_integer] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
=> ( ord_le3102999989581377725nteger @ ( groups5393439102872367408nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7317_sum__bounded__above,axiom,
! [A: set_nat,F: nat > rat,K4: rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7318_sum__bounded__above,axiom,
! [A: set_int,F: int > rat,K4: rat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7319_sum__bounded__above,axiom,
! [A: set_o,F: $o > rat,K4: rat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
=> ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7320_sum__bounded__above,axiom,
! [A: set_Product_unit,F: product_unit > rat,K4: rat] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
=> ( ord_less_eq_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7321_sum__bounded__above,axiom,
! [A: set_int,F: int > nat,K4: nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ K4 ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7322_sum__bounded__above,axiom,
! [A: set_o,F: $o > nat,K4: nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ K4 ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% sum_bounded_above
thf(fact_7323_sum__bounded__below,axiom,
! [A: set_nat,K4: code_integer,F: nat > code_integer] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A ) ) @ K4 ) @ ( groups7501900531339628137nteger @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7324_sum__bounded__below,axiom,
! [A: set_int,K4: code_integer,F: int > code_integer] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups7873554091576472773nteger @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7325_sum__bounded__below,axiom,
! [A: set_o,K4: code_integer,F: $o > code_integer] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups4406642042086082107nteger @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7326_sum__bounded__below,axiom,
! [A: set_Product_unit,K4: code_integer,F: product_unit > code_integer] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
=> ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) @ ( groups5393439102872367408nteger @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7327_sum__bounded__below,axiom,
! [A: set_nat,K4: rat,F: nat > rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) @ ( groups2906978787729119204at_rat @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7328_sum__bounded__below,axiom,
! [A: set_int,K4: rat,F: int > rat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups3906332499630173760nt_rat @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7329_sum__bounded__below,axiom,
! [A: set_o,K4: rat,F: $o > rat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups7872700643590313910_o_rat @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7330_sum__bounded__below,axiom,
! [A: set_Product_unit,K4: rat,F: product_unit > rat] :
( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) @ ( groups8131667117488461483it_rat @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7331_sum__bounded__below,axiom,
! [A: set_int,K4: nat,F: int > nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ord_less_eq_nat @ K4 @ ( F @ I3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups4541462559716669496nt_nat @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7332_sum__bounded__below,axiom,
! [A: set_o,K4: nat,F: $o > nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_nat @ K4 @ ( F @ I3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups8507830703676809646_o_nat @ F @ A ) ) ) ).
% sum_bounded_below
thf(fact_7333_gbinomial__absorb__comp,axiom,
! [A2: rat,K3: nat] :
( ( times_times_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( gbinomial_rat @ A2 @ K3 ) )
= ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% gbinomial_absorb_comp
thf(fact_7334_gbinomial__mult__1,axiom,
! [A2: rat,K3: nat] :
( ( times_times_rat @ A2 @ ( gbinomial_rat @ A2 @ K3 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( gbinomial_rat @ A2 @ K3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_7335_gbinomial__mult__1_H,axiom,
! [A2: rat,K3: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A2 @ K3 ) @ A2 )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( gbinomial_rat @ A2 @ K3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_7336_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
( ( ( inf_in1906310914598751387_nat_o @ X @ Y )
= bot_bo7824918357723582233_nat_o )
=> ( ( ( sup_su5209123915105501825_nat_o @ X @ Y )
= top_to4869887016220417533_nat_o )
=> ( ( uminus5254974436814262692_nat_o @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7337_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X @ Y )
= bot_bo3957492148770167129t_unit )
=> ( ( ( sup_su793286257634532545t_unit @ X @ Y )
= top_to1996260823553986621t_unit )
=> ( ( uminus5944136376168626660t_unit @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7338_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
=> ( ( ( sup_sup_set_o @ X @ Y )
= top_top_set_o )
=> ( ( uminus_uminus_set_o @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7339_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7340_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_int,Y: set_int] :
( ( ( inf_inf_set_int @ X @ Y )
= bot_bot_set_int )
=> ( ( ( sup_sup_set_int @ X @ Y )
= top_top_set_int )
=> ( ( uminus1532241313380277803et_int @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7341_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: assn,Y: assn] :
( ( ( inf_inf_assn @ X @ Y )
= bot_bot_assn )
=> ( ( ( sup_sup_assn @ X @ Y )
= top_top_assn )
=> ( ( uminus_uminus_assn @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7342_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( inf_inf_Product_unit @ X @ Y )
= bot_bot_Product_unit )
=> ( ( ( sup_sup_Product_unit @ X @ Y )
= top_top_Product_unit )
=> ( ( uminus2952777764628376836t_unit @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_7343_prod__mono__strict,axiom,
! [A: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ord_le6747313008572928689nteger @ ( groups3674199335183972705nteger @ F @ A ) @ ( groups3674199335183972705nteger @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7344_prod__mono__strict,axiom,
! [A: set_Product_unit,F: product_unit > code_integer,G: product_unit > code_integer] :
( ( finite4290736615968046902t_unit @ A )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_le6747313008572928689nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( groups8185359811944078981nteger @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7345_prod__mono__strict,axiom,
! [A: set_o,F: $o > code_integer,G: $o > code_integer] :
( ( finite_finite_o @ A )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_le6747313008572928689nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( groups7694694392188491536nteger @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7346_prod__mono__strict,axiom,
! [A: set_nat,F: nat > code_integer,G: nat > code_integer] :
( ( finite_finite_nat @ A )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bot_set_nat )
=> ( ord_le6747313008572928689nteger @ ( groups3455450783089532116nteger @ F @ A ) @ ( groups3455450783089532116nteger @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7347_prod__mono__strict,axiom,
! [A: set_int,F: int > code_integer,G: int > code_integer] :
( ( finite_finite_int @ A )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
& ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bot_set_int )
=> ( ord_le6747313008572928689nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( groups3827104343326376752nteger @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7348_prod__mono__strict,axiom,
! [A: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ! [I3: code_integer] :
( ( member_Code_integer @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bo3990330152332043303nteger )
=> ( ord_less_rat @ ( groups2555765274223993564er_rat @ F @ A ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7349_prod__mono__strict,axiom,
! [A: set_Product_unit,F: product_unit > rat,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ! [I3: product_unit] :
( ( member_Product_unit @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bo3957492148770167129t_unit )
=> ( ord_less_rat @ ( groups8984663225794451456it_rat @ F @ A ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7350_prod__mono__strict,axiom,
! [A: set_o,F: $o > rat,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7351_prod__mono__strict,axiom,
! [A: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7352_prod__mono__strict,axiom,
! [A: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ).
