TPTP Problem File: SLH0655^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Dedekind_Real/0000_Dedekind_Real/prob_01222_040091__5740220_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1479 ( 814 unt; 203 typ; 0 def)
% Number of atoms : 2944 (1434 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8919 ( 307 ~; 30 |; 107 &;7609 @)
% ( 0 <=>; 866 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 326 ( 326 >; 0 *; 0 +; 0 <<)
% Number of symbols : 169 ( 168 usr; 20 con; 0-2 aty)
% Number of variables : 2814 ( 127 ^;2657 !; 30 ?;2814 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:16.873
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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% Explicit typings (168)
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thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J,type,
remove5625372653177682828_preal: produc5063651213027957927_preal > set_Pr3959665804823141767_preal > set_Pr3959665804823141767_preal ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J,type,
remove616724958711913164_preal: produc7488332866442972391_preal > set_Pr6738245714641013959_preal > set_Pr6738245714641013959_preal ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J,type,
remove3864610947230394476_preal: set_Pr3959665804823141767_preal > set_se7472626482866788967_preal > set_se7472626482866788967_preal ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J,type,
the_el8575259181799709340_preal: set_Pr3959665804823141767_preal > produc5063651213027957927_preal ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_member_001t__Dedekind____Real__Opreal,type,
member6871284927547481791_preal: dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J,type,
member7052443626638417872_preal: produc5063651213027957927_preal > set_Pr3959665804823141767_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J,type,
member7369391392762924807_preal: produc1476315154867276144_preal > set_Pr2166019032385455654_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_J,type,
member6334355225518030407_preal: produc2369682017031648176_preal > set_Pr1083209295347970406_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_J,type,
member2335964382059793383_preal: produc867701225723568720_preal > set_Pr661714660250796806_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Int__Oint_J,type,
member1795383350042252825al_int: produc5125679149001379970al_int > set_Pr5788225742885250104al_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J,type,
member7273733640249002768_preal: produc7488332866442972391_preal > set_Pr6738245714641013959_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_J,type,
member2376753870807258448_preal: produc5596323128810183719_preal > set_Pr8594670500262599431_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_J,type,
member2272691600839418864_preal: produc41299236205021639_preal > set_Pr1512189777605002663_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_J,type,
member7581708263213832976_preal: produc7976057826613339879_preal > set_Pr5164066273998628039_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_Mt__Int__Oint_J,type,
member4206120071959551737al_int: produc2737856915623327074al_int > set_Pr388394143813266712al_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_Mt__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J,type,
member1913120636565518896_preal: produc8905100308785897479_preal > set_Pr2351743508290812903_preal > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J_J,type,
member4141329389715116816_preal: produc120301155200965863_preal > set_Pr3180418311277671623_preal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Dedekind____Real__Opreal_Mt__Dedekind____Real__Opreal_J_J,type,
member6785577539073195184_preal: set_Pr3959665804823141767_preal > set_se7472626482866788967_preal > $o ).
thf(sy_v_x1,type,
x1: dedekind_preal ).
thf(sy_v_x2,type,
x2: dedekind_preal ).
thf(sy_v_y1,type,
y1: dedekind_preal ).
thf(sy_v_y2,type,
y2: dedekind_preal ).
% Relevant facts (1275)
thf(fact_0_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_left
thf(fact_1_preal__add__le__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_right
thf(fact_2_eq__Abs__Real,axiom,
! [Z: dedekind_real] :
~ ! [X: dedekind_preal,Y: dedekind_preal] :
( Z
!= ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X @ Y ) @ bot_bo5832510953819287539_preal ) ) ) ) ).
% eq_Abs_Real
thf(fact_3_real__le__refl,axiom,
! [W: dedekind_real] : ( ord_le2716243287969276982d_real @ W @ W ) ).
% real_le_refl
thf(fact_4_add__eq__exists,axiom,
! [A: dedekind_real,B: dedekind_real] :
? [X: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ X )
= B ) ).
% add_eq_exists
thf(fact_5_add__eq__exists,axiom,
! [A: int,B: int] :
? [X: int] :
( ( plus_plus_int @ A @ X )
= B ) ).
% add_eq_exists
thf(fact_6_real__le__lemma,axiom,
! [U1: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,V1: dedekind_preal,X1: dedekind_preal,Y1: dedekind_preal,X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U1 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X1 @ V1 )
= ( plus_p3173629198307831117_preal @ U1 @ Y1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X2 @ V2 )
= ( plus_p3173629198307831117_preal @ U2 @ Y2 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X1 @ Y2 ) @ ( plus_p3173629198307831117_preal @ X2 @ Y1 ) ) ) ) ) ).
% real_le_lemma
thf(fact_7_preal__add__assoc,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) @ Z )
= ( plus_p3173629198307831117_preal @ X3 @ ( plus_p3173629198307831117_preal @ Y3 @ Z ) ) ) ).
% preal_add_assoc
thf(fact_8_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X4: dedekind_preal,Y4: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y4 @ X4 ) ) ) ).
% preal_add_commute
thf(fact_9_preal__trans__lemma,axiom,
! [X3: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y3: dedekind_preal,Y2: dedekind_preal,X2: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X3 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y3 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X3 @ Y2 )
= ( plus_p3173629198307831117_preal @ X2 @ Y3 ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y2 )
= ( plus_p3173629198307831117_preal @ X2 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_10_preal__eq__le__imp__le,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A @ B )
= ( plus_p3173629198307831117_preal @ C @ D ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A )
=> ( ord_le5604041210740703414_preal @ B @ D ) ) ) ).
% preal_eq_le_imp_le
thf(fact_11_preal__add__left__cancel,axiom,
! [C: dedekind_preal,A: dedekind_preal,B: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C @ A )
= ( plus_p3173629198307831117_preal @ C @ B ) )
=> ( A = B ) ) ).
% preal_add_left_cancel
thf(fact_12_preal__add__right__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R @ T )
= ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( R = S ) ) ).
% preal_add_right_cancel
thf(fact_13_Image__singleton__iff,axiom,
! [B: dedekind_preal,R: set_Pr3959665804823141767_preal,A: dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( image_1423628590549589971_preal @ R @ ( insert4264952892349296358_preal @ A @ bot_bo4848840305443107682_preal ) ) )
= ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_14_Image__singleton__iff,axiom,
! [B: set_Pr3959665804823141767_preal,R: set_Pr1512189777605002663_preal,A: produc5063651213027957927_preal] :
( ( member6785577539073195184_preal @ B @ ( image_3527746029208777843_preal @ R @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) )
= ( member2272691600839418864_preal @ ( produc7964486806676790327_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_15_Image__singleton__iff,axiom,
! [B: produc7488332866442972391_preal,R: set_Pr8594670500262599431_preal,A: produc5063651213027957927_preal] :
( ( member7273733640249002768_preal @ B @ ( image_3542466196807407571_preal @ R @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) )
= ( member2376753870807258448_preal @ ( produc9080166963393387415_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_16_Image__singleton__iff,axiom,
! [B: int,R: set_Pr5788225742885250104al_int,A: produc5063651213027957927_preal] :
( ( member_int @ B @ ( image_3463864234023239744al_int @ R @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) )
= ( member1795383350042252825al_int @ ( produc774093805678497660al_int @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_17_Image__singleton__iff,axiom,
! [B: produc5063651213027957927_preal,R: set_Pr6738245714641013959_preal,A: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ B @ ( image_7293650632215208851_preal @ R @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) )
= ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_18_Image__singleton__iff,axiom,
! [B: set_Pr3959665804823141767_preal,R: set_Pr661714660250796806_preal,A: int] :
( ( member6785577539073195184_preal @ B @ ( image_3250679698536620230_preal @ R @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( member2335964382059793383_preal @ ( produc4050541007652503554_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_19_Image__singleton__iff,axiom,
! [B: produc7488332866442972391_preal,R: set_Pr1083209295347970406_preal,A: int] :
( ( member7273733640249002768_preal @ B @ ( image_412476031391729958_preal @ R @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( member6334355225518030407_preal @ ( produc4971909114269240418_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_20_Image__singleton__iff,axiom,
! [B: produc5063651213027957927_preal,R: set_Pr2166019032385455654_preal,A: int] :
( ( member7052443626638417872_preal @ B @ ( image_7455349637590883302_preal @ R @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( member7369391392762924807_preal @ ( produc4765579209246141218_preal @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_21_Image__singleton__iff,axiom,
! [B: int,R: set_Pr958786334691620121nt_int,A: int] :
( ( member_int @ B @ ( image_int_int @ R @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R ) ) ).
% Image_singleton_iff
thf(fact_22_Image__empty1,axiom,
! [X5: set_Pr3959665804823141767_preal] :
( ( image_7293650632215208851_preal @ bot_bo3249957383623713587_preal @ X5 )
= bot_bo5832510953819287539_preal ) ).
% Image_empty1
thf(fact_23_Image__empty1,axiom,
! [X5: set_Dedekind_preal] :
( ( image_1423628590549589971_preal @ bot_bo5832510953819287539_preal @ X5 )
= bot_bo4848840305443107682_preal ) ).
% Image_empty1
thf(fact_24_Image__empty2,axiom,
! [R2: set_Pr6738245714641013959_preal] :
( ( image_7293650632215208851_preal @ R2 @ bot_bo5832510953819287539_preal )
= bot_bo5832510953819287539_preal ) ).
% Image_empty2
thf(fact_25_Image__empty2,axiom,
! [R2: set_Pr5788225742885250104al_int] :
( ( image_3463864234023239744al_int @ R2 @ bot_bo5832510953819287539_preal )
= bot_bot_set_int ) ).
% Image_empty2
thf(fact_26_Image__empty2,axiom,
! [R2: set_Pr2166019032385455654_preal] :
( ( image_7455349637590883302_preal @ R2 @ bot_bot_set_int )
= bot_bo5832510953819287539_preal ) ).
% Image_empty2
thf(fact_27_Image__empty2,axiom,
! [R2: set_Pr958786334691620121nt_int] :
( ( image_int_int @ R2 @ bot_bot_set_int )
= bot_bot_set_int ) ).
% Image_empty2
thf(fact_28_ImageI,axiom,
! [A: dedekind_preal,B: dedekind_preal,R: set_Pr3959665804823141767_preal,A2: set_Dedekind_preal] :
( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ A @ B ) @ R )
=> ( ( member6871284927547481791_preal @ A @ A2 )
=> ( member6871284927547481791_preal @ B @ ( image_1423628590549589971_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_29_ImageI,axiom,
! [A: int,B: int,R: set_Pr958786334691620121nt_int,A2: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
=> ( ( member_int @ A @ A2 )
=> ( member_int @ B @ ( image_int_int @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_30_ImageI,axiom,
! [A: produc5063651213027957927_preal,B: int,R: set_Pr5788225742885250104al_int,A2: set_Pr3959665804823141767_preal] :
( ( member1795383350042252825al_int @ ( produc774093805678497660al_int @ A @ B ) @ R )
=> ( ( member7052443626638417872_preal @ A @ A2 )
=> ( member_int @ B @ ( image_3463864234023239744al_int @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_31_ImageI,axiom,
! [A: int,B: produc5063651213027957927_preal,R: set_Pr2166019032385455654_preal,A2: set_int] :
( ( member7369391392762924807_preal @ ( produc4765579209246141218_preal @ A @ B ) @ R )
=> ( ( member_int @ A @ A2 )
=> ( member7052443626638417872_preal @ B @ ( image_7455349637590883302_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_32_ImageI,axiom,
! [A: set_Pr3959665804823141767_preal,B: int,R: set_Pr388394143813266712al_int,A2: set_se7472626482866788967_preal] :
( ( member4206120071959551737al_int @ ( produc713202198316658780al_int @ A @ B ) @ R )
=> ( ( member6785577539073195184_preal @ A @ A2 )
=> ( member_int @ B @ ( image_9136712926055551264al_int @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_33_ImageI,axiom,
! [A: int,B: set_Pr3959665804823141767_preal,R: set_Pr661714660250796806_preal,A2: set_int] :
( ( member2335964382059793383_preal @ ( produc4050541007652503554_preal @ A @ B ) @ R )
=> ( ( member_int @ A @ A2 )
=> ( member6785577539073195184_preal @ B @ ( image_3250679698536620230_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_34_ImageI,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal,R: set_Pr6738245714641013959_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ A @ B ) @ R )
=> ( ( member7052443626638417872_preal @ A @ A2 )
=> ( member7052443626638417872_preal @ B @ ( image_7293650632215208851_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_35_ImageI,axiom,
! [A: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal,R: set_Pr2351743508290812903_preal,A2: set_se7472626482866788967_preal] :
( ( member1913120636565518896_preal @ ( produc814898704014258807_preal @ A @ B ) @ R )
=> ( ( member6785577539073195184_preal @ A @ A2 )
=> ( member7052443626638417872_preal @ B @ ( image_5601529963401022131_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_36_ImageI,axiom,
! [A: produc5063651213027957927_preal,B: set_Pr3959665804823141767_preal,R: set_Pr1512189777605002663_preal,A2: set_Pr3959665804823141767_preal] :
( ( member2272691600839418864_preal @ ( produc7964486806676790327_preal @ A @ B ) @ R )
=> ( ( member7052443626638417872_preal @ A @ A2 )
=> ( member6785577539073195184_preal @ B @ ( image_3527746029208777843_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_37_ImageI,axiom,
! [A: set_Pr3959665804823141767_preal,B: set_Pr3959665804823141767_preal,R: set_Pr3180418311277671623_preal,A2: set_se7472626482866788967_preal] :
( ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ A @ B ) @ R )
=> ( ( member6785577539073195184_preal @ A @ A2 )
=> ( member6785577539073195184_preal @ B @ ( image_4793389045358533523_preal @ R @ A2 ) ) ) ) ).
% ImageI
thf(fact_38_singletonI,axiom,
! [A: set_Pr3959665804823141767_preal] : ( member6785577539073195184_preal @ A @ ( insert5864466517235936087_preal @ A @ bot_bo6482683055655779539_preal ) ) ).
% singletonI
thf(fact_39_singletonI,axiom,
! [A: produc7488332866442972391_preal] : ( member7273733640249002768_preal @ A @ ( insert8416738607890598583_preal @ A @ bot_bo3249957383623713587_preal ) ) ).
% singletonI
thf(fact_40_singletonI,axiom,
! [A: produc5063651213027957927_preal] : ( member7052443626638417872_preal @ A @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) ).
% singletonI
thf(fact_41_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_42_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_43_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_44_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_45_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_46_realrel__in__real,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] : ( member6785577539073195184_preal @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X3 @ Y3 ) @ bot_bo5832510953819287539_preal ) ) @ dedekind_Real ) ).
% realrel_in_real
thf(fact_47_real__add,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,U: dedekind_preal,V: dedekind_preal] :
( ( plus_p4060926892116697567d_real @ ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X3 @ Y3 ) @ bot_bo5832510953819287539_preal ) ) ) @ ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ U @ V ) @ bot_bo5832510953819287539_preal ) ) ) )
= ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ ( plus_p3173629198307831117_preal @ X3 @ U ) @ ( plus_p3173629198307831117_preal @ Y3 @ V ) ) @ bot_bo5832510953819287539_preal ) ) ) ) ).
% real_add
thf(fact_48_insertCI,axiom,
! [A: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal,B: set_Pr3959665804823141767_preal] :
( ( ~ ( member6785577539073195184_preal @ A @ B2 )
=> ( A = B ) )
=> ( member6785577539073195184_preal @ A @ ( insert5864466517235936087_preal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_49_insertCI,axiom,
! [A: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal,B: produc7488332866442972391_preal] :
( ( ~ ( member7273733640249002768_preal @ A @ B2 )
=> ( A = B ) )
=> ( member7273733640249002768_preal @ A @ ( insert8416738607890598583_preal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_50_insertCI,axiom,
! [A: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal] :
( ( ~ ( member7052443626638417872_preal @ A @ B2 )
=> ( A = B ) )
=> ( member7052443626638417872_preal @ A @ ( insert2800550009750661751_preal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_51_insertCI,axiom,
! [A: int,B2: set_int,B: int] :
( ( ~ ( member_int @ A @ B2 )
=> ( A = B ) )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_52_insert__iff,axiom,
! [A: set_Pr3959665804823141767_preal,B: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ A @ ( insert5864466517235936087_preal @ B @ A2 ) )
= ( ( A = B )
| ( member6785577539073195184_preal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_53_insert__iff,axiom,
! [A: produc7488332866442972391_preal,B: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ A @ ( insert8416738607890598583_preal @ B @ A2 ) )
= ( ( A = B )
| ( member7273733640249002768_preal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_54_insert__iff,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ A @ ( insert2800550009750661751_preal @ B @ A2 ) )
= ( ( A = B )
| ( member7052443626638417872_preal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_55_insert__iff,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
= ( ( A = B )
| ( member_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_56_insert__absorb2,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( insert2800550009750661751_preal @ X3 @ ( insert2800550009750661751_preal @ X3 @ A2 ) )
= ( insert2800550009750661751_preal @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_57_insert__absorb2,axiom,
! [X3: int,A2: set_int] :
( ( insert_int @ X3 @ ( insert_int @ X3 @ A2 ) )
= ( insert_int @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_58_add__right__cancel,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_59_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_60_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_61_add__left__cancel,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_62_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_63_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_64_empty__subsetI,axiom,
! [A2: set_Pr3959665804823141767_preal] : ( ord_le845437053512250663_preal @ bot_bo5832510953819287539_preal @ A2 ) ).
% empty_subsetI
thf(fact_65_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_66_subset__empty,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ A2 @ bot_bo5832510953819287539_preal )
= ( A2 = bot_bo5832510953819287539_preal ) ) ).
% subset_empty
thf(fact_67_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_68_empty__Collect__eq,axiom,
! [P: produc5063651213027957927_preal > $o] :
( ( bot_bo5832510953819287539_preal
= ( collec5689422505606869010_preal @ P ) )
= ( ! [X4: produc5063651213027957927_preal] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_69_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X4: int] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_70_Collect__empty__eq,axiom,
! [P: produc5063651213027957927_preal > $o] :
( ( ( collec5689422505606869010_preal @ P )
= bot_bo5832510953819287539_preal )
= ( ! [X4: produc5063651213027957927_preal] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_71_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X4: int] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_72_all__not__in__conv,axiom,
! [A2: set_se7472626482866788967_preal] :
( ( ! [X4: set_Pr3959665804823141767_preal] :
~ ( member6785577539073195184_preal @ X4 @ A2 ) )
= ( A2 = bot_bo6482683055655779539_preal ) ) ).
% all_not_in_conv
thf(fact_73_all__not__in__conv,axiom,
! [A2: set_Pr6738245714641013959_preal] :
( ( ! [X4: produc7488332866442972391_preal] :
~ ( member7273733640249002768_preal @ X4 @ A2 ) )
= ( A2 = bot_bo3249957383623713587_preal ) ) ).
% all_not_in_conv
thf(fact_74_all__not__in__conv,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( ! [X4: produc5063651213027957927_preal] :
~ ( member7052443626638417872_preal @ X4 @ A2 ) )
= ( A2 = bot_bo5832510953819287539_preal ) ) ).
% all_not_in_conv
thf(fact_75_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X4: int] :
~ ( member_int @ X4 @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_76_empty__iff,axiom,
! [C: set_Pr3959665804823141767_preal] :
~ ( member6785577539073195184_preal @ C @ bot_bo6482683055655779539_preal ) ).
% empty_iff
thf(fact_77_empty__iff,axiom,
! [C: produc7488332866442972391_preal] :
~ ( member7273733640249002768_preal @ C @ bot_bo3249957383623713587_preal ) ).
% empty_iff
thf(fact_78_empty__iff,axiom,
! [C: produc5063651213027957927_preal] :
~ ( member7052443626638417872_preal @ C @ bot_bo5832510953819287539_preal ) ).
% empty_iff
thf(fact_79_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_80_insert__subset,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ( ord_le6746582767776264711_preal @ ( insert5864466517235936087_preal @ X3 @ A2 ) @ B2 )
= ( ( member6785577539073195184_preal @ X3 @ B2 )
& ( ord_le6746582767776264711_preal @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_81_insert__subset,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ( ord_le6462023905233536103_preal @ ( insert8416738607890598583_preal @ X3 @ A2 ) @ B2 )
= ( ( member7273733640249002768_preal @ X3 @ B2 )
& ( ord_le6462023905233536103_preal @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_82_insert__subset,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ ( insert2800550009750661751_preal @ X3 @ A2 ) @ B2 )
= ( ( member7052443626638417872_preal @ X3 @ B2 )
& ( ord_le845437053512250663_preal @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_83_insert__subset,axiom,
! [X3: int,A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
= ( ( member_int @ X3 @ B2 )
& ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_84_singleton__insert__inj__eq_H,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal] :
( ( ( insert2800550009750661751_preal @ A @ A2 )
= ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal ) )
= ( ( A = B )
& ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_85_singleton__insert__inj__eq_H,axiom,
! [A: int,A2: set_int,B: int] :
( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_86_singleton__insert__inj__eq,axiom,
! [B: produc5063651213027957927_preal,A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal )
= ( insert2800550009750661751_preal @ A @ A2 ) )
= ( ( A = B )
& ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_87_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A2: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_88_Abs__Real__inject,axiom,
! [X3: set_Pr3959665804823141767_preal,Y3: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ X3 @ dedekind_Real )
=> ( ( member6785577539073195184_preal @ Y3 @ dedekind_Real )
=> ( ( ( dedekind_Abs_Real @ X3 )
= ( dedekind_Abs_Real @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% Abs_Real_inject
thf(fact_89_realrel__iff,axiom,
! [X1: dedekind_preal,Y1: dedekind_preal,X2: dedekind_preal,Y2: dedekind_preal] :
( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ ( produc5992578386459872151_preal @ X1 @ Y1 ) @ ( produc5992578386459872151_preal @ X2 @ Y2 ) ) @ dedekind_realrel )
= ( ( plus_p3173629198307831117_preal @ X1 @ Y2 )
= ( plus_p3173629198307831117_preal @ X2 @ Y1 ) ) ) ).