% prod_mono_strict
thf(fact_7353_prod_Oinsert__remove,axiom,
! [A: set_Code_integer,G: code_integer > assn,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7354_prod_Oinsert__remove,axiom,
! [A: set_Product_unit,G: product_unit > assn,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7355_prod_Oinsert__remove,axiom,
! [A: set_o,G: $o > assn,X: $o] :
( ( finite_finite_o @ A )
=> ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7356_prod_Oinsert__remove,axiom,
! [A: set_int,G: int > assn,X: int] :
( ( finite_finite_int @ A )
=> ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7357_prod_Oinsert__remove,axiom,
! [A: set_nat,G: nat > assn,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
= ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7358_prod_Oinsert__remove,axiom,
! [A: set_Code_integer,G: code_integer > rat,X: code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7359_prod_Oinsert__remove,axiom,
! [A: set_Product_unit,G: product_unit > rat,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7360_prod_Oinsert__remove,axiom,
! [A: set_o,G: $o > rat,X: $o] :
( ( finite_finite_o @ A )
=> ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7361_prod_Oinsert__remove,axiom,
! [A: set_int,G: int > rat,X: int] :
( ( finite_finite_int @ A )
=> ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7362_prod_Oinsert__remove,axiom,
! [A: set_nat,G: nat > rat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
= ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_7363_prod_Oremove,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( groups1304777262505850412r_assn @ G @ A )
= ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7364_prod_Oremove,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( groups5360233303959380816t_assn @ G @ A )
= ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7365_prod_Oremove,axiom,
! [A: set_o,X: $o,G: $o > assn] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( groups5301882518646026715o_assn @ G @ A )
= ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7366_prod_Oremove,axiom,
! [A: set_int,X: int,G: int > assn] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( groups7882442080178216443t_assn @ G @ A )
= ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7367_prod_Oremove,axiom,
! [A: set_nat,X: nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( groups6906906614972039071t_assn @ G @ A )
= ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7368_prod_Oremove,axiom,
! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( member_Code_integer @ X @ A )
=> ( ( groups2555765274223993564er_rat @ G @ A )
= ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7369_prod_Oremove,axiom,
! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( member_Product_unit @ X @ A )
=> ( ( groups8984663225794451456it_rat @ G @ A )
= ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7370_prod_Oremove,axiom,
! [A: set_o,X: $o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( groups2869687844427037835_o_rat @ G @ A )
= ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7371_prod_Oremove,axiom,
! [A: set_int,X: int,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( groups1072433553688619179nt_rat @ G @ A )
= ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7372_prod_Oremove,axiom,
! [A: set_nat,X: nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( groups73079841787564623at_rat @ G @ A )
= ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_7373_prod_Ounion__inter__neutral,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_integer] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_Code_integer ) )
=> ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A ) @ ( groups3674199335183972705nteger @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7374_prod_Ounion__inter__neutral,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_natural] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_Code_natural ) )
=> ( ( groups2497794486940690220atural @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_2397367101498566445atural @ ( groups2497794486940690220atural @ G @ A ) @ ( groups2497794486940690220atural @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7375_prod_Ounion__inter__neutral,axiom,
! [A: set_nat,B4: set_nat,G: nat > code_integer] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_Code_integer ) )
=> ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A ) @ ( groups3455450783089532116nteger @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7376_prod_Ounion__inter__neutral,axiom,
! [A: set_nat,B4: set_nat,G: nat > code_natural] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_Code_natural ) )
=> ( ( groups2279045934846249631atural @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_2397367101498566445atural @ ( groups2279045934846249631atural @ G @ A ) @ ( groups2279045934846249631atural @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7377_prod_Ounion__inter__neutral,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7378_prod_Ounion__inter__neutral,axiom,
! [A: set_nat,B4: set_nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_assn ) )
=> ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7379_prod_Ounion__inter__neutral,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7380_prod_Ounion__inter__neutral,axiom,
! [A: set_nat,B4: set_nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7381_prod_Ounion__inter__neutral,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7382_prod_Ounion__inter__neutral,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ! [X3: code_integer] :
( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_int @ ( groups3188404863801439024er_int @ G @ A ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_7383_prod_Ounion__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7384_prod_Ounion__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( groups5360233303959380816t_assn @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( times_times_assn @ ( groups5360233303959380816t_assn @ G @ A ) @ ( groups5360233303959380816t_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7385_prod_Ounion__disjoint,axiom,
! [A: set_o,B4: set_o,G: $o > assn] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
=> ( ( groups5301882518646026715o_assn @ G @ ( sup_sup_set_o @ A @ B4 ) )
= ( times_times_assn @ ( groups5301882518646026715o_assn @ G @ A ) @ ( groups5301882518646026715o_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7386_prod_Ounion__disjoint,axiom,
! [A: set_nat,B4: set_nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
=> ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7387_prod_Ounion__disjoint,axiom,
! [A: set_int,B4: set_int,G: int > assn] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
=> ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A @ B4 ) )
= ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A ) @ ( groups7882442080178216443t_assn @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7388_prod_Ounion__disjoint,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
= bot_bo3990330152332043303nteger )
=> ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7389_prod_Ounion__disjoint,axiom,
! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > rat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( groups8984663225794451456it_rat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
= ( times_times_rat @ ( groups8984663225794451456it_rat @ G @ A ) @ ( groups8984663225794451456it_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7390_prod_Ounion__disjoint,axiom,
! [A: set_o,B4: set_o,G: $o > rat] :
( ( finite_finite_o @ A )
=> ( ( finite_finite_o @ B4 )
=> ( ( ( inf_inf_set_o @ A @ B4 )
= bot_bot_set_o )
=> ( ( groups2869687844427037835_o_rat @ G @ ( sup_sup_set_o @ A @ B4 ) )
= ( times_times_rat @ ( groups2869687844427037835_o_rat @ G @ A ) @ ( groups2869687844427037835_o_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7391_prod_Ounion__disjoint,axiom,
! [A: set_nat,B4: set_nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
=> ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7392_prod_Ounion__disjoint,axiom,
! [A: set_int,B4: set_int,G: int > rat] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
=> ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A @ B4 ) )
= ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A ) @ ( groups1072433553688619179nt_rat @ G @ B4 ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_7393_prod_Ounion__diff2,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_assn @ ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7394_prod_Ounion__diff2,axiom,
! [A: set_nat,B4: set_nat,G: nat > assn] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_assn @ ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7395_prod_Ounion__diff2,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7396_prod_Ounion__diff2,axiom,
! [A: set_nat,B4: set_nat,G: nat > rat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7397_prod_Ounion__diff2,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_nat @ ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7398_prod_Ounion__diff2,axiom,
! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
( ( finite6017078050557962740nteger @ A )
=> ( ( finite6017078050557962740nteger @ B4 )
=> ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
= ( times_times_int @ ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7399_prod_Ounion__diff2,axiom,
! [A: set_nat,B4: set_nat,G: nat > int] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( groups705719431365010083at_int @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_int @ ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7400_prod_Ounion__diff2,axiom,
! [A: set_nat,B4: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
= ( times_times_nat @ ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_7401_Suc__times__gbinomial,axiom,
! [K3: nat,A2: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) ) )
= ( times_times_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( gbinomial_rat @ A2 @ K3 ) ) ) ).
% Suc_times_gbinomial
thf(fact_7402_gbinomial__absorption,axiom,
! [K3: nat,A2: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) )
= ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% gbinomial_absorption
thf(fact_7403_gbinomial__trinomial__revision,axiom,
! [K3: nat,M4: nat,A2: rat] :
( ( ord_less_eq_nat @ K3 @ M4 )
=> ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M4 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M4 ) @ K3 ) )
= ( times_times_rat @ ( gbinomial_rat @ A2 @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( minus_minus_nat @ M4 @ K3 ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_7404_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K3: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K3 ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7405_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K3: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K3 ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7406_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7407_gbinomial__absorption_H,axiom,
! [K3: nat,A2: rat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( gbinomial_rat @ A2 @ K3 )
= ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_7408_pochhammer__minus,axiom,
! [B: rat,K3: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K3 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) ) ) ).
% pochhammer_minus
thf(fact_7409_pochhammer__minus,axiom,
! [B: int,K3: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K3 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K3 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K3 ) ) @ one_one_int ) @ K3 ) ) ) ).
% pochhammer_minus
thf(fact_7410_pochhammer__minus,axiom,
! [B: code_integer,K3: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K3 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K3 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K3 ) ) @ one_one_Code_integer ) @ K3 ) ) ) ).
% pochhammer_minus
thf(fact_7411_pochhammer__minus_H,axiom,
! [B: rat,K3: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K3 ) ) ) ).
% pochhammer_minus'
thf(fact_7412_pochhammer__minus_H,axiom,
! [B: int,K3: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K3 ) ) @ one_one_int ) @ K3 )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K3 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K3 ) ) ) ).
% pochhammer_minus'
thf(fact_7413_pochhammer__minus_H,axiom,
! [B: code_integer,K3: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K3 ) ) @ one_one_Code_integer ) @ K3 )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K3 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K3 ) ) ) ).
% pochhammer_minus'
thf(fact_7414_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K3: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_7415_nat__mult__div__cancel__disj,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ( K3 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= zero_zero_nat ) )
& ( ( K3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( divide_divide_nat @ M4 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_7416_pos__zmult__eq__1__iff__lemma,axiom,
! [M4: int,N2: int] :
( ( ( times_times_int @ M4 @ N2 )
= one_one_int )
=> ( ( M4 = one_one_int )
| ( M4
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_7417_zmult__eq__1__iff,axiom,
! [M4: int,N2: int] :
( ( ( times_times_int @ M4 @ N2 )
= one_one_int )
= ( ( ( M4 = one_one_int )
& ( N2 = one_one_int ) )
| ( ( M4
= ( uminus_uminus_int @ one_one_int ) )
& ( N2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_7418_zdvd__mono,axiom,
! [K3: int,M4: int,T4: int] :
( ( K3 != zero_zero_int )
=> ( ( dvd_dvd_int @ M4 @ T4 )
= ( dvd_dvd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ T4 ) ) ) ) ).
% zdvd_mono
thf(fact_7419_zdvd__mult__cancel,axiom,
! [K3: int,M4: int,N2: int] :
( ( dvd_dvd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ N2 ) )
=> ( ( K3 != zero_zero_int )
=> ( dvd_dvd_int @ M4 @ N2 ) ) ) ).
% zdvd_mult_cancel
thf(fact_7420_zdvd__reduce,axiom,
! [K3: int,N2: int,M4: int] :
( ( dvd_dvd_int @ K3 @ ( plus_plus_int @ N2 @ ( times_times_int @ K3 @ M4 ) ) )
= ( dvd_dvd_int @ K3 @ N2 ) ) ).
% zdvd_reduce
thf(fact_7421_zdvd__period,axiom,
! [A2: int,D2: int,X: int,T4: int,C: int] :
( ( dvd_dvd_int @ A2 @ D2 )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T4 ) )
= ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T4 ) ) ) ) ).