% realrel_iff
thf(fact_90_equiv__realrel__iff,axiom,
! [X3: produc5063651213027957927_preal,Y3: produc5063651213027957927_preal] :
( ( ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) )
= ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ Y3 @ bot_bo5832510953819287539_preal ) ) )
= ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X3 @ Y3 ) @ dedekind_realrel ) ) ).
% equiv_realrel_iff
thf(fact_91_Image__mono,axiom,
! [R3: set_Pr6738245714641013959_preal,R: set_Pr6738245714641013959_preal,A3: set_Pr3959665804823141767_preal,A2: set_Pr3959665804823141767_preal] :
( ( ord_le6462023905233536103_preal @ R3 @ R )
=> ( ( ord_le845437053512250663_preal @ A3 @ A2 )
=> ( ord_le845437053512250663_preal @ ( image_7293650632215208851_preal @ R3 @ A3 ) @ ( image_7293650632215208851_preal @ R @ A2 ) ) ) ) ).
% Image_mono
thf(fact_92_subrelI,axiom,
! [R: set_Pr3959665804823141767_preal,S: set_Pr3959665804823141767_preal] :
( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X @ Y ) @ R )
=> ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X @ Y ) @ S ) )
=> ( ord_le845437053512250663_preal @ R @ S ) ) ).
% subrelI
thf(fact_93_subrelI,axiom,
! [R: set_Pr6738245714641013959_preal,S: set_Pr6738245714641013959_preal] :
( ! [X: produc5063651213027957927_preal,Y: produc5063651213027957927_preal] :
( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X @ Y ) @ R )
=> ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X @ Y ) @ S ) )
=> ( ord_le6462023905233536103_preal @ R @ S ) ) ).
% subrelI
thf(fact_94_subrelI,axiom,
! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ! [X: int,Y: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) )
=> ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% subrelI
thf(fact_95_subset__insertI2,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ A2 @ B2 )
=> ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_96_subset__insertI2,axiom,
! [A2: set_int,B2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_97_subset__insertI,axiom,
! [B2: set_Pr3959665804823141767_preal,A: produc5063651213027957927_preal] : ( ord_le845437053512250663_preal @ B2 @ ( insert2800550009750661751_preal @ A @ B2 ) ) ).
% subset_insertI
thf(fact_98_subset__insertI,axiom,
! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% subset_insertI
thf(fact_99_subset__insert,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ~ ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ( ord_le6746582767776264711_preal @ A2 @ ( insert5864466517235936087_preal @ X3 @ B2 ) )
= ( ord_le6746582767776264711_preal @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_100_subset__insert,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ~ ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ( ord_le6462023905233536103_preal @ A2 @ ( insert8416738607890598583_preal @ X3 @ B2 ) )
= ( ord_le6462023905233536103_preal @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_101_subset__insert,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ~ ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ B2 ) )
= ( ord_le845437053512250663_preal @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_102_subset__insert,axiom,
! [X3: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
= ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_103_insert__mono,axiom,
! [C2: set_Pr3959665804823141767_preal,D2: set_Pr3959665804823141767_preal,A: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ C2 @ D2 )
=> ( ord_le845437053512250663_preal @ ( insert2800550009750661751_preal @ A @ C2 ) @ ( insert2800550009750661751_preal @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_104_insert__mono,axiom,
! [C2: set_int,D2: set_int,A: int] :
( ( ord_less_eq_set_int @ C2 @ D2 )
=> ( ord_less_eq_set_int @ ( insert_int @ A @ C2 ) @ ( insert_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_105_subset__singleton__iff,axiom,
! [X5: set_Pr3959665804823141767_preal,A: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ X5 @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) )
= ( ( X5 = bot_bo5832510953819287539_preal )
| ( X5
= ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) ) ) ).
% subset_singleton_iff
thf(fact_106_subset__singleton__iff,axiom,
! [X5: set_int,A: int] :
( ( ord_less_eq_set_int @ X5 @ ( insert_int @ A @ bot_bot_set_int ) )
= ( ( X5 = bot_bot_set_int )
| ( X5
= ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_107_mem__Collect__eq,axiom,
! [A: set_Pr3959665804823141767_preal,P: set_Pr3959665804823141767_preal > $o] :
( ( member6785577539073195184_preal @ A @ ( collec9146870221497238770_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A: produc7488332866442972391_preal,P: produc7488332866442972391_preal > $o] :
( ( member7273733640249002768_preal @ A @ ( collec1285689335269777746_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A: produc5063651213027957927_preal,P: produc5063651213027957927_preal > $o] :
( ( member7052443626638417872_preal @ A @ ( collec5689422505606869010_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A2: set_se7472626482866788967_preal] :
( ( collec9146870221497238770_preal
@ ^ [X4: set_Pr3959665804823141767_preal] : ( member6785577539073195184_preal @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A2: set_Pr6738245714641013959_preal] :
( ( collec1285689335269777746_preal
@ ^ [X4: produc7488332866442972391_preal] : ( member7273733640249002768_preal @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( collec5689422505606869010_preal
@ ^ [X4: produc5063651213027957927_preal] : ( member7052443626638417872_preal @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_115_subset__singletonD,axiom,
! [A2: set_Pr3959665804823141767_preal,X3: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) )
=> ( ( A2 = bot_bo5832510953819287539_preal )
| ( A2
= ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) ) ) ) ).
% subset_singletonD
thf(fact_116_subset__singletonD,axiom,
! [A2: set_int,X3: int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) )
=> ( ( A2 = bot_bot_set_int )
| ( A2
= ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_117_Abs__Real__induct,axiom,
! [P: dedekind_real > $o,X3: dedekind_real] :
( ! [Y: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ Y @ dedekind_Real )
=> ( P @ ( dedekind_Abs_Real @ Y ) ) )
=> ( P @ X3 ) ) ).
% Abs_Real_induct
thf(fact_118_Abs__Real__cases,axiom,
! [X3: dedekind_real] :
~ ! [Y: set_Pr3959665804823141767_preal] :
( ( X3
= ( dedekind_Abs_Real @ Y ) )
=> ~ ( member6785577539073195184_preal @ Y @ dedekind_Real ) ) ).
% Abs_Real_cases
thf(fact_119_add__right__imp__eq,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_120_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_121_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_122_add__left__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_123_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_124_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_125_add_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ B @ ( plus_p3173629198307831117_preal @ A @ C ) )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.left_commute
thf(fact_126_add_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_127_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_128_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_129_add_Ocommute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] : ( plus_p3173629198307831117_preal @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_130_add_Ocommute,axiom,
( plus_p4060926892116697567d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] : ( plus_p4060926892116697567d_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_131_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_132_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_133_add_Oright__cancel,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_134_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_135_add_Oleft__cancel,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_136_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_137_add_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.assoc
thf(fact_138_add_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_139_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_140_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_141_group__cancel_Oadd2,axiom,
! [B2: dedekind_real,K: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( B2
= ( plus_p4060926892116697567d_real @ K @ B ) )
=> ( ( plus_p4060926892116697567d_real @ A @ B2 )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_142_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_143_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_144_group__cancel_Oadd1,axiom,
! [A2: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( plus_p4060926892116697567d_real @ A2 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_145_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_146_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_147_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p3173629198307831117_preal @ I @ K )
= ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_148_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_149_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_150_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_151_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_152_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_153_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_154_ex__in__conv,axiom,
! [A2: set_se7472626482866788967_preal] :
( ( ? [X4: set_Pr3959665804823141767_preal] : ( member6785577539073195184_preal @ X4 @ A2 ) )
= ( A2 != bot_bo6482683055655779539_preal ) ) ).
% ex_in_conv
thf(fact_155_ex__in__conv,axiom,
! [A2: set_Pr6738245714641013959_preal] :
( ( ? [X4: produc7488332866442972391_preal] : ( member7273733640249002768_preal @ X4 @ A2 ) )
= ( A2 != bot_bo3249957383623713587_preal ) ) ).
% ex_in_conv
thf(fact_156_ex__in__conv,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( ? [X4: produc5063651213027957927_preal] : ( member7052443626638417872_preal @ X4 @ A2 ) )
= ( A2 != bot_bo5832510953819287539_preal ) ) ).
% ex_in_conv
thf(fact_157_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X4: int] : ( member_int @ X4 @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_158_equals0I,axiom,
! [A2: set_se7472626482866788967_preal] :
( ! [Y: set_Pr3959665804823141767_preal] :
~ ( member6785577539073195184_preal @ Y @ A2 )
=> ( A2 = bot_bo6482683055655779539_preal ) ) ).
% equals0I
thf(fact_159_equals0I,axiom,
! [A2: set_Pr6738245714641013959_preal] :
( ! [Y: produc7488332866442972391_preal] :
~ ( member7273733640249002768_preal @ Y @ A2 )
=> ( A2 = bot_bo3249957383623713587_preal ) ) ).
% equals0I
thf(fact_160_equals0I,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ! [Y: produc5063651213027957927_preal] :
~ ( member7052443626638417872_preal @ Y @ A2 )
=> ( A2 = bot_bo5832510953819287539_preal ) ) ).
% equals0I
thf(fact_161_equals0I,axiom,
! [A2: set_int] :
( ! [Y: int] :
~ ( member_int @ Y @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_162_equals0D,axiom,
! [A2: set_se7472626482866788967_preal,A: set_Pr3959665804823141767_preal] :
( ( A2 = bot_bo6482683055655779539_preal )
=> ~ ( member6785577539073195184_preal @ A @ A2 ) ) ).
% equals0D
thf(fact_163_equals0D,axiom,
! [A2: set_Pr6738245714641013959_preal,A: produc7488332866442972391_preal] :
( ( A2 = bot_bo3249957383623713587_preal )
=> ~ ( member7273733640249002768_preal @ A @ A2 ) ) ).
% equals0D
thf(fact_164_equals0D,axiom,
! [A2: set_Pr3959665804823141767_preal,A: produc5063651213027957927_preal] :
( ( A2 = bot_bo5832510953819287539_preal )
=> ~ ( member7052443626638417872_preal @ A @ A2 ) ) ).
% equals0D
thf(fact_165_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_166_emptyE,axiom,
! [A: set_Pr3959665804823141767_preal] :
~ ( member6785577539073195184_preal @ A @ bot_bo6482683055655779539_preal ) ).
% emptyE
thf(fact_167_emptyE,axiom,
! [A: produc7488332866442972391_preal] :
~ ( member7273733640249002768_preal @ A @ bot_bo3249957383623713587_preal ) ).
% emptyE
thf(fact_168_emptyE,axiom,
! [A: produc5063651213027957927_preal] :
~ ( member7052443626638417872_preal @ A @ bot_bo5832510953819287539_preal ) ).