% zdvd_period
thf(fact_7422_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R6: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( ord_less_int @ B @ R6 )
=> ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_7423_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R6: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R6 )
=> ( ( ord_less_int @ R6 @ B )
=> ( ( ord_less_int @ R2 @ B )
=> ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_7424_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q4: int,R2: int,B3: int,Q5: int,R6: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R6 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_7425_zdiv__mono2__lemma,axiom,
! [B: int,Q4: int,R2: int,B3: int,Q5: int,R6: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
=> ( ( ord_less_int @ R6 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ B3 @ B )
=> ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_7426_int__div__pos__eq,axiom,
! [A2: int,B: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A2 @ B )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_7427_int__div__neg__eq,axiom,
! [A2: int,B: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A2 @ B )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_7428_q__pos__lemma,axiom,
! [B3: int,Q5: int,R6: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
=> ( ( ord_less_int @ R6 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_7429_split__zdiv,axiom,
! [P: int > $o,N2: int,K3: int] :
( ( P @ ( divide_divide_int @ N2 @ K3 ) )
= ( ( ( K3 = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K3 )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K3 )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less_int @ K3 @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K3 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_7430_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K3: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K3 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_7431_pos__zmult__eq__1__iff,axiom,
! [M4: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M4 )
=> ( ( ( times_times_int @ M4 @ N2 )
= one_one_int )
= ( ( M4 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_7432_zmult__zless__mono2,axiom,
! [I2: int,J: int,K3: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K3 )
=> ( ord_less_int @ ( times_times_int @ K3 @ I2 ) @ ( times_times_int @ K3 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_7433_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_7434_plusinfinity,axiom,
! [D2: int,P7: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [X_12: int] : ( P7 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_7435_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K3: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K3 )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_7436_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_7437_times__int__code_I1_J,axiom,
! [K3: int] :
( ( times_times_int @ K3 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_7438_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_7439_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_7440_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_7441_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_7442_nat__mult__div__cancel1,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
= ( divide_divide_nat @ M4 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_7443_div__mult__self__is__m,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M4 @ N2 ) @ N2 )
= M4 ) ) ).
% div_mult_self_is_m
thf(fact_7444_div__mult__self1__is__m,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M4 ) @ N2 )
= M4 ) ) ).
% div_mult_self1_is_m
thf(fact_7445_div__mult2__eq,axiom,
! [M4: nat,N2: nat,Q4: nat] :
( ( divide_divide_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M4 @ N2 ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_7446_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_7447_less__mult__imp__div__less,axiom,
! [M4: nat,I2: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( times_times_nat @ I2 @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M4 @ N2 ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_7448_div__times__less__eq__dividend,axiom,
! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N2 ) @ N2 ) @ M4 ) ).
% div_times_less_eq_dividend
thf(fact_7449_times__div__less__eq__dividend,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M4 @ N2 ) ) @ M4 ) ).
% times_div_less_eq_dividend
thf(fact_7450_div__less__iff__less__mult,axiom,
! [Q4: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M4 @ Q4 ) @ N2 )
= ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_7451_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M4 @ ( divide_divide_nat @ N2 @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M4 @ Q4 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_7452_div__nat__eqI,axiom,
! [N2: nat,Q4: nat,M4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M4 )
=> ( ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
=> ( ( divide_divide_nat @ M4 @ N2 )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_7453_dividend__less__times__div,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M4 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M4 @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_7454_dividend__less__div__times,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M4 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_7455_split__div,axiom,
! [P: nat > $o,M4: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M4 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( M4
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_7456_split__div_H,axiom,
! [P: nat > $o,M4: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M4 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q6: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q6 ) @ M4 )
& ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_7457_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_7458_eucl__rel__int__iff,axiom,
! [K3: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
= ( ( K3
= ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L )
=> ( ( ( ord_less_int @ L @ zero_zero_int )
=> ( ( ord_less_int @ L @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L @ zero_zero_int )
=> ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_7459_eucl__rel__int__dividesI,axiom,
! [L: int,K3: int,Q4: int] :
( ( L != zero_zero_int )
=> ( ( K3
= ( times_times_int @ Q4 @ L ) )
=> ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_7460_split__pos__lemma,axiom,
! [K3: int,P: int > int > $o,N2: int] :
( ( ord_less_int @ zero_zero_int @ K3 )
=> ( ( P @ ( divide_divide_int @ N2 @ K3 ) @ ( modulo_modulo_int @ N2 @ K3 ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K3 )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_7461_split__neg__lemma,axiom,
! [K3: int,P: int > int > $o,N2: int] :
( ( ord_less_int @ K3 @ zero_zero_int )
=> ( ( P @ ( divide_divide_int @ N2 @ K3 ) @ ( modulo_modulo_int @ N2 @ K3 ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_int @ K3 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_7462_decr__lemma,axiom,
! [D2: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_7463_incr__lemma,axiom,
! [D2: int,Z: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_7464_zmod__eq__0__iff,axiom,
! [M4: int,D2: int] :
( ( ( modulo_modulo_int @ M4 @ D2 )
= zero_zero_int )
= ( ? [Q6: int] :
( M4
= ( times_times_int @ D2 @ Q6 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_7465_zmod__eq__0D,axiom,
! [M4: int,D2: int] :
( ( ( modulo_modulo_int @ M4 @ D2 )
= zero_zero_int )
=> ? [Q7: int] :
( M4
= ( times_times_int @ D2 @ Q7 ) ) ) ).
% zmod_eq_0D
thf(fact_7466_abs__zmult__eq__1,axiom,
! [M4: int,N2: int] :
( ( ( abs_abs_int @ ( times_times_int @ M4 @ N2 ) )
= one_one_int )
=> ( ( abs_abs_int @ M4 )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_7467_div__mod__decomp__int,axiom,
! [A: int,N2: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A @ N2 ) ) ) ).
% div_mod_decomp_int
thf(fact_7468_zdvd__mult__cancel1,axiom,
! [M4: int,N2: int] :
( ( M4 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M4 @ N2 ) @ M4 )
= ( ( abs_abs_int @ N2 )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_7469_split__zmod,axiom,
! [P: int > $o,N2: int,K3: int] :
( ( P @ ( modulo_modulo_int @ N2 @ K3 ) )
= ( ( ( K3 = zero_zero_int )
=> ( P @ N2 ) )
& ( ( ord_less_int @ zero_zero_int @ K3 )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K3 )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less_int @ K3 @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K3 @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N2
= ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_7470_int__mod__neg__eq,axiom,
! [A2: int,B: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( modulo_modulo_int @ A2 @ B )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_7471_int__mod__pos__eq,axiom,
! [A2: int,B: int,Q4: int,R2: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( modulo_modulo_int @ A2 @ B )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_7472_zmod__zmult2__eq,axiom,
! [C: int,A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_7473_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A13: int,A24: int,A32: product_prod_int_int] :
( ? [K5: int] :
( ( A13 = K5 )
& ( A24 = zero_zero_int )
& ( A32
= ( product_Pair_int_int @ zero_zero_int @ K5 ) ) )
| ? [L2: int,K5: int,Q6: int] :
( ( A13 = K5 )
& ( A24 = L2 )
& ( A32
= ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
& ( L2 != zero_zero_int )
& ( K5
= ( times_times_int @ Q6 @ L2 ) ) )
| ? [R4: int,L2: int,K5: int,Q6: int] :
( ( A13 = K5 )
& ( A24 = L2 )
& ( A32
= ( product_Pair_int_int @ Q6 @ R4 ) )
& ( ( sgn_sgn_int @ R4 )
= ( sgn_sgn_int @ L2 ) )
& ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L2 ) )
& ( K5
= ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R4 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_7474_Suc__mod__mult__self4,axiom,
! [N2: nat,K3: nat,M4: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K3 ) @ M4 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_mod_mult_self4
thf(fact_7475_Suc__mod__mult__self3,axiom,
! [K3: nat,N2: nat,M4: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K3 @ N2 ) @ M4 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_mod_mult_self3
thf(fact_7476_Suc__mod__mult__self2,axiom,
! [M4: nat,N2: nat,K3: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M4 @ ( times_times_nat @ N2 @ K3 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_mod_mult_self2
thf(fact_7477_Suc__mod__mult__self1,axiom,
! [M4: nat,K3: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M4 @ ( times_times_nat @ K3 @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% Suc_mod_mult_self1
thf(fact_7478_Suc__times__numeral__mod__eq,axiom,
! [K3: num,N2: nat] :
( ( ( numeral_numeral_nat @ K3 )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K3 ) @ N2 ) ) @ ( numeral_numeral_nat @ K3 ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_7479_dvd__mult__sgn__iff,axiom,
! [L: int,K3: int,R2: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ K3 @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L @ K3 )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_7480_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K3: int] :
( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K3 ) )
= ( ( dvd_dvd_int @ L @ K3 )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_7481_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K3: int] :
( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K3 )
= ( ( dvd_dvd_int @ L @ K3 )
& ( ( R2 = zero_zero_int )
=> ( K3 = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_7482_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K3: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K3 )
= ( ( dvd_dvd_int @ L @ K3 )
& ( ( R2 = zero_zero_int )
=> ( K3 = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_7483_int__sgnE,axiom,
! [K3: int] :
~ ! [N3: nat,L3: int] :
( K3
!= ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_sgnE
thf(fact_7484_mod__eq__0D,axiom,
! [M4: nat,D2: nat] :
( ( ( modulo_modulo_nat @ M4 @ D2 )
= zero_zero_nat )
=> ? [Q7: nat] :
( M4
= ( times_times_nat @ D2 @ Q7 ) ) ) ).
% mod_eq_0D
thf(fact_7485_nat__mod__eq__iff,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N2 )
= ( modulo_modulo_nat @ Y @ N2 ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_7486_nat__mod__eq__lemma,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N2 )
= ( modulo_modulo_nat @ Y @ N2 ) )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ? [Q7: nat] :
( X
= ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q7 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_7487_mod__eq__nat2E,axiom,
! [M4: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo_nat @ M4 @ Q4 )
= ( modulo_modulo_nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq_nat @ M4 @ N2 )
=> ~ ! [S8: nat] :
( N2
!= ( plus_plus_nat @ M4 @ ( times_times_nat @ Q4 @ S8 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_7488_mod__eq__nat1E,axiom,
! [M4: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo_nat @ M4 @ Q4 )
= ( modulo_modulo_nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq_nat @ N2 @ M4 )
=> ~ ! [S8: nat] :
( M4
!= ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S8 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_7489_div__mod__decomp,axiom,
! [A: nat,N2: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A @ N2 ) ) ) ).
% div_mod_decomp
thf(fact_7490_mod__mult2__eq,axiom,
! [M4: nat,N2: nat,Q4: nat] :
( ( modulo_modulo_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) )
= ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M4 @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ).
% mod_mult2_eq
thf(fact_7491_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N4: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N4 ) @ N4 ) ) ) ) ).
% modulo_nat_def
thf(fact_7492_div__sgn__abs__cancel,axiom,
! [V: int,K3: int,L: int] :
( ( V != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K3 ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
= ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_7493_div__dvd__sgn__abs,axiom,
! [L: int,K3: int] :
( ( dvd_dvd_int @ L @ K3 )
=> ( ( divide_divide_int @ K3 @ L )
= ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_7494_split__mod,axiom,
! [P: nat > $o,M4: nat,N2: nat] :
( ( P @ ( modulo_modulo_nat @ M4 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ M4 ) )
& ( ( N2 != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N2 )
=> ( ( M4
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_7495_Suc__times__mod__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M4 @ N2 ) ) @ M4 )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_7496_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K3: int,Q4: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
=> ( ( K3
= ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_7497_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod_int_int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22 = zero_zero_int )
=> ( A33
!= ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
=> ( ! [Q7: int] :
( ( A33
= ( product_Pair_int_int @ Q7 @ zero_zero_int ) )
=> ( ( A22 != zero_zero_int )
=> ( A1
!= ( times_times_int @ Q7 @ A22 ) ) ) )
=> ~ ! [R7: int,Q7: int] :
( ( A33
= ( product_Pair_int_int @ Q7 @ R7 ) )
=> ( ( ( sgn_sgn_int @ R7 )
= ( sgn_sgn_int @ A22 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R7 ) @ ( abs_abs_int @ A22 ) )
=> ( A1
!= ( plus_plus_int @ ( times_times_int @ Q7 @ A22 ) @ R7 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_7498_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_7499_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_7500_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_7501_divide__int__unfold,axiom,
! [L: int,K3: int,N2: nat,M4: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K3 )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K3 )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M4 @ N2 ) ) ) )
& ( ( ( sgn_sgn_int @ K3 )
!= ( sgn_sgn_int @ L ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M4 @ N2 )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_7502_modulo__int__unfold,axiom,
! [L: int,K3: int,N2: nat,M4: nat] :
( ( ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K3 )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L )
= zero_zero_int )
| ( ( sgn_sgn_int @ K3 )
= zero_zero_int )
| ( N2 = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ) )
& ( ( ( sgn_sgn_int @ K3 )
!= ( sgn_sgn_int @ L ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N2
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N2 @ M4 ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_7503_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod_int_int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa = zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa != zero_zero_nat )
=> ( Y
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_7504_semiring__norm_I12_J,axiom,
! [N2: num] :
( ( times_times_num @ one @ N2 )
= N2 ) ).
% semiring_norm(12)
thf(fact_7505_semiring__norm_I11_J,axiom,
! [M4: num] :
( ( times_times_num @ M4 @ one )
= M4 ) ).
% semiring_norm(11)
thf(fact_7506_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A2: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_7507_semiring__norm_I13_J,axiom,
! [M4: num,N2: num] :
( ( times_times_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M4 @ N2 ) ) ) ) ).
% semiring_norm(13)
thf(fact_7508_num__double,axiom,
! [N2: num] :
( ( times_times_num @ ( bit0 @ one ) @ N2 )
= ( bit0 @ N2 ) ) ).
% num_double
thf(fact_7509_zmod__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_7510_odd__two__times__div__two__nat,axiom,
! [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_7511_Suc__double__not__eq__double,axiom,
! [M4: nat,N2: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% Suc_double_not_eq_double
thf(fact_7512_double__not__eq__Suc__double,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% double_not_eq_Suc_double
thf(fact_7513_nat__bit__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_7514_int__bit__induct,axiom,
! [P: int > $o,K3: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K: int] :
( ( P @ K )
=> ( ( K != zero_zero_int )
=> ( P @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K: int] :
( ( P @ K )
=> ( ( K
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K3 ) ) ) ) ) ).
% int_bit_induct
thf(fact_7515_pos__zdiv__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ B @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_7516_neg__zdiv__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_7517_pos__zmod__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A2 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_7518_neg__zmod__mult__2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).
% neg_zmod_mult_2
thf(fact_7519_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A2: int,Q4: int,R2: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A2 @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_7520_signed__take__bit__Suc__bit0,axiom,
! [N2: nat,K3: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_7521_signed__take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K3: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_7522_signed__take__bit__mult,axiom,
! [N2: nat,K3: int,L: int] :
( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K3 ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
= ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K3 @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_7523_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa )
= Y )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_7524_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K5: int,L2: int] :
( if_int
@ ( ( member_int @ K5 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_7525_signed__take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K3: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_7526_concat__bit__Suc,axiom,
! [N2: nat,K3: int,L: int] :
( ( bit_concat_bit @ ( suc @ N2 ) @ K3 @ L )
= ( plus_plus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_7527_semiring__norm_I14_J,axiom,
! [M4: num,N2: num] :
( ( times_times_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) )
= ( bit0 @ ( times_times_num @ M4 @ ( bit1 @ N2 ) ) ) ) ).
% semiring_norm(14)
thf(fact_7528_semiring__norm_I15_J,axiom,
! [M4: num,N2: num] :
( ( times_times_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M4 ) @ N2 ) ) ) ).
% semiring_norm(15)
thf(fact_7529_semiring__norm_I16_J,axiom,
! [M4: num,N2: num] :
( ( times_times_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M4 @ N2 ) @ ( bit0 @ ( times_times_num @ M4 @ N2 ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_7530_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_7531_signed__take__bit__Suc__bit1,axiom,
! [N2: nat,K3: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_7532_take__bit__mult,axiom,
! [N2: nat,K3: int,L: int] :
( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
= ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K3 @ L ) ) ) ).
% take_bit_mult
thf(fact_7533_take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K3: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_7534_and__int__rec,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K5: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_7535_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K5: int,L2: int] :
( if_int
@ ( ( K5 = zero_zero_int )
| ( L2 = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K5
= ( uminus_uminus_int @ one_one_int ) )
@ L2
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ K5
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_7536_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K3: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_7537_signed__take__bit__numeral__bit1,axiom,
! [L: num,K3: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_7538_and__int_Opsimps,axiom,
! [K3: int,L: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L ) )
=> ( ( ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K3 @ L )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K3 @ L )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_7539_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_7540_signed__take__bit__numeral__bit0,axiom,
! [L: num,K3: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_7541_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K3: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_7542_take__bit__numeral__minus__bit0,axiom,
! [L: num,K3: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_7543_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M3: nat,N4: nat] :
( if_nat
@ ( ( M3 = zero_zero_nat )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_7544_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M3: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_7545_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [K: int,L3: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L3 ) )
=> ( ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_7546_take__bit__numeral__minus__bit1,axiom,
! [L: num,K3: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K3 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_7547_take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K3: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K3 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_7548_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K5: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K5 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K5 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K5 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K5 ) ) ) ) ) ) ).
% binomial_code
thf(fact_7549_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K5: int,L2: int] :
( if_int @ ( L2 = zero_zero_int ) @ K5
@ ( if_int
@ ( ( sgn_sgn_int @ K5 )
= ( sgn_sgn_int @ L2 ) )
@ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K5 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L2 )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L2 )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L2 @ K5 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K5 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_7550_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K5: int,L2: int] :
( if_int
@ ( ( K5
= ( uminus_uminus_int @ one_one_int ) )
| ( L2
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K5 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K5 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_7551_Suc__times__binomial__eq,axiom,
! [N2: nat,K3: nat] :
( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) )
= ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) ) @ ( suc @ K3 ) ) ) ).
% Suc_times_binomial_eq
thf(fact_7552_Suc__times__binomial,axiom,
! [K3: nat,N2: nat] :
( ( times_times_nat @ ( suc @ K3 ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) ) )
= ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) ) ) ).