% emptyE
thf(fact_169_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_170_mk__disjoint__insert,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ A @ A2 )
=> ? [B4: set_se7472626482866788967_preal] :
( ( A2
= ( insert5864466517235936087_preal @ A @ B4 ) )
& ~ ( member6785577539073195184_preal @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_171_mk__disjoint__insert,axiom,
! [A: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ A @ A2 )
=> ? [B4: set_Pr6738245714641013959_preal] :
( ( A2
= ( insert8416738607890598583_preal @ A @ B4 ) )
& ~ ( member7273733640249002768_preal @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_172_mk__disjoint__insert,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ A @ A2 )
=> ? [B4: set_Pr3959665804823141767_preal] :
( ( A2
= ( insert2800550009750661751_preal @ A @ B4 ) )
& ~ ( member7052443626638417872_preal @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_173_mk__disjoint__insert,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ? [B4: set_int] :
( ( A2
= ( insert_int @ A @ B4 ) )
& ~ ( member_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_174_insert__commute,axiom,
! [X3: produc5063651213027957927_preal,Y3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( insert2800550009750661751_preal @ X3 @ ( insert2800550009750661751_preal @ Y3 @ A2 ) )
= ( insert2800550009750661751_preal @ Y3 @ ( insert2800550009750661751_preal @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_175_insert__commute,axiom,
! [X3: int,Y3: int,A2: set_int] :
( ( insert_int @ X3 @ ( insert_int @ Y3 @ A2 ) )
= ( insert_int @ Y3 @ ( insert_int @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_176_insert__eq__iff,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal] :
( ~ ( member6785577539073195184_preal @ A @ A2 )
=> ( ~ ( member6785577539073195184_preal @ B @ B2 )
=> ( ( ( insert5864466517235936087_preal @ A @ A2 )
= ( insert5864466517235936087_preal @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_se7472626482866788967_preal] :
( ( A2
= ( insert5864466517235936087_preal @ B @ C3 ) )
& ~ ( member6785577539073195184_preal @ B @ C3 )
& ( B2
= ( insert5864466517235936087_preal @ A @ C3 ) )
& ~ ( member6785577539073195184_preal @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_177_insert__eq__iff,axiom,
! [A: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal] :
( ~ ( member7273733640249002768_preal @ A @ A2 )
=> ( ~ ( member7273733640249002768_preal @ B @ B2 )
=> ( ( ( insert8416738607890598583_preal @ A @ A2 )
= ( insert8416738607890598583_preal @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Pr6738245714641013959_preal] :
( ( A2
= ( insert8416738607890598583_preal @ B @ C3 ) )
& ~ ( member7273733640249002768_preal @ B @ C3 )
& ( B2
= ( insert8416738607890598583_preal @ A @ C3 ) )
& ~ ( member7273733640249002768_preal @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_178_insert__eq__iff,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal] :
( ~ ( member7052443626638417872_preal @ A @ A2 )
=> ( ~ ( member7052443626638417872_preal @ B @ B2 )
=> ( ( ( insert2800550009750661751_preal @ A @ A2 )
= ( insert2800550009750661751_preal @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Pr3959665804823141767_preal] :
( ( A2
= ( insert2800550009750661751_preal @ B @ C3 ) )
& ~ ( member7052443626638417872_preal @ B @ C3 )
& ( B2
= ( insert2800550009750661751_preal @ A @ C3 ) )
& ~ ( member7052443626638417872_preal @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_179_insert__eq__iff,axiom,
! [A: int,A2: set_int,B: int,B2: set_int] :
( ~ ( member_int @ A @ A2 )
=> ( ~ ( member_int @ B @ B2 )
=> ( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_int] :
( ( A2
= ( insert_int @ B @ C3 ) )
& ~ ( member_int @ B @ C3 )
& ( B2
= ( insert_int @ A @ C3 ) )
& ~ ( member_int @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_180_insert__absorb,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ A @ A2 )
=> ( ( insert5864466517235936087_preal @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_181_insert__absorb,axiom,
! [A: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ A @ A2 )
=> ( ( insert8416738607890598583_preal @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_182_insert__absorb,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ A @ A2 )
=> ( ( insert2800550009750661751_preal @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_183_insert__absorb,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_184_insert__ident,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ~ ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ~ ( member6785577539073195184_preal @ X3 @ B2 )
=> ( ( ( insert5864466517235936087_preal @ X3 @ A2 )
= ( insert5864466517235936087_preal @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_185_insert__ident,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ~ ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ~ ( member7273733640249002768_preal @ X3 @ B2 )
=> ( ( ( insert8416738607890598583_preal @ X3 @ A2 )
= ( insert8416738607890598583_preal @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_186_insert__ident,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ~ ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ~ ( member7052443626638417872_preal @ X3 @ B2 )
=> ( ( ( insert2800550009750661751_preal @ X3 @ A2 )
= ( insert2800550009750661751_preal @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_187_insert__ident,axiom,
! [X3: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ~ ( member_int @ X3 @ B2 )
=> ( ( ( insert_int @ X3 @ A2 )
= ( insert_int @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_188_Set_Oset__insert,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ X3 @ A2 )
=> ~ ! [B4: set_se7472626482866788967_preal] :
( ( A2
= ( insert5864466517235936087_preal @ X3 @ B4 ) )
=> ( member6785577539073195184_preal @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_189_Set_Oset__insert,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ X3 @ A2 )
=> ~ ! [B4: set_Pr6738245714641013959_preal] :
( ( A2
= ( insert8416738607890598583_preal @ X3 @ B4 ) )
=> ( member7273733640249002768_preal @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_190_Set_Oset__insert,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ X3 @ A2 )
=> ~ ! [B4: set_Pr3959665804823141767_preal] :
( ( A2
= ( insert2800550009750661751_preal @ X3 @ B4 ) )
=> ( member7052443626638417872_preal @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_191_Set_Oset__insert,axiom,
! [X3: int,A2: set_int] :
( ( member_int @ X3 @ A2 )
=> ~ ! [B4: set_int] :
( ( A2
= ( insert_int @ X3 @ B4 ) )
=> ( member_int @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_192_insertI2,axiom,
! [A: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal,B: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ A @ B2 )
=> ( member6785577539073195184_preal @ A @ ( insert5864466517235936087_preal @ B @ B2 ) ) ) ).
% insertI2
thf(fact_193_insertI2,axiom,
! [A: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal,B: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ A @ B2 )
=> ( member7273733640249002768_preal @ A @ ( insert8416738607890598583_preal @ B @ B2 ) ) ) ).
% insertI2
thf(fact_194_insertI2,axiom,
! [A: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ A @ B2 )
=> ( member7052443626638417872_preal @ A @ ( insert2800550009750661751_preal @ B @ B2 ) ) ) ).
% insertI2
thf(fact_195_insertI2,axiom,
! [A: int,B2: set_int,B: int] :
( ( member_int @ A @ B2 )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_196_insertI1,axiom,
! [A: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal] : ( member6785577539073195184_preal @ A @ ( insert5864466517235936087_preal @ A @ B2 ) ) ).
% insertI1
thf(fact_197_insertI1,axiom,
! [A: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal] : ( member7273733640249002768_preal @ A @ ( insert8416738607890598583_preal @ A @ B2 ) ) ).
% insertI1
thf(fact_198_insertI1,axiom,
! [A: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal] : ( member7052443626638417872_preal @ A @ ( insert2800550009750661751_preal @ A @ B2 ) ) ).
% insertI1
thf(fact_199_insertI1,axiom,
! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% insertI1
thf(fact_200_insertE,axiom,
! [A: set_Pr3959665804823141767_preal,B: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ A @ ( insert5864466517235936087_preal @ B @ A2 ) )
=> ( ( A != B )
=> ( member6785577539073195184_preal @ A @ A2 ) ) ) ).
% insertE
thf(fact_201_insertE,axiom,
! [A: produc7488332866442972391_preal,B: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ A @ ( insert8416738607890598583_preal @ B @ A2 ) )
=> ( ( A != B )
=> ( member7273733640249002768_preal @ A @ A2 ) ) ) ).
% insertE
thf(fact_202_insertE,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ A @ ( insert2800550009750661751_preal @ B @ A2 ) )
=> ( ( A != B )
=> ( member7052443626638417872_preal @ A @ A2 ) ) ) ).
% insertE
thf(fact_203_insertE,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_204_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_205_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_206_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_207_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_208_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_209_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_210_add__right__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add_right_mono
thf(fact_211_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_212_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C5: nat] :
( B
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_213_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_214_add__left__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ C @ A ) @ ( plus_p3173629198307831117_preal @ C @ B ) ) ) ).
% add_left_mono
thf(fact_215_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_216_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_217_add__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ C @ D )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ D ) ) ) ) ).
% add_mono
thf(fact_218_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_219_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_220_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_221_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_222_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_223_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_224_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_225_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_226_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( K = L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_227_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_228_singleton__inject,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal] :
( ( ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal )
= ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_229_singleton__inject,axiom,
! [A: int,B: int] :
( ( ( insert_int @ A @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_230_insert__not__empty,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( insert2800550009750661751_preal @ A @ A2 )
!= bot_bo5832510953819287539_preal ) ).