% Suc_times_binomial
thf(fact_7553_choose__mult__lemma,axiom,
! [M4: nat,R2: nat,K3: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M4 @ R2 ) @ K3 ) @ ( plus_plus_nat @ M4 @ K3 ) ) @ ( binomial @ ( plus_plus_nat @ M4 @ K3 ) @ K3 ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M4 @ R2 ) @ K3 ) @ K3 ) @ ( binomial @ ( plus_plus_nat @ M4 @ R2 ) @ M4 ) ) ) ).
% choose_mult_lemma
thf(fact_7554_Suc__times__binomial__add,axiom,
! [A2: nat,B: nat] :
( ( times_times_nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B ) ) @ ( suc @ A2 ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B ) ) @ A2 ) ) ) ).
% Suc_times_binomial_add
thf(fact_7555_binomial__fact__lemma,axiom,
! [K3: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K3 ) ) ) @ ( binomial @ N2 @ K3 ) )
= ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% binomial_fact_lemma
thf(fact_7556_choose__mult,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( times_times_nat @ ( binomial @ N2 @ M4 ) @ ( binomial @ M4 @ K3 ) )
= ( times_times_nat @ ( binomial @ N2 @ K3 ) @ ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M4 @ K3 ) ) ) ) ) ) ).
% choose_mult
thf(fact_7557_binomial__absorb__comp,axiom,
! [N2: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N2 @ K3 ) @ ( binomial @ N2 @ K3 ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K3 ) ) ) ).
% binomial_absorb_comp
thf(fact_7558_binomial__Suc__Suc__eq__times,axiom,
! [N2: nat,K3: nat] :
( ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) ) @ ( suc @ K3 ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_7559_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_7560_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times_num @ X @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_7561_binomial__absorption,axiom,
! [K3: nat,N2: nat] :
( ( times_times_nat @ ( suc @ K3 ) @ ( binomial @ N2 @ ( suc @ K3 ) ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K3 ) ) ) ).
% binomial_absorption
thf(fact_7562_nat__mult__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_7563_binomial__altdef__nat,axiom,
! [K3: nat,N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ( binomial @ N2 @ K3 )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_7564_times__binomial__minus1__eq,axiom,
! [K3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( times_times_nat @ K3 @ ( binomial @ N2 @ K3 ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_7565_nat__mult__distrib__neg,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_7566_choose__two,axiom,
! [N2: nat] :
( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_7567_or__int__rec,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K5: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
| ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_7568_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N4: nat,K5: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K5 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K5 @ N4 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_7569_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_7570_push__bit__int__def,axiom,
( bit_se545348938243370406it_int
= ( ^ [N4: nat,K5: int] : ( times_times_int @ K5 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% push_bit_int_def
thf(fact_7571_push__bit__nat__def,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N4: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% push_bit_nat_def
thf(fact_7572_set__bit__eq,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N4: nat,K5: int] :
( plus_plus_int @ K5
@ ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ~ ( bit_se1146084159140164899it_int @ K5 @ N4 ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% set_bit_eq
thf(fact_7573_unset__bit__eq,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N4: nat,K5: int] : ( minus_minus_int @ K5 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K5 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_7574_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M3: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_7575_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_7576_take__bit__Suc__from__most,axiom,
! [N2: nat,K3: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) ) ) ).
% take_bit_Suc_from_most
thf(fact_7577_xor__int__rec,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K5: int,L2: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_7578_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M3: nat,N4: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_7579_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K5: int,L2: int] :
( if_int
@ ( K5
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L2 )
@ ( if_int
@ ( L2
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K5 )
@ ( if_int @ ( K5 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K5 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_7580_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K5: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_7581_or__not__numerals_I8_J,axiom,
! [M4: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_7582_or__not__numerals_I9_J,axiom,
! [M4: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_7583_and__not__numerals_I8_J,axiom,
! [M4: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_7584_bool__assn__proper_I4_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H4: produc3658429121746597890et_nat] :
( ( P @ H4 )
& ( Q2 @ H4 ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_7585_bool__assn__proper_I3_J,axiom,
! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( ( proper @ Q2 )
=> ( proper
@ ^ [H4: produc3658429121746597890et_nat] :
( ( P @ H4 )
| ( Q2 @ H4 ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_7586_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu: produc3658429121746597890et_nat] : $false ) ).
% bool_assn_proper(2)
thf(fact_7587_bool__assn__proper_I5_J,axiom,
! [P: produc3658429121746597890et_nat > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H4: produc3658429121746597890et_nat] :
( ( in_range @ H4 )
& ~ ( P @ H4 ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_7588_bot__assn__def,axiom,
( bot_bot_assn
= ( abs_assn
@ ^ [Uu: produc3658429121746597890et_nat] : $false ) ) ).
% bot_assn_def
thf(fact_7589_sup__assn__def,axiom,
( sup_sup_assn
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H4: produc3658429121746597890et_nat] :
( ( rep_assn @ P2 @ H4 )
| ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% sup_assn_def
thf(fact_7590_inf__assn__def,axiom,
( inf_inf_assn
= ( ^ [P2: assn,Q: assn] :
( abs_assn
@ ^ [H4: produc3658429121746597890et_nat] :
( ( rep_assn @ P2 @ H4 )
& ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% inf_assn_def
thf(fact_7591_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A6: nat,B6: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_7592_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M3: nat,N4: nat] :
( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K5: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K5 @ N4 ) @ M3 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_7593_uminus__assn__def,axiom,
( uminus_uminus_assn
= ( ^ [P2: assn] :
( abs_assn
@ ^ [H4: produc3658429121746597890et_nat] :
( ( in_range @ H4 )
& ~ ( rep_assn @ P2 @ H4 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_7594_Gcd__eq__Max,axiom,
! [M: set_nat] :
( ( finite_finite_nat @ M )
=> ( ( M != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M )
=> ( ( gcd_Gcd_nat @ M )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M3: nat] :
( collect_nat
@ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M3 ) )
@ M ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_7595_and__not__numerals_I5_J,axiom,
! [M4: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_7596_and__not__numerals_I6_J,axiom,
! [M4: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_7597_and__not__numerals_I9_J,axiom,
! [M4: num,N2: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_7598_or__not__numerals_I6_J,axiom,
! [M4: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_7599_or__not__numerals_I5_J,axiom,
! [M4: num,N2: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_7600_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_7601_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_7602_vandermonde,axiom,
! [M4: nat,N2: nat,R2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K5: nat] : ( times_times_nat @ ( binomial @ M4 @ K5 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K5 ) ) )
@ ( set_ord_atMost_nat @ R2 ) )
= ( binomial @ ( plus_plus_nat @ M4 @ N2 ) @ R2 ) ) ).
% vandermonde
thf(fact_7603_fact__eq__fact__times,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ N2 @ M4 )
=> ( ( semiri1408675320244567234ct_nat @ M4 )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M4 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_7604_binomial,axiom,
! [A2: nat,B: nat,N2: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A2 @ B ) @ N2 )
= ( groups3542108847815614940at_nat
@ ^ [K5: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K5 ) ) @ ( power_power_nat @ A2 @ K5 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K5 ) ) )
@ ( set_ord_atMost_nat @ N2 ) ) ) ).
% binomial
thf(fact_7605_choose__square__sum,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K5: nat] : ( power_power_nat @ ( binomial @ N2 @ K5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% choose_square_sum
thf(fact_7606_gauss__sum__nat,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_7607_binomial__r__part__sum,axiom,
! [M4: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M4 ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) ) ) ).
% binomial_r_part_sum
thf(fact_7608_choose__linear__sum,axiom,
! [N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N2 @ I4 ) )
@ ( set_ord_atMost_nat @ N2 ) )
= ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% choose_linear_sum
thf(fact_7609_arith__series__nat,axiom,
! [A2: nat,D2: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D2 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N2 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_7610_Sum__Icc__nat,axiom,
! [M4: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_7611_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X3: int,K: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_7612_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_7613_cpmi,axiom,
! [D3: int,P: int > $o,P7: int > $o,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P7 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P7 @ X2 ) )
| ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ B4 )
& ( P @ ( plus_plus_int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_7614_cppi,axiom,
! [D3: int,P: int > $o,P7: int > $o,A: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less_int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P7 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A )
=> ( X3
!= ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P7 @ X2 ) )
| ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ A )
& ( P @ ( minus_minus_int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_7615_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L2: num] :
( produc2626176000494625587at_nat
@ ^ [Q6: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_7616_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L2: num] :
( produc4245557441103728435nt_int
@ ^ [Q6: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_7617_Sum__Icc__int,axiom,
! [M4: int,N2: int] :
( ( ord_less_eq_int @ M4 @ N2 )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( set_or1266510415728281911st_int @ M4 @ N2 ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M4 @ ( minus_minus_int @ M4 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_7618_Sum__Ico__nat,axiom,
! [M4: nat,N2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ M4 @ N2 ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_7619_atLeastLessThan__singleton,axiom,
! [M4: nat] :
( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ M4 ) )
= ( insert_nat @ M4 @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_7620_atLeastLessThan0,axiom,
! [M4: nat] :
( ( set_or4665077453230672383an_nat @ M4 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_7621_atLeastLessThanSuc,axiom,
! [M4: nat,N2: nat] :
( ( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ N2 ) )
= ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ N2 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_7622_atLeastLessThan__nat__numeral,axiom,
! [M4: nat,K3: num] :
( ( ( ord_less_eq_nat @ M4 @ ( pred_numeral @ K3 ) )
=> ( ( set_or4665077453230672383an_nat @ M4 @ ( numeral_numeral_nat @ K3 ) )
= ( insert_nat @ ( pred_numeral @ K3 ) @ ( set_or4665077453230672383an_nat @ M4 @ ( pred_numeral @ K3 ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M4 @ ( pred_numeral @ K3 ) )
=> ( ( set_or4665077453230672383an_nat @ M4 @ ( numeral_numeral_nat @ K3 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_7623_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y: nat,X: nat] :
( ( ( ord_less_nat @ C @ Y )
=> ( ( image_nat_nat2
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat2
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( image_nat_nat2
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_7624_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L2: num] :
( produc6916734918728496179nteger
@ ^ [Q6: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_7625_triangle__def,axiom,
( nat_triangle
= ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_7626_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image_4470545334726330049nteger
@ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L )
@ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
= ( set_or8404916559141939852nteger @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_7627_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
= ( set_or189985376899183464nteger @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_7628_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(2)
thf(fact_7629_times__integer__code_I1_J,axiom,
! [K3: code_integer] :
( ( times_3573771949741848930nteger @ K3 @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(1)
thf(fact_7630_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K5: int] :
( if_Code_integer @ ( ord_less_int @ K5 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K5 ) ) )
@ ( if_Code_integer @ ( K5 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_7631_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).
% finite_atLeastLessThan_integer
thf(fact_7632_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).
% finite_atLeastAtMost_integer
thf(fact_7633_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).
% finite_atLeastZeroLessThan_integer
thf(fact_7634_times__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
@ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
@ times_times_int
@ times_times_int ) ).
% times_integer.rsp
thf(fact_7635_times__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
@ times_times_nat
@ times_times_nat ) ).
% times_natural.rsp
thf(fact_7636_times__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_7637_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K5: code_integer,L2: code_integer] :
( if_Pro6119634080678213985nteger @ ( K5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K5 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K5 )
= ( sgn_sgn_Code_integer @ L2 ) )
@ ( code_divmod_abs @ K5 @ L2 )
@ ( produc6916734918728496179nteger
@ ^ [R4: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S7 ) ) )
@ ( code_divmod_abs @ K5 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_7638_plus__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
@ plus_plus_rat ) ).
% plus_rat.transfer
thf(fact_7639_times__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
@ times_times_rat ) ).
% times_rat.transfer
thf(fact_7640_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K5: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K5 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K5 ) ) )
@ ( if_int @ ( K5 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_7641_positive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
@ positive ) ).
% positive.transfer
thf(fact_7642_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_7643_times__int_Oabs__eq,axiom,
! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_7644_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
= ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% times_integer.rep_eq
thf(fact_7645_positive__mult,axiom,
! [X: rat,Y: rat] :
( ( positive @ X )
=> ( ( positive @ Y )
=> ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).
% positive_mult
thf(fact_7646_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
= ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_7647_type__definition__integer,axiom,
type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).
% type_definition_integer
thf(fact_7648_positive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% positive.rep_eq
thf(fact_7649_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
=> ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
= ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_7650_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_7651_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_7652_min__Suc__gt_I1_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_min_nat @ ( suc @ A2 ) @ B )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(1)
thf(fact_7653_min__Suc__gt_I2_J,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_min_nat @ B @ ( suc @ A2 ) )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(2)
thf(fact_7654_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_7655_nat__mult__min__right,axiom,
! [M4: nat,N2: nat,Q4: nat] :
( ( times_times_nat @ M4 @ ( ord_min_nat @ N2 @ Q4 ) )
= ( ord_min_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_7656_nat__mult__min__left,axiom,
! [M4: nat,N2: nat,Q4: nat] :
( ( times_times_nat @ ( ord_min_nat @ M4 @ N2 ) @ Q4 )
= ( ord_min_nat @ ( times_times_nat @ M4 @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_7657_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_7658_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
=> ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
= ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_integer
thf(fact_7659_bezw__aux,axiom,
! [X: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_7660_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_7661_gcd__mult__distrib__nat,axiom,
! [K3: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K3 @ ( gcd_gcd_nat @ M4 @ N2 ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_7662_Gcd__in,axiom,
! [A: set_nat] :
( ! [A4: nat,B2: nat] :
( ( member_nat @ A4 @ A )
=> ( ( member_nat @ B2 @ A )
=> ( member_nat @ ( gcd_gcd_nat @ A4 @ B2 ) @ A ) ) )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( gcd_Gcd_nat @ A ) @ A ) ) ) ).
% Gcd_in
thf(fact_7663_bezout__nat,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ? [X3: nat,Y2: nat] :
( ( times_times_nat @ A2 @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ ( gcd_gcd_nat @ A2 @ B ) ) ) ) ).
% bezout_nat
thf(fact_7664_bezout__gcd__nat_H,axiom,
! [B: nat,A2: nat] :
? [X3: nat,Y2: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y2 ) @ ( times_times_nat @ A2 @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
= ( gcd_gcd_nat @ A2 @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ Y2 ) @ ( times_times_nat @ B @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y2 ) )
= ( gcd_gcd_nat @ A2 @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_7665_greaterThan__Suc,axiom,
! [K3: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K3 ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K3 ) @ ( insert_nat @ ( suc @ K3 ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_7666_plus__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% plus_rat.rsp
thf(fact_7667_positive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% positive.rsp
thf(fact_7668_ratrel__iff,axiom,
( ratrel
= ( ^ [X2: product_prod_int_int,Y3: product_prod_int_int] :
( ( ( product_snd_int_int @ X2 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y3 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) )
= ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_7669_bezout__int,axiom,
! [X: int,Y: int] :
? [U4: int,V3: int] :
( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V3 @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
% bezout_int
thf(fact_7670_gcd__mult__distrib__int,axiom,
! [K3: int,M4: int,N2: int] :
( ( times_times_int @ ( abs_abs_int @ K3 ) @ ( gcd_gcd_int @ M4 @ N2 ) )
= ( gcd_gcd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ N2 ) ) ) ).
% gcd_mult_distrib_int
thf(fact_7671_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod_int_int,Y3: product_prod_int_int] :
( ( ( product_snd_int_int @ X2 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y3 )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) )
= ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_7672_times__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% times_rat.rsp
thf(fact_7673_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_7674_positive_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% positive.abs_eq
thf(fact_7675_times__rat_Oabs__eq,axiom,
! [Xa: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_7676_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).
% finite_greaterThanLessThan_integer
thf(fact_7677_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
= ( set_or4266950643985792945nteger @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_7678_positive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% positive_def
thf(fact_7679_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).
% finite_greaterThanAtMost_integer
thf(fact_7680_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
= ( set_or2715278749043346189nteger @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_7681_atLeast__Suc,axiom,
! [K3: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K3 ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K3 ) @ ( insert_nat @ K3 @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_7682_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_Suc
thf(fact_7683_eventually__sequentially__seg,axiom,
! [P: nat > $o,K3: nat] :
( ( eventually_nat
@ ^ [N4: nat] : ( P @ ( plus_plus_nat @ N4 @ K3 ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_7684_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_7685_eventually__sequentiallyI,axiom,
! [C: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( P @ X3 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_7686_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N7: nat] :
! [N4: nat] :
( ( ord_less_eq_nat @ N7 @ N4 )
=> ( P @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_7687_le__sequentially,axiom,
! [F2: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
= ( ! [N7: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N7 ) @ F2 ) ) ) ).
% le_sequentially
thf(fact_7688_filterlim__atMost__at__top,axiom,
filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% filterlim_atMost_at_top
thf(fact_7689_filterlim__lessThan__at__top,axiom,
filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% filterlim_lessThan_at_top
thf(fact_7690_trivial__limit__sequentially,axiom,
at_top_nat != bot_bot_filter_nat ).
% trivial_limit_sequentially
thf(fact_7691_eventually__False__sequentially,axiom,
~ ( eventually_nat
@ ^ [N4: nat] : $false
@ at_top_nat ) ).
% eventually_False_sequentially
thf(fact_7692_plus__rat__def,axiom,
( plus_plus_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% plus_rat_def
thf(fact_7693_eventually__prod__sequentially,axiom,
! [P: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N7: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ N7 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq_nat @ N7 @ N4 )
=> ( P @ ( product_Pair_nat_nat @ N4 @ M3 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_7694_times__rat__def,axiom,
( times_times_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% times_rat_def
thf(fact_7695_add__rat,axiom,
! [B: int,D2: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% add_rat
thf(fact_7696_divide__rat,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( divide_divide_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
= ( fract @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ).