% insert_not_empty
thf(fact_231_insert__not__empty,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ A2 )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_232_doubleton__eq__iff,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal,C: produc5063651213027957927_preal,D: produc5063651213027957927_preal] :
( ( ( insert2800550009750661751_preal @ A @ ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal ) )
= ( insert2800550009750661751_preal @ C @ ( insert2800550009750661751_preal @ D @ bot_bo5832510953819287539_preal ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_233_doubleton__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_234_singleton__iff,axiom,
! [B: set_Pr3959665804823141767_preal,A: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ B @ ( insert5864466517235936087_preal @ A @ bot_bo6482683055655779539_preal ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_235_singleton__iff,axiom,
! [B: produc7488332866442972391_preal,A: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ B @ ( insert8416738607890598583_preal @ A @ bot_bo3249957383623713587_preal ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_236_singleton__iff,axiom,
! [B: produc5063651213027957927_preal,A: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ B @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_237_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_238_singletonD,axiom,
! [B: set_Pr3959665804823141767_preal,A: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ B @ ( insert5864466517235936087_preal @ A @ bot_bo6482683055655779539_preal ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_239_singletonD,axiom,
! [B: produc7488332866442972391_preal,A: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ B @ ( insert8416738607890598583_preal @ A @ bot_bo3249957383623713587_preal ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_240_singletonD,axiom,
! [B: produc5063651213027957927_preal,A: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ B @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_241_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_242_rev__ImageI,axiom,
! [A: dedekind_preal,A2: set_Dedekind_preal,B: dedekind_preal,R: set_Pr3959665804823141767_preal] :
( ( member6871284927547481791_preal @ A @ A2 )
=> ( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ A @ B ) @ R )
=> ( member6871284927547481791_preal @ B @ ( image_1423628590549589971_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_243_rev__ImageI,axiom,
! [A: int,A2: set_int,B: int,R: set_Pr958786334691620121nt_int] :
( ( member_int @ A @ A2 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
=> ( member_int @ B @ ( image_int_int @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_244_rev__ImageI,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B: int,R: set_Pr5788225742885250104al_int] :
( ( member7052443626638417872_preal @ A @ A2 )
=> ( ( member1795383350042252825al_int @ ( produc774093805678497660al_int @ A @ B ) @ R )
=> ( member_int @ B @ ( image_3463864234023239744al_int @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_245_rev__ImageI,axiom,
! [A: int,A2: set_int,B: produc5063651213027957927_preal,R: set_Pr2166019032385455654_preal] :
( ( member_int @ A @ A2 )
=> ( ( member7369391392762924807_preal @ ( produc4765579209246141218_preal @ A @ B ) @ R )
=> ( member7052443626638417872_preal @ B @ ( image_7455349637590883302_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_246_rev__ImageI,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B: int,R: set_Pr388394143813266712al_int] :
( ( member6785577539073195184_preal @ A @ A2 )
=> ( ( member4206120071959551737al_int @ ( produc713202198316658780al_int @ A @ B ) @ R )
=> ( member_int @ B @ ( image_9136712926055551264al_int @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_247_rev__ImageI,axiom,
! [A: int,A2: set_int,B: set_Pr3959665804823141767_preal,R: set_Pr661714660250796806_preal] :
( ( member_int @ A @ A2 )
=> ( ( member2335964382059793383_preal @ ( produc4050541007652503554_preal @ A @ B ) @ R )
=> ( member6785577539073195184_preal @ B @ ( image_3250679698536620230_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_248_rev__ImageI,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal,R: set_Pr6738245714641013959_preal] :
( ( member7052443626638417872_preal @ A @ A2 )
=> ( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ A @ B ) @ R )
=> ( member7052443626638417872_preal @ B @ ( image_7293650632215208851_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_249_rev__ImageI,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B: produc5063651213027957927_preal,R: set_Pr2351743508290812903_preal] :
( ( member6785577539073195184_preal @ A @ A2 )
=> ( ( member1913120636565518896_preal @ ( produc814898704014258807_preal @ A @ B ) @ R )
=> ( member7052443626638417872_preal @ B @ ( image_5601529963401022131_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_250_rev__ImageI,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B: set_Pr3959665804823141767_preal,R: set_Pr1512189777605002663_preal] :
( ( member7052443626638417872_preal @ A @ A2 )
=> ( ( member2272691600839418864_preal @ ( produc7964486806676790327_preal @ A @ B ) @ R )
=> ( member6785577539073195184_preal @ B @ ( image_3527746029208777843_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_251_rev__ImageI,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B: set_Pr3959665804823141767_preal,R: set_Pr3180418311277671623_preal] :
( ( member6785577539073195184_preal @ A @ A2 )
=> ( ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ A @ B ) @ R )
=> ( member6785577539073195184_preal @ B @ ( image_4793389045358533523_preal @ R @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_252_Image__iff,axiom,
! [B: dedekind_preal,R: set_Pr3959665804823141767_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( image_1423628590549589971_preal @ R @ A2 ) )
= ( ? [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ A2 )
& ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X4 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_253_Image__iff,axiom,
! [B: produc5063651213027957927_preal,R: set_Pr6738245714641013959_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ B @ ( image_7293650632215208851_preal @ R @ A2 ) )
= ( ? [X4: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ X4 @ A2 )
& ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X4 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_254_Image__iff,axiom,
! [B: int,R: set_Pr958786334691620121nt_int,A2: set_int] :
( ( member_int @ B @ ( image_int_int @ R @ A2 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ B ) @ R ) ) ) ) ).
% Image_iff
thf(fact_255_ImageE,axiom,
! [B: dedekind_preal,R: set_Pr3959665804823141767_preal,A2: set_Dedekind_preal] :
( ( member6871284927547481791_preal @ B @ ( image_1423628590549589971_preal @ R @ A2 ) )
=> ~ ! [X: dedekind_preal] :
( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X @ B ) @ R )
=> ~ ( member6871284927547481791_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_256_ImageE,axiom,
! [B: int,R: set_Pr958786334691620121nt_int,A2: set_int] :
( ( member_int @ B @ ( image_int_int @ R @ A2 ) )
=> ~ ! [X: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ B ) @ R )
=> ~ ( member_int @ X @ A2 ) ) ) ).
% ImageE
thf(fact_257_ImageE,axiom,
! [B: produc5063651213027957927_preal,R: set_Pr2166019032385455654_preal,A2: set_int] :
( ( member7052443626638417872_preal @ B @ ( image_7455349637590883302_preal @ R @ A2 ) )
=> ~ ! [X: int] :
( ( member7369391392762924807_preal @ ( produc4765579209246141218_preal @ X @ B ) @ R )
=> ~ ( member_int @ X @ A2 ) ) ) ).
% ImageE
thf(fact_258_ImageE,axiom,
! [B: int,R: set_Pr5788225742885250104al_int,A2: set_Pr3959665804823141767_preal] :
( ( member_int @ B @ ( image_3463864234023239744al_int @ R @ A2 ) )
=> ~ ! [X: produc5063651213027957927_preal] :
( ( member1795383350042252825al_int @ ( produc774093805678497660al_int @ X @ B ) @ R )
=> ~ ( member7052443626638417872_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_259_ImageE,axiom,
! [B: set_Pr3959665804823141767_preal,R: set_Pr661714660250796806_preal,A2: set_int] :
( ( member6785577539073195184_preal @ B @ ( image_3250679698536620230_preal @ R @ A2 ) )
=> ~ ! [X: int] :
( ( member2335964382059793383_preal @ ( produc4050541007652503554_preal @ X @ B ) @ R )
=> ~ ( member_int @ X @ A2 ) ) ) ).
% ImageE
thf(fact_260_ImageE,axiom,
! [B: int,R: set_Pr388394143813266712al_int,A2: set_se7472626482866788967_preal] :
( ( member_int @ B @ ( image_9136712926055551264al_int @ R @ A2 ) )
=> ~ ! [X: set_Pr3959665804823141767_preal] :
( ( member4206120071959551737al_int @ ( produc713202198316658780al_int @ X @ B ) @ R )
=> ~ ( member6785577539073195184_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_261_ImageE,axiom,
! [B: produc5063651213027957927_preal,R: set_Pr6738245714641013959_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ B @ ( image_7293650632215208851_preal @ R @ A2 ) )
=> ~ ! [X: produc5063651213027957927_preal] :
( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X @ B ) @ R )
=> ~ ( member7052443626638417872_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_262_ImageE,axiom,
! [B: set_Pr3959665804823141767_preal,R: set_Pr1512189777605002663_preal,A2: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ B @ ( image_3527746029208777843_preal @ R @ A2 ) )
=> ~ ! [X: produc5063651213027957927_preal] :
( ( member2272691600839418864_preal @ ( produc7964486806676790327_preal @ X @ B ) @ R )
=> ~ ( member7052443626638417872_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_263_ImageE,axiom,
! [B: produc5063651213027957927_preal,R: set_Pr2351743508290812903_preal,A2: set_se7472626482866788967_preal] :
( ( member7052443626638417872_preal @ B @ ( image_5601529963401022131_preal @ R @ A2 ) )
=> ~ ! [X: set_Pr3959665804823141767_preal] :
( ( member1913120636565518896_preal @ ( produc814898704014258807_preal @ X @ B ) @ R )
=> ~ ( member6785577539073195184_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_264_ImageE,axiom,
! [B: set_Pr3959665804823141767_preal,R: set_Pr3180418311277671623_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ B @ ( image_4793389045358533523_preal @ R @ A2 ) )
=> ~ ! [X: set_Pr3959665804823141767_preal] :
( ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ X @ B ) @ R )
=> ~ ( member6785577539073195184_preal @ X @ A2 ) ) ) ).
% ImageE
thf(fact_265_the__elem__eq,axiom,
! [X3: produc5063651213027957927_preal] :
( ( the_el8575259181799709340_preal @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) )
= X3 ) ).
% the_elem_eq
thf(fact_266_the__elem__eq,axiom,
! [X3: int] :
( ( the_elem_int @ ( insert_int @ X3 @ bot_bot_set_int ) )
= X3 ) ).
% the_elem_eq
thf(fact_267_real__le__def,axiom,
( ord_le2716243287969276982d_real
= ( ^ [Z2: dedekind_real,W2: dedekind_real] :
? [X4: dedekind_preal,Y4: dedekind_preal,U3: dedekind_preal,V3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X4 @ V3 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y4 ) )
& ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X4 @ Y4 ) @ ( dedekind_Rep_Real @ Z2 ) )
& ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ U3 @ V3 ) @ ( dedekind_Rep_Real @ W2 ) ) ) ) ) ).
% real_le_def
thf(fact_268_old_Oprod_Oinject,axiom,
! [A: dedekind_preal,B: dedekind_preal,A5: dedekind_preal,B5: dedekind_preal] :
( ( ( produc5992578386459872151_preal @ A @ B )
= ( produc5992578386459872151_preal @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_269_old_Oprod_Oinject,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal,A5: produc5063651213027957927_preal,B5: produc5063651213027957927_preal] :
( ( ( produc2372827984048905047_preal @ A @ B )
= ( produc2372827984048905047_preal @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_270_old_Oprod_Oinject,axiom,
! [A: int,B: int,A5: int,B5: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_271_prod_Oinject,axiom,
! [X1: dedekind_preal,X2: dedekind_preal,Y1: dedekind_preal,Y2: dedekind_preal] :
( ( ( produc5992578386459872151_preal @ X1 @ X2 )
= ( produc5992578386459872151_preal @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_272_prod_Oinject,axiom,
! [X1: produc5063651213027957927_preal,X2: produc5063651213027957927_preal,Y1: produc5063651213027957927_preal,Y2: produc5063651213027957927_preal] :
( ( ( produc2372827984048905047_preal @ X1 @ X2 )
= ( produc2372827984048905047_preal @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_273_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_274_dual__order_Orefl,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% dual_order.refl
thf(fact_275_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_276_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_277_order__refl,axiom,
! [X3: dedekind_preal] : ( ord_le5604041210740703414_preal @ X3 @ X3 ) ).
% order_refl
thf(fact_278_order__refl,axiom,
! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% order_refl
thf(fact_279_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_280_real__zero__iff,axiom,
! [X3: dedekind_preal] :
( ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X3 @ X3 ) @ bot_bo5832510953819287539_preal ) ) )
= zero_z580800474297136991d_real ) ).
% real_zero_iff
thf(fact_281_real__of__preal__def,axiom,
( dedeki829075424511776921_preal
= ( ^ [M: dedekind_preal] : ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ ( plus_p3173629198307831117_preal @ M @ one_on9143529541772854033_preal ) @ one_on9143529541772854033_preal ) @ bot_bo5832510953819287539_preal ) ) ) ) ) ).
% real_of_preal_def
thf(fact_282_real__minus,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( uminus7714077491378687647d_real @ ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X3 @ Y3 ) @ bot_bo5832510953819287539_preal ) ) ) )
= ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ Y3 @ X3 ) @ bot_bo5832510953819287539_preal ) ) ) ) ).
% real_minus
thf(fact_283_is__singletonI,axiom,
! [X3: produc5063651213027957927_preal] : ( is_sin7730553767288049563_preal @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) ) ).
% is_singletonI
thf(fact_284_is__singletonI,axiom,
! [X3: int] : ( is_singleton_int @ ( insert_int @ X3 @ bot_bot_set_int ) ) ).
% is_singletonI
thf(fact_285_add_Oinverse__inverse,axiom,
! [A: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( uminus7714077491378687647d_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_286_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_287_neg__equal__iff__equal,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= ( uminus7714077491378687647d_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_288_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_289_subsetI,axiom,
! [A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ! [X: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ X @ A2 )
=> ( member6785577539073195184_preal @ X @ B2 ) )
=> ( ord_le6746582767776264711_preal @ A2 @ B2 ) ) ).