% divide_rat
thf(fact_7697_sgn__rat,axiom,
! [A2: int,B: int] :
( ( sgn_sgn_rat @ ( fract @ A2 @ B ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% sgn_rat
thf(fact_7698_mult__rat,axiom,
! [A2: int,B: int,C: int,D2: int] :
( ( times_times_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
= ( fract @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ).
% mult_rat
thf(fact_7699_le__rat,axiom,
! [B: int,D2: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% le_rat
thf(fact_7700_less__rat,axiom,
! [B: int,D2: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
= ( ord_less_int @ ( times_times_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% less_rat
thf(fact_7701_diff__rat,axiom,
! [B: int,D2: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% diff_rat
thf(fact_7702_eq__rat_I1_J,axiom,
! [B: int,D2: int,A2: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D2 != zero_zero_int )
=> ( ( ( fract @ A2 @ B )
= ( fract @ C @ D2 ) )
= ( ( times_times_int @ A2 @ D2 )
= ( times_times_int @ C @ B ) ) ) ) ) ).
% eq_rat(1)
thf(fact_7703_mult__rat__cancel,axiom,
! [C: int,A2: int,B: int] :
( ( C != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
= ( fract @ A2 @ B ) ) ) ).
% mult_rat_cancel
thf(fact_7704_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_7705_positive__rat,axiom,
! [A2: int,B: int] :
( ( positive @ ( fract @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% positive_rat
thf(fact_7706_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_7707_normalize__crossproduct,axiom,
! [Q4: int,S2: int,P3: int,R2: int] :
( ( Q4 != zero_zero_int )
=> ( ( S2 != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P3 @ Q4 ) )
= ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
=> ( ( times_times_int @ P3 @ S2 )
= ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_7708_rat__minus__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus_rat @ P3 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A6: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ B6 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_minus_code
thf(fact_7709_rat__plus__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus_rat @ P3 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A6: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ B6 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_plus_code
thf(fact_7710_rat__times__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( times_times_rat @ P3 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A6: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ B6 ) @ ( times_times_int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_times_code
thf(fact_7711_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P5: rat,Q6: rat] :
( produc4947309494688390418_int_o
@ ^ [A6: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B6: int,D4: int] : ( ord_less_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_7712_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P5: rat,Q6: rat] :
( produc4947309494688390418_int_o
@ ^ [A6: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B6: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_7713_rat__divide__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide_rat @ P3 @ Q4 ) )
= ( produc4245557441103728435nt_int
@ ^ [A6: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_divide_code
thf(fact_7714_rat__inverse__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P3 ) )
= ( produc4245557441103728435nt_int
@ ^ [A6: int,B6: int] : ( if_Pro3027730157355071871nt_int @ ( A6 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A6 ) @ B6 ) @ ( abs_abs_int @ A6 ) ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_inverse_code
thf(fact_7715_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q6: rat,R4: rat] : ( times_times_rat @ Q6 @ ( inverse_inverse_rat @ R4 ) ) ) ) ).
% divide_rat_def
thf(fact_7716_sub__BitM__One__eq,axiom,
! [N2: num] :
( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_7717_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_7718_cofinite__eq__sequentially,axiom,
cofinite_nat = at_top_nat ).
% cofinite_eq_sequentially
thf(fact_7719_INFM__nat__inductI,axiom,
! [P: nat > $o,Q2: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [I3: nat] :
( ( P @ I3 )
=> ? [J4: nat] :
( ( ord_less_nat @ I3 @ J4 )
& ( P @ J4 )
& ( Q2 @ J4 ) ) )
=> ( frequently_nat @ Q2 @ cofinite_nat ) ) ) ).
% INFM_nat_inductI
thf(fact_7720_lcm__altdef__int,axiom,
( gcd_lcm_int
= ( ^ [A6: int,B6: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A6 ) @ ( abs_abs_int @ B6 ) ) @ ( gcd_gcd_int @ A6 @ B6 ) ) ) ) ).
% lcm_altdef_int
thf(fact_7721_times__num__def,axiom,
( times_times_num
= ( ^ [M3: num,N4: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).
% times_num_def
thf(fact_7722_nat__of__num__mult,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( times_times_num @ X @ Y ) )
= ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_mult
thf(fact_7723_prod__gcd__lcm__int,axiom,
! [M4: int,N2: int] :
( ( times_times_int @ ( abs_abs_int @ M4 ) @ ( abs_abs_int @ N2 ) )
= ( times_times_int @ ( gcd_gcd_int @ M4 @ N2 ) @ ( gcd_lcm_int @ M4 @ N2 ) ) ) ).
% prod_gcd_lcm_int
thf(fact_7724_prod__gcd__lcm__nat,axiom,
( times_times_nat
= ( ^ [M3: nat,N4: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M3 @ N4 ) @ ( gcd_lcm_nat @ M3 @ N4 ) ) ) ) ).
% prod_gcd_lcm_nat
thf(fact_7725_lcm__code__integer,axiom,
( gcd_lcm_Code_integer
= ( ^ [A6: code_integer,B6: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A6 ) @ ( abs_abs_Code_integer @ B6 ) ) @ ( gcd_gcd_Code_integer @ A6 @ B6 ) ) ) ) ).
% lcm_code_integer
thf(fact_7726_lcm__nat__def,axiom,
( gcd_lcm_nat
= ( ^ [X2: nat,Y3: nat] : ( divide_divide_nat @ ( times_times_nat @ X2 @ Y3 ) @ ( gcd_gcd_nat @ X2 @ Y3 ) ) ) ) ).
% lcm_nat_def
thf(fact_7727_Lcm__eq__Max__nat,axiom,
! [M: set_nat] :
( ( finite_finite_nat @ M )
=> ( ( M != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M )
=> ( ! [M2: nat,N3: nat] :
( ( member_nat @ M2 @ M )
=> ( ( member_nat @ N3 @ M )
=> ( member_nat @ ( gcd_lcm_nat @ M2 @ N3 ) @ M ) ) )
=> ( ( gcd_Lcm_nat @ M )
= ( lattic8265883725875713057ax_nat @ M ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_7728_nat__of__num__sqr,axiom,
! [X: num] :
( ( nat_of_num @ ( sqr @ X ) )
= ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_sqr
thf(fact_7729_Lcm__nat__empty,axiom,
( ( gcd_Lcm_nat @ bot_bot_set_nat )
= one_one_nat ) ).
% Lcm_nat_empty
thf(fact_7730_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_7731_Lcm__in__lcm__closed__set__nat,axiom,
! [M: set_nat] :
( ( finite_finite_nat @ M )
=> ( ( M != bot_bot_set_nat )
=> ( ! [M2: nat,N3: nat] :
( ( member_nat @ M2 @ M )
=> ( ( member_nat @ N3 @ M )
=> ( member_nat @ ( gcd_lcm_nat @ M2 @ N3 ) @ M ) ) )
=> ( member_nat @ ( gcd_Lcm_nat @ M ) @ M ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_7732_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit1 @ Y ) )
= ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_7733_coprime__crossproduct__nat,axiom,
! [A2: nat,D2: nat,B: nat,C: nat] :
( ( algebr934650988132801477me_nat @ A2 @ D2 )
=> ( ( algebr934650988132801477me_nat @ B @ C )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ D2 ) )
= ( ( A2 = B )
& ( C = D2 ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_7734_coprime__crossproduct__int,axiom,
! [A2: int,D2: int,B: int,C: int] :
( ( algebr932160517623751201me_int @ A2 @ D2 )
=> ( ( algebr932160517623751201me_int @ B @ C )
=> ( ( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ C ) )
= ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D2 ) ) )
= ( ( ( abs_abs_int @ A2 )
= ( abs_abs_int @ B ) )
& ( ( abs_abs_int @ C )
= ( abs_abs_int @ D2 ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_7735_in__range_Oelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ ( in_range @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ As2 )
=> ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_7736_in__range_Osimps,axiom,
! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ As )
=> ( ord_less_nat @ X2 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).
% in_range.simps
thf(fact_7737_in__range_Oelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( Y
= ( ~ ! [X2: nat] :
( ( member_nat @ X2 @ As2 )
=> ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_7738_in__range_Oelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ( in_range @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ As2 )
=> ( ord_less_nat @ X6 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_7739_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_7740_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_7741_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( ( product_Abs_unit @ X )
= ( product_Abs_unit @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_unit_inject
thf(fact_7742_Rep__unit__inject,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( product_Rep_unit @ X )
= ( product_Rep_unit @ Y ) )
= ( X = Y ) ) ).
% Rep_unit_inject
thf(fact_7743_Rep__unit__inverse,axiom,
! [X: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
= X ) ).
% Rep_unit_inverse
thf(fact_7744_Abs__unit__inverse,axiom,
! [Y: $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
= Y ) ) ).
% Abs_unit_inverse
thf(fact_7745_type__definition__unit,axiom,
type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% type_definition_unit
thf(fact_7746_Rep__unit,axiom,
! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% Rep_unit
thf(fact_7747_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X
= ( product_Abs_unit @ Y2 ) )
=> ~ ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% Abs_unit_cases
thf(fact_7748_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ~ ! [X3: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).