% subsetI
thf(fact_290_subsetI,axiom,
! [A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ! [X: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ X @ A2 )
=> ( member7273733640249002768_preal @ X @ B2 ) )
=> ( ord_le6462023905233536103_preal @ A2 @ B2 ) ) ).
% subsetI
thf(fact_291_subsetI,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ! [X: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ X @ A2 )
=> ( member7052443626638417872_preal @ X @ B2 ) )
=> ( ord_le845437053512250663_preal @ A2 @ B2 ) ) ).
% subsetI
thf(fact_292_subsetI,axiom,
! [A2: set_int,B2: set_int] :
( ! [X: int] :
( ( member_int @ X @ A2 )
=> ( member_int @ X @ B2 ) )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_293_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_294_add_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.right_neutral
thf(fact_295_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_296_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_297_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_298_add__cancel__left__left,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= A )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_left
thf(fact_299_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_300_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_301_add__cancel__left__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= A )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_right
thf(fact_302_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_303_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_304_add__cancel__right__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( plus_p4060926892116697567d_real @ B @ A ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_left
thf(fact_305_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_306_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_307_add__cancel__right__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( plus_p4060926892116697567d_real @ A @ B ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_right
thf(fact_308_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_309_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y3: nat] :
( ( ( plus_plus_nat @ X3 @ Y3 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_310_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y3 ) )
= ( ( X3 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_311_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_312_add__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add_0
thf(fact_313_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_314_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_315_add_Oinverse__neutral,axiom,
( ( uminus7714077491378687647d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% add.inverse_neutral
thf(fact_316_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_317_neg__0__equal__iff__equal,axiom,
! [A: dedekind_real] :
( ( zero_z580800474297136991d_real
= ( uminus7714077491378687647d_real @ A ) )
= ( zero_z580800474297136991d_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_318_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_319_neg__equal__0__iff__equal,axiom,
! [A: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_320_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_321_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_322_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_323_add__minus__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_324_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_325_minus__add__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ ( plus_p4060926892116697567d_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_326_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_327_minus__add__distrib,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_328_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_329_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_330_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_331_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_332_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_333_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_334_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_335_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_336_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_337_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_338_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_339_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_340_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_341_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_342_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_343_ab__left__minus,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= zero_z580800474297136991d_real ) ).
% ab_left_minus
thf(fact_344_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_345_add_Oright__inverse,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( uminus7714077491378687647d_real @ A ) )
= zero_z580800474297136991d_real ) ).
% add.right_inverse
thf(fact_346_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_347_Abs__Real__inverse,axiom,
! [Y3: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ Y3 @ dedekind_Real )
=> ( ( dedekind_Rep_Real @ ( dedekind_Abs_Real @ Y3 ) )
= Y3 ) ) ).
% Abs_Real_inverse
thf(fact_348_equation__minus__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ B ) )
= ( B
= ( uminus7714077491378687647d_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_349_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_350_minus__equation__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= B )
= ( ( uminus7714077491378687647d_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_351_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_352_in__mono,axiom,
! [A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal,X3: set_Pr3959665804823141767_preal] :
( ( ord_le6746582767776264711_preal @ A2 @ B2 )
=> ( ( member6785577539073195184_preal @ X3 @ A2 )
=> ( member6785577539073195184_preal @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_353_in__mono,axiom,
! [A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal,X3: produc7488332866442972391_preal] :
( ( ord_le6462023905233536103_preal @ A2 @ B2 )
=> ( ( member7273733640249002768_preal @ X3 @ A2 )
=> ( member7273733640249002768_preal @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_354_in__mono,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal,X3: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ A2 @ B2 )
=> ( ( member7052443626638417872_preal @ X3 @ A2 )
=> ( member7052443626638417872_preal @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_355_in__mono,axiom,
! [A2: set_int,B2: set_int,X3: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_356_subsetD,axiom,
! [A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal,C: set_Pr3959665804823141767_preal] :
( ( ord_le6746582767776264711_preal @ A2 @ B2 )
=> ( ( member6785577539073195184_preal @ C @ A2 )
=> ( member6785577539073195184_preal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_357_subsetD,axiom,
! [A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal,C: produc7488332866442972391_preal] :
( ( ord_le6462023905233536103_preal @ A2 @ B2 )
=> ( ( member7273733640249002768_preal @ C @ A2 )
=> ( member7273733640249002768_preal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_358_subsetD,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal,C: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ A2 @ B2 )
=> ( ( member7052443626638417872_preal @ C @ A2 )
=> ( member7052443626638417872_preal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_359_subsetD,axiom,
! [A2: set_int,B2: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% subsetD
thf(fact_360_subset__eq,axiom,
( ord_le6746582767776264711_preal
= ( ^ [A6: set_se7472626482866788967_preal,B6: set_se7472626482866788967_preal] :
! [X4: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ X4 @ A6 )
=> ( member6785577539073195184_preal @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_361_subset__eq,axiom,
( ord_le6462023905233536103_preal
= ( ^ [A6: set_Pr6738245714641013959_preal,B6: set_Pr6738245714641013959_preal] :
! [X4: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ X4 @ A6 )
=> ( member7273733640249002768_preal @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_362_subset__eq,axiom,
( ord_le845437053512250663_preal
= ( ^ [A6: set_Pr3959665804823141767_preal,B6: set_Pr3959665804823141767_preal] :
! [X4: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ X4 @ A6 )
=> ( member7052443626638417872_preal @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_363_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( member_int @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_364_subset__iff,axiom,
( ord_le6746582767776264711_preal
= ( ^ [A6: set_se7472626482866788967_preal,B6: set_se7472626482866788967_preal] :
! [T2: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ T2 @ A6 )
=> ( member6785577539073195184_preal @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_365_subset__iff,axiom,
( ord_le6462023905233536103_preal
= ( ^ [A6: set_Pr6738245714641013959_preal,B6: set_Pr6738245714641013959_preal] :
! [T2: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ T2 @ A6 )
=> ( member7273733640249002768_preal @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_366_subset__iff,axiom,
( ord_le845437053512250663_preal
= ( ^ [A6: set_Pr3959665804823141767_preal,B6: set_Pr3959665804823141767_preal] :
! [T2: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ T2 @ A6 )
=> ( member7052443626638417872_preal @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_367_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_368_one__reorient,axiom,
! [X3: dedekind_preal] :
( ( one_on9143529541772854033_preal = X3 )
= ( X3 = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_369_one__reorient,axiom,
! [X3: dedekind_real] :
( ( one_on6069100329679821595d_real = X3 )
= ( X3 = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_370_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_371_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_372_zero__reorient,axiom,
! [X3: dedekind_real] :
( ( zero_z580800474297136991d_real = X3 )
= ( X3 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_373_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_374_Rep__Real__inject,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( ( dedekind_Rep_Real @ X3 )
= ( dedekind_Rep_Real @ Y3 ) )
= ( X3 = Y3 ) ) ).
% Rep_Real_inject
thf(fact_375_neg__eq__iff__add__eq__0,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= B )
= ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_376_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_377_eq__neg__iff__add__eq__0,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ B ) )
= ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_378_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_379_add_Oinverse__unique,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_380_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_381_ab__group__add__class_Oab__left__minus,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= zero_z580800474297136991d_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_382_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_383_add__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( B
= ( uminus7714077491378687647d_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_384_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_385_bot__set__def,axiom,
( bot_bo5832510953819287539_preal
= ( collec5689422505606869010_preal @ bot_bo3400249293420203690real_o ) ) ).
% bot_set_def
thf(fact_386_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_387_bot__empty__eq,axiom,
( bot_bo4782584465478625738real_o
= ( ^ [X4: set_Pr3959665804823141767_preal] : ( member6785577539073195184_preal @ X4 @ bot_bo6482683055655779539_preal ) ) ) ).
% bot_empty_eq
thf(fact_388_bot__empty__eq,axiom,
( bot_bo1391430648228228970real_o
= ( ^ [X4: produc7488332866442972391_preal] : ( member7273733640249002768_preal @ X4 @ bot_bo3249957383623713587_preal ) ) ) ).
% bot_empty_eq
thf(fact_389_bot__empty__eq,axiom,
( bot_bo3400249293420203690real_o
= ( ^ [X4: produc5063651213027957927_preal] : ( member7052443626638417872_preal @ X4 @ bot_bo5832510953819287539_preal ) ) ) ).
% bot_empty_eq
thf(fact_390_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X4: int] : ( member_int @ X4 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_391_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_392_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_393_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_394_group__cancel_Oneg1,axiom,
! [A2: dedekind_real,K: dedekind_real,A: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( uminus7714077491378687647d_real @ A2 )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ K ) @ ( uminus7714077491378687647d_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_395_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_396_add_Oinverse__distrib__swap,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_397_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_398_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_399_comm__monoid__add__class_Oadd__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_400_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_401_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_402_add_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.comm_neutral
thf(fact_403_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_404_add_Ogroup__left__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_405_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_406_Rep__Real__inverse,axiom,
! [X3: dedekind_real] :
( ( dedekind_Abs_Real @ ( dedekind_Rep_Real @ X3 ) )
= X3 ) ).
% Rep_Real_inverse
thf(fact_407_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_408_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_409_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_410_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_411_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_412_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_413_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_414_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_415_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_416_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_417_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_418_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_419_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X3 @ Y3 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_420_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
=> ( ( ( plus_plus_int @ X3 @ Y3 )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_421_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y3 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_422_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y3 )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y3 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_423_is__singleton__the__elem,axiom,
( is_sin7730553767288049563_preal
= ( ^ [A6: set_Pr3959665804823141767_preal] :
( A6
= ( insert2800550009750661751_preal @ ( the_el8575259181799709340_preal @ A6 ) @ bot_bo5832510953819287539_preal ) ) ) ) ).
% is_singleton_the_elem
thf(fact_424_is__singleton__the__elem,axiom,
( is_singleton_int
= ( ^ [A6: set_int] :
( A6
= ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).
% is_singleton_the_elem
thf(fact_425_Rep__Real__induct,axiom,
! [Y3: set_Pr3959665804823141767_preal,P: set_Pr3959665804823141767_preal > $o] :
( ( member6785577539073195184_preal @ Y3 @ dedekind_Real )
=> ( ! [X: dedekind_real] : ( P @ ( dedekind_Rep_Real @ X ) )
=> ( P @ Y3 ) ) ) ).
% Rep_Real_induct
thf(fact_426_Rep__Real__cases,axiom,
! [Y3: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ Y3 @ dedekind_Real )
=> ~ ! [X: dedekind_real] :
( Y3
!= ( dedekind_Rep_Real @ X ) ) ) ).
% Rep_Real_cases
thf(fact_427_Rep__Real,axiom,
! [X3: dedekind_real] : ( member6785577539073195184_preal @ ( dedekind_Rep_Real @ X3 ) @ dedekind_Real ) ).
% Rep_Real
thf(fact_428_is__singletonI_H,axiom,
! [A2: set_se7472626482866788967_preal] :
( ( A2 != bot_bo6482683055655779539_preal )
=> ( ! [X: set_Pr3959665804823141767_preal,Y: set_Pr3959665804823141767_preal] :
( ( member6785577539073195184_preal @ X @ A2 )
=> ( ( member6785577539073195184_preal @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_sin5917698295481368699_preal @ A2 ) ) ) ).
% is_singletonI'
thf(fact_429_is__singletonI_H,axiom,
! [A2: set_Pr6738245714641013959_preal] :
( ( A2 != bot_bo3249957383623713587_preal )
=> ( ! [X: produc7488332866442972391_preal,Y: produc7488332866442972391_preal] :
( ( member7273733640249002768_preal @ X @ A2 )
=> ( ( member7273733640249002768_preal @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_sin4600180186396777435_preal @ A2 ) ) ) ).
% is_singletonI'
thf(fact_430_is__singletonI_H,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( A2 != bot_bo5832510953819287539_preal )
=> ( ! [X: produc5063651213027957927_preal,Y: produc5063651213027957927_preal] :
( ( member7052443626638417872_preal @ X @ A2 )
=> ( ( member7052443626638417872_preal @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_sin7730553767288049563_preal @ A2 ) ) ) ).
% is_singletonI'
thf(fact_431_is__singletonI_H,axiom,
! [A2: set_int] :
( ( A2 != bot_bot_set_int )
=> ( ! [X: int,Y: int] :
( ( member_int @ X @ A2 )
=> ( ( member_int @ Y @ A2 )
=> ( X = Y ) ) )
=> ( is_singleton_int @ A2 ) ) ) ).
% is_singletonI'
thf(fact_432_nle__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A @ B ) )
= ( ( ord_le5604041210740703414_preal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_433_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_434_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_435_le__cases3,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ~ ( ord_le5604041210740703414_preal @ Y3 @ Z ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ~ ( ord_le5604041210740703414_preal @ X3 @ Z ) )
=> ( ( ( ord_le5604041210740703414_preal @ X3 @ Z )
=> ~ ( ord_le5604041210740703414_preal @ Z @ Y3 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z @ Y3 )
=> ~ ( ord_le5604041210740703414_preal @ Y3 @ X3 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y3 @ Z )
=> ~ ( ord_le5604041210740703414_preal @ Z @ X3 ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z @ X3 )
=> ~ ( ord_le5604041210740703414_preal @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_436_le__cases3,axiom,
! [X3: num,Y3: num,Z: num] :
( ( ( ord_less_eq_num @ X3 @ Y3 )
=> ~ ( ord_less_eq_num @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_num @ Y3 @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Z ) )
=> ( ( ( ord_less_eq_num @ X3 @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_num @ Z @ Y3 )
=> ~ ( ord_less_eq_num @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_num @ Y3 @ Z )
=> ~ ( ord_less_eq_num @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_num @ Z @ X3 )
=> ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_437_le__cases3,axiom,
! [X3: int,Y3: int,Z: int] :
( ( ( ord_less_eq_int @ X3 @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_int @ Y3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_int @ Z @ Y3 )
=> ~ ( ord_less_eq_int @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y3 @ Z )
=> ~ ( ord_less_eq_int @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_438_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z3: dedekind_preal] : ( Y5 = Z3 ) )
= ( ^ [X4: dedekind_preal,Y4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X4 @ Y4 )
& ( ord_le5604041210740703414_preal @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_439_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
& ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_440_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_441_ord__eq__le__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A = B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_442_ord__eq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_443_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_444_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_445_ord__le__eq__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_446_ord__le__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_447_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_448_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_449_order__antisym,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_450_order__antisym,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ord_less_eq_num @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_451_order__antisym,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_452_order_Otrans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% order.trans
thf(fact_453_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_454_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_455_order__trans,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ( ord_le5604041210740703414_preal @ Y3 @ Z )
=> ( ord_le5604041210740703414_preal @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_456_order__trans,axiom,
! [X3: num,Y3: num,Z: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ( ord_less_eq_num @ Y3 @ Z )
=> ( ord_less_eq_num @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_457_order__trans,axiom,
! [X3: int,Y3: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ Y3 @ Z )
=> ( ord_less_eq_int @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_458_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A7: dedekind_preal,B7: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: dedekind_preal,B7: dedekind_preal] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_459_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A7: num,B7: num] :
( ( ord_less_eq_num @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: num,B7: num] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_460_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A7: int,B7: int] :
( ( ord_less_eq_int @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: int,B7: int] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_461_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z3: dedekind_preal] : ( Y5 = Z3 ) )
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B3 @ A4 )
& ( ord_le5604041210740703414_preal @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_462_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_463_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_464_dual__order_Oantisym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_465_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_466_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_467_dual__order_Otrans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5604041210740703414_preal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_468_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_469_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_470_antisym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_471_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_472_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_473_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: dedekind_preal,Z3: dedekind_preal] : ( Y5 = Z3 ) )
= ( ^ [A4: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B3 )
& ( ord_le5604041210740703414_preal @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_474_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: num,Z3: num] : ( Y5 = Z3 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_475_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_476_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_477_order__subst1,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_478_order__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_479_order__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_480_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_481_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_482_order__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_483_order__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_484_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_485_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_486_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_487_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_488_order__subst2,axiom,
! [A: num,B: num,F: num > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_489_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_490_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_491_order__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_492_order__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_493_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_494_order__eq__refl,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( X3 = Y3 )
=> ( ord_le5604041210740703414_preal @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_495_order__eq__refl,axiom,
! [X3: num,Y3: num] :
( ( X3 = Y3 )
=> ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_496_order__eq__refl,axiom,
! [X3: int,Y3: int] :
( ( X3 = Y3 )
=> ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_497_linorder__linear,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
| ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_498_linorder__linear,axiom,
! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
| ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_499_linorder__linear,axiom,
! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
| ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_500_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_501_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_502_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_503_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_504_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_505_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_506_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_507_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_508_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_509_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_510_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_511_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_512_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_513_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_514_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_515_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_516_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_517_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_518_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_519_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_linorder__le__cases,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X3 @ Y3 )
=> ( ord_le5604041210740703414_preal @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_521_linorder__le__cases,axiom,
! [X3: num,Y3: num] :
( ~ ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_522_linorder__le__cases,axiom,
! [X3: int,Y3: int] :
( ~ ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_523_order__antisym__conv,axiom,
! [Y3: dedekind_preal,X3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y3 @ X3 )
=> ( ( ord_le5604041210740703414_preal @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_524_order__antisym__conv,axiom,
! [Y3: num,X3: num] :
( ( ord_less_eq_num @ Y3 @ X3 )
=> ( ( ord_less_eq_num @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_525_order__antisym__conv,axiom,
! [Y3: int,X3: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_526_old_Oprod_Oexhaust,axiom,
! [Y3: produc5063651213027957927_preal] :
~ ! [A7: dedekind_preal,B7: dedekind_preal] :
( Y3
!= ( produc5992578386459872151_preal @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_527_old_Oprod_Oexhaust,axiom,
! [Y3: produc7488332866442972391_preal] :
~ ! [A7: produc5063651213027957927_preal,B7: produc5063651213027957927_preal] :
( Y3
!= ( produc2372827984048905047_preal @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_528_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_int_int] :
~ ! [A7: int,B7: int] :
( Y3
!= ( product_Pair_int_int @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_529_surj__pair,axiom,
! [P2: produc5063651213027957927_preal] :
? [X: dedekind_preal,Y: dedekind_preal] :
( P2
= ( produc5992578386459872151_preal @ X @ Y ) ) ).
% surj_pair
thf(fact_530_surj__pair,axiom,
! [P2: produc7488332866442972391_preal] :
? [X: produc5063651213027957927_preal,Y: produc5063651213027957927_preal] :
( P2
= ( produc2372827984048905047_preal @ X @ Y ) ) ).
% surj_pair
thf(fact_531_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X: int,Y: int] :
( P2
= ( product_Pair_int_int @ X @ Y ) ) ).
% surj_pair
thf(fact_532_prod__cases,axiom,
! [P: produc5063651213027957927_preal > $o,P2: produc5063651213027957927_preal] :
( ! [A7: dedekind_preal,B7: dedekind_preal] : ( P @ ( produc5992578386459872151_preal @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_533_prod__cases,axiom,
! [P: produc7488332866442972391_preal > $o,P2: produc7488332866442972391_preal] :
( ! [A7: produc5063651213027957927_preal,B7: produc5063651213027957927_preal] : ( P @ ( produc2372827984048905047_preal @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_534_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A7: int,B7: int] : ( P @ ( product_Pair_int_int @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_535_Pair__inject,axiom,
! [A: dedekind_preal,B: dedekind_preal,A5: dedekind_preal,B5: dedekind_preal] :
( ( ( produc5992578386459872151_preal @ A @ B )
= ( produc5992578386459872151_preal @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_536_Pair__inject,axiom,
! [A: produc5063651213027957927_preal,B: produc5063651213027957927_preal,A5: produc5063651213027957927_preal,B5: produc5063651213027957927_preal] :
( ( ( produc2372827984048905047_preal @ A @ B )
= ( produc2372827984048905047_preal @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_537_Pair__inject,axiom,
! [A: int,B: int,A5: int,B5: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_538_prod__cases3,axiom,
! [Y3: produc7488332866442972391_preal] :
~ ! [A7: produc5063651213027957927_preal,B7: dedekind_preal,C5: dedekind_preal] :
( Y3
!= ( produc2372827984048905047_preal @ A7 @ ( produc5992578386459872151_preal @ B7 @ C5 ) ) ) ).
% prod_cases3
thf(fact_539_prod__induct3,axiom,
! [P: produc7488332866442972391_preal > $o,X3: produc7488332866442972391_preal] :
( ! [A7: produc5063651213027957927_preal,B7: dedekind_preal,C5: dedekind_preal] : ( P @ ( produc2372827984048905047_preal @ A7 @ ( produc5992578386459872151_preal @ B7 @ C5 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_540_real__zero__def,axiom,
( zero_z580800474297136991d_real
= ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ one_on9143529541772854033_preal @ one_on9143529541772854033_preal ) @ bot_bo5832510953819287539_preal ) ) ) ) ).
% real_zero_def
thf(fact_541_is__singletonE,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( is_sin7730553767288049563_preal @ A2 )
=> ~ ! [X: produc5063651213027957927_preal] :
( A2
!= ( insert2800550009750661751_preal @ X @ bot_bo5832510953819287539_preal ) ) ) ).
% is_singletonE
thf(fact_542_is__singletonE,axiom,
! [A2: set_int] :
( ( is_singleton_int @ A2 )
=> ~ ! [X: int] :
( A2
!= ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% is_singletonE
thf(fact_543_is__singleton__def,axiom,
( is_sin7730553767288049563_preal
= ( ^ [A6: set_Pr3959665804823141767_preal] :
? [X4: produc5063651213027957927_preal] :
( A6
= ( insert2800550009750661751_preal @ X4 @ bot_bo5832510953819287539_preal ) ) ) ) ).
% is_singleton_def
thf(fact_544_is__singleton__def,axiom,
( is_singleton_int
= ( ^ [A6: set_int] :
? [X4: int] :
( A6
= ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).
% is_singleton_def
thf(fact_545_bot_Oextremum,axiom,
! [A: set_Pr3959665804823141767_preal] : ( ord_le845437053512250663_preal @ bot_bo5832510953819287539_preal @ A ) ).