% Rep_unit_cases
thf(fact_7749_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( P @ ( product_Abs_unit @ Y2 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_7750_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_7751_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ^ [B6: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ) ).
% sum_encode_def
thf(fact_7752_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( ( upt @ zero_zero_nat @ J )
= nil_nat )
= ( J = zero_zero_nat ) ) ).
% upt_0_eq_Nil_conv
thf(fact_7753_upt__merge,axiom,
! [I2: nat,J: nat,K3: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ J @ K3 ) )
=> ( ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ K3 ) )
= ( upt @ I2 @ K3 ) ) ) ).
% upt_merge
thf(fact_7754_map__add__upt_H,axiom,
! [Ofs: nat,A2: nat,B: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ Ofs )
@ ( upt @ A2 @ B ) )
= ( upt @ ( plus_plus_nat @ A2 @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_7755_upt__eq__append__conv,axiom,
! [I2: nat,J: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( upt @ I2 @ J )
= ( append_nat @ Xs @ Ys ) )
= ( ? [K5: nat] :
( ( ord_less_eq_nat @ I2 @ K5 )
& ( ord_less_eq_nat @ K5 @ J )
& ( ( upt @ I2 @ K5 )
= Xs )
& ( ( upt @ K5 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_7756_upt__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( append_nat @ ( upt @ zero_zero_nat @ I2 ) @ ( upt @ I2 @ J ) )
= ( upt @ zero_zero_nat @ J ) ) ) ).
% upt_append
thf(fact_7757_butlast__upt,axiom,
! [M4: nat,N2: nat] :
( ( butlast_nat @ ( upt @ M4 @ N2 ) )
= ( upt @ M4 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% butlast_upt
thf(fact_7758_upt__eq__lel__conv,axiom,
! [L: nat,H2: nat,Is1: list_nat,I2: nat,Is2: list_nat] :
( ( ( upt @ L @ H2 )
= ( append_nat @ Is1 @ ( cons_nat @ I2 @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I2 ) )
& ( Is2
= ( upt @ ( suc @ I2 ) @ H2 ) )
& ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_7759_upt__filter__extend,axiom,
! [U: nat,U5: nat,P: nat > $o] :
( ( ord_less_eq_nat @ U @ U5 )
=> ( ! [I3: nat] :
( ( ( ord_less_eq_nat @ U @ I3 )
& ( ord_less_nat @ I3 @ U5 ) )
=> ~ ( P @ I3 ) )
=> ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
= ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U5 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_7760_in__range_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ ( in_range @ X )
=> ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ As2 )
=> ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% in_range.pelims(3)
thf(fact_7761_in__range_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( ( in_range @ X )
= Y )
=> ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( Y
= ( ! [X2: nat] :
( ( member_nat @ X2 @ As2 )
=> ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) )
=> ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ).
% in_range.pelims(1)
thf(fact_7762_in__range_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ( in_range @ X )
=> ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ~ ! [X6: nat] :
( ( member_nat @ X6 @ As2 )
=> ( ord_less_nat @ X6 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% in_range.pelims(2)
thf(fact_7763_one__assn__raw_Opelims_I3_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ~ ( one_assn_raw @ X )
=> ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( As2 = bot_bot_set_nat ) ) ) ) ) ).
% one_assn_raw.pelims(3)
thf(fact_7764_one__assn__raw_Opelims_I2_J,axiom,
! [X: produc3658429121746597890et_nat] :
( ( one_assn_raw @ X )
=> ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( As2 != bot_bot_set_nat ) ) ) ) ) ).
% one_assn_raw.pelims(2)
thf(fact_7765_one__assn__raw_Opelims_I1_J,axiom,
! [X: produc3658429121746597890et_nat,Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
=> ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
( ( X
= ( produc7507926704131184380et_nat @ H @ As2 ) )
=> ( ( Y
= ( As2 = bot_bot_set_nat ) )
=> ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ).
% one_assn_raw.pelims(1)
thf(fact_7766_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y3: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_7767_next_Osimps,axiom,
! [V: code_natural,W: code_natural] :
( ( next @ ( produc3574140220909816553atural @ V @ W ) )
= ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_7768_Random_Orange__def,axiom,
( range
= ( ^ [K5: code_natural] :
( produc5538323210962509403atural
@ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K5 )
@ ^ [L2: code_natural] :
( produc5538323210962509403atural @ next
@ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ one_one_Code_natural )
@ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K5 ) ) ) ) ) ).
% Random.range_def
thf(fact_7769_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_7770_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_7771_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : ( Y = product_Unity ) ).
% old.unit.exhaust
thf(fact_7772_Inf__unit__def,axiom,
( comple2584293577114468500t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_7773_uminus__unit__def,axiom,
( uminus2952777764628376836t_unit
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_7774_top__unit__def,axiom,
top_top_Product_unit = product_Unity ).
% top_unit_def
thf(fact_7775_Unity__def,axiom,
( product_Unity
= ( product_Abs_unit @ $true ) ) ).
% Unity_def
thf(fact_7776_Sup__unit__def,axiom,
( comple4687483117567863418t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_7777_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_7778_UNIV__unit,axiom,
( top_to1996260823553986621t_unit
= ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% UNIV_unit
thf(fact_7779_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
thf(fact_7780_cone__def,axiom,
( bNF_Cardinal_cone
= ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).
% cone_def
thf(fact_7781_times__integer__code_I3_J,axiom,
! [M4: num,N2: num] :
( ( times_3573771949741848930nteger @ ( code_Pos @ M4 ) @ ( code_Pos @ N2 ) )
= ( code_Pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_integer_code(3)
thf(fact_7782_times__integer__code_I6_J,axiom,
! [M4: num,N2: num] :
( ( times_3573771949741848930nteger @ ( code_Neg @ M4 ) @ ( code_Neg @ N2 ) )
= ( code_Pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_integer_code(6)
thf(fact_7783_times__integer__code_I5_J,axiom,
! [M4: num,N2: num] :
( ( times_3573771949741848930nteger @ ( code_Neg @ M4 ) @ ( code_Pos @ N2 ) )
= ( code_Neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_integer_code(5)
thf(fact_7784_times__integer__code_I4_J,axiom,
! [M4: num,N2: num] :
( ( times_3573771949741848930nteger @ ( code_Pos @ M4 ) @ ( code_Neg @ N2 ) )
= ( code_Neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_integer_code(4)
thf(fact_7785_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa ) )
= ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% times_natural.rep_eq
thf(fact_7786_times__int__code_I3_J,axiom,
! [M4: num,N2: num] :
( ( times_times_int @ ( pos @ M4 ) @ ( pos @ N2 ) )
= ( pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_int_code(3)
thf(fact_7787_times__int__code_I4_J,axiom,
! [M4: num,N2: num] :
( ( times_times_int @ ( pos @ M4 ) @ ( neg @ N2 ) )
= ( neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_int_code(4)
thf(fact_7788_times__int__code_I5_J,axiom,
! [M4: num,N2: num] :
( ( times_times_int @ ( neg @ M4 ) @ ( pos @ N2 ) )
= ( neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_int_code(5)
thf(fact_7789_times__int__code_I6_J,axiom,
! [M4: num,N2: num] :
( ( times_times_int @ ( neg @ M4 ) @ ( neg @ N2 ) )
= ( pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% times_int_code(6)
thf(fact_7790_times__natural_Otransfer,axiom,
bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).
% times_natural.transfer
thf(fact_7791_times__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( times_times_nat @ Xa @ X ) ) ) ).
% times_natural.abs_eq
thf(fact_7792_type__definition__natural,axiom,
type_d4410041424927559462al_nat @ code_nat_of_natural @ code_natural_of_nat @ top_top_set_nat ).
% type_definition_natural
thf(fact_7793_times__natural__def,axiom,
( times_2397367101498566445atural
= ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).
% times_natural_def
thf(fact_7794_times__integer_Otransfer,axiom,
bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).
% times_integer.transfer
thf(fact_7795_times__integer__def,axiom,
( times_3573771949741848930nteger
= ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).
% times_integer_def
% Helper facts (29)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] :
( ( if_rat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X: set_o,Y: set_o] :
( ( if_set_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X: set_o,Y: set_o] :
( ( if_set_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
! [X: assn,Y: assn] :
( ( if_assn @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
! [X: assn,Y: assn] :
( ( if_assn @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( if_Code_integer @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
! [X: code_natural,Y: code_natural] :
( ( if_Code_natural @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
! [X: code_natural,Y: code_natural] :
( ( if_Code_natural @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y: set_int] :
( ( if_set_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( if_filter_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
! [X: filter_nat,Y: filter_nat] :
( ( if_filter_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Ounit_J_T,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( if_set_Product_unit @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Ounit_J_T,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( if_set_Product_unit @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= ( ( produc8586169260539613262et_nat @ h )
= bot_bot_set_nat ) ) ).
%------------------------------------------------------------------------------