% bot.extremum
thf(fact_546_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_547_bot_Oextremum__unique,axiom,
! [A: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ A @ bot_bo5832510953819287539_preal )
= ( A = bot_bo5832510953819287539_preal ) ) ).
% bot.extremum_unique
thf(fact_548_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_549_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ A @ bot_bo5832510953819287539_preal )
=> ( A = bot_bo5832510953819287539_preal ) ) ).
% bot.extremum_uniqueI
thf(fact_550_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_551_add__neg__numeral__special_I8_J,axiom,
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ one_on6069100329679821595d_real )
= zero_z580800474297136991d_real ) ).
% add_neg_numeral_special(8)
thf(fact_552_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_553_add__neg__numeral__special_I7_J,axiom,
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= zero_z580800474297136991d_real ) ).
% add_neg_numeral_special(7)
thf(fact_554_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_555_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_556_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_557_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_558_real__one__def,axiom,
( one_on6069100329679821595d_real
= ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ ( plus_p3173629198307831117_preal @ one_on9143529541772854033_preal @ one_on9143529541772854033_preal ) @ one_on9143529541772854033_preal ) @ bot_bo5832510953819287539_preal ) ) ) ) ).
% real_one_def
thf(fact_559_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_560_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_561_subset__Compl__singleton,axiom,
! [A2: set_se7472626482866788967_preal,B: set_Pr3959665804823141767_preal] :
( ( ord_le6746582767776264711_preal @ A2 @ ( uminus6658923725895887966_preal @ ( insert5864466517235936087_preal @ B @ bot_bo6482683055655779539_preal ) ) )
= ( ~ ( member6785577539073195184_preal @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_562_subset__Compl__singleton,axiom,
! [A2: set_Pr6738245714641013959_preal,B: produc7488332866442972391_preal] :
( ( ord_le6462023905233536103_preal @ A2 @ ( uminus2031495666071150526_preal @ ( insert8416738607890598583_preal @ B @ bot_bo3249957383623713587_preal ) ) )
= ( ~ ( member7273733640249002768_preal @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_563_subset__Compl__singleton,axiom,
! [A2: set_Pr3959665804823141767_preal,B: produc5063651213027957927_preal] :
( ( ord_le845437053512250663_preal @ A2 @ ( uminus7914843409941547902_preal @ ( insert2800550009750661751_preal @ B @ bot_bo5832510953819287539_preal ) ) )
= ( ~ ( member7052443626638417872_preal @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_564_subset__Compl__singleton,axiom,
! [A2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_565_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_566_subset__Compl__self__eq,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ A2 @ ( uminus7914843409941547902_preal @ A2 ) )
= ( A2 = bot_bo5832510953819287539_preal ) ) ).
% subset_Compl_self_eq
thf(fact_567_subset__Compl__self__eq,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_568_is__num__normalize_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_569_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_570_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_571_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_572_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_573_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_574_is__num__normalize_I8_J,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_575_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_576_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_577_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_578_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_579_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_580_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_581_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_582_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_583_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7286324685880884523d_real @ zero_z580800474297136991d_real )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% dbl_dec_simps(2)
thf(fact_584_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_585_verit__minus__simplify_I4_J,axiom,
! [B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( uminus7714077491378687647d_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_586_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_587_Collect__empty__eq__bot,axiom,
! [P: produc5063651213027957927_preal > $o] :
( ( ( collec5689422505606869010_preal @ P )
= bot_bo5832510953819287539_preal )
= ( P = bot_bo3400249293420203690real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_588_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_589_ComplI,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ~ ( member6785577539073195184_preal @ C @ A2 )
=> ( member6785577539073195184_preal @ C @ ( uminus6658923725895887966_preal @ A2 ) ) ) ).
% ComplI
thf(fact_590_ComplI,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ~ ( member7273733640249002768_preal @ C @ A2 )
=> ( member7273733640249002768_preal @ C @ ( uminus2031495666071150526_preal @ A2 ) ) ) ).
% ComplI
thf(fact_591_ComplI,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ~ ( member7052443626638417872_preal @ C @ A2 )
=> ( member7052443626638417872_preal @ C @ ( uminus7914843409941547902_preal @ A2 ) ) ) ).
% ComplI
thf(fact_592_ComplI,axiom,
! [C: int,A2: set_int] :
( ~ ( member_int @ C @ A2 )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% ComplI
thf(fact_593_Compl__iff,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ ( uminus6658923725895887966_preal @ A2 ) )
= ( ~ ( member6785577539073195184_preal @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_594_Compl__iff,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ ( uminus2031495666071150526_preal @ A2 ) )
= ( ~ ( member7273733640249002768_preal @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_595_Compl__iff,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ ( uminus7914843409941547902_preal @ A2 ) )
= ( ~ ( member7052443626638417872_preal @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_596_Compl__iff,axiom,
! [C: int,A2: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
= ( ~ ( member_int @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_597_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7286324685880884523d_real @ one_on6069100329679821595d_real )
= one_on6069100329679821595d_real ) ).
% dbl_dec_simps(3)
thf(fact_598_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_599_ComplD,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ ( uminus6658923725895887966_preal @ A2 ) )
=> ~ ( member6785577539073195184_preal @ C @ A2 ) ) ).
% ComplD
thf(fact_600_ComplD,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ ( uminus2031495666071150526_preal @ A2 ) )
=> ~ ( member7273733640249002768_preal @ C @ A2 ) ) ).
% ComplD
thf(fact_601_ComplD,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ ( uminus7914843409941547902_preal @ A2 ) )
=> ~ ( member7052443626638417872_preal @ C @ A2 ) ) ).
% ComplD
thf(fact_602_ComplD,axiom,
! [C: int,A2: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
=> ~ ( member_int @ C @ A2 ) ) ).
% ComplD
thf(fact_603_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_604_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_605_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_606_verit__la__disequality,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A = B )
| ~ ( ord_le5604041210740703414_preal @ A @ B )
| ~ ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_607_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_608_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_609_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_610_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_611_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_612_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_613_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_614_verit__sum__simplify,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% verit_sum_simplify
thf(fact_615_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_616_less__by__empty,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( A2 = bot_bo5832510953819287539_preal )
=> ( ord_le845437053512250663_preal @ A2 @ B2 ) ) ).
% less_by_empty
thf(fact_617_insert__subsetI,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,X5: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ( ord_le6746582767776264711_preal @ X5 @ A2 )
=> ( ord_le6746582767776264711_preal @ ( insert5864466517235936087_preal @ X3 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_618_insert__subsetI,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,X5: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ( ord_le6462023905233536103_preal @ X5 @ A2 )
=> ( ord_le6462023905233536103_preal @ ( insert8416738607890598583_preal @ X3 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_619_insert__subsetI,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,X5: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ( ord_le845437053512250663_preal @ X5 @ A2 )
=> ( ord_le845437053512250663_preal @ ( insert2800550009750661751_preal @ X3 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_620_insert__subsetI,axiom,
! [X3: int,A2: set_int,X5: set_int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_set_int @ X5 @ A2 )
=> ( ord_less_eq_set_int @ ( insert_int @ X3 @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_621_subset__emptyI,axiom,
! [A2: set_se7472626482866788967_preal] :
( ! [X: set_Pr3959665804823141767_preal] :
~ ( member6785577539073195184_preal @ X @ A2 )
=> ( ord_le6746582767776264711_preal @ A2 @ bot_bo6482683055655779539_preal ) ) ).
% subset_emptyI
thf(fact_622_subset__emptyI,axiom,
! [A2: set_Pr6738245714641013959_preal] :
( ! [X: produc7488332866442972391_preal] :
~ ( member7273733640249002768_preal @ X @ A2 )
=> ( ord_le6462023905233536103_preal @ A2 @ bot_bo3249957383623713587_preal ) ) ).
% subset_emptyI
thf(fact_623_subset__emptyI,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ! [X: produc5063651213027957927_preal] :
~ ( member7052443626638417872_preal @ X @ A2 )
=> ( ord_le845437053512250663_preal @ A2 @ bot_bo5832510953819287539_preal ) ) ).
% subset_emptyI
thf(fact_624_subset__emptyI,axiom,
! [A2: set_int] :
( ! [X: int] :
~ ( member_int @ X @ A2 )
=> ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_625_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_626_add__0__iff,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B
= ( plus_p4060926892116697567d_real @ B @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% add_0_iff
thf(fact_627_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_628_dbl__inc__simps_I4_J,axiom,
( ( neg_nu2849763295382964967d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% dbl_inc_simps(4)
thf(fact_629_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_630_dbl__inc__simps_I2_J,axiom,
( ( neg_nu2849763295382964967d_real @ zero_z580800474297136991d_real )
= one_on6069100329679821595d_real ) ).
% dbl_inc_simps(2)
thf(fact_631_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_632_refl__on__singleton,axiom,
! [X3: dedekind_preal] : ( refl_o1982607829261058197_preal @ ( insert4264952892349296358_preal @ X3 @ bot_bo4848840305443107682_preal ) @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X3 @ X3 ) @ bot_bo5832510953819287539_preal ) ) ).
% refl_on_singleton
thf(fact_633_refl__on__singleton,axiom,
! [X3: produc5063651213027957927_preal] : ( refl_o513130203077362342_preal @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) @ ( insert8416738607890598583_preal @ ( produc2372827984048905047_preal @ X3 @ X3 ) @ bot_bo3249957383623713587_preal ) ) ).
% refl_on_singleton
thf(fact_634_refl__on__singleton,axiom,
! [X3: int] : ( refl_on_int @ ( insert_int @ X3 @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ bot_bo1796632182523588997nt_int ) ) ).
% refl_on_singleton
thf(fact_635_refl__onD2,axiom,
! [A2: set_se7472626482866788967_preal,R: set_Pr3180418311277671623_preal,X3: set_Pr3959665804823141767_preal,Y3: set_Pr3959665804823141767_preal] :
( ( refl_o8885767970008106886_preal @ A2 @ R )
=> ( ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ X3 @ Y3 ) @ R )
=> ( member6785577539073195184_preal @ Y3 @ A2 ) ) ) ).
% refl_onD2
thf(fact_636_refl__onD2,axiom,
! [A2: set_Pr6738245714641013959_preal,R: set_Pr5164066273998628039_preal,X3: produc7488332866442972391_preal,Y3: produc7488332866442972391_preal] :
( ( refl_o4543496873498767334_preal @ A2 @ R )
=> ( ( member7581708263213832976_preal @ ( produc4416245248500966231_preal @ X3 @ Y3 ) @ R )
=> ( member7273733640249002768_preal @ Y3 @ A2 ) ) ) ).
% refl_onD2
thf(fact_637_refl__onD2,axiom,
! [A2: set_Dedekind_preal,R: set_Pr3959665804823141767_preal,X3: dedekind_preal,Y3: dedekind_preal] :
( ( refl_o1982607829261058197_preal @ A2 @ R )
=> ( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X3 @ Y3 ) @ R )
=> ( member6871284927547481791_preal @ Y3 @ A2 ) ) ) ).
% refl_onD2
thf(fact_638_refl__onD2,axiom,
! [A2: set_Pr3959665804823141767_preal,R: set_Pr6738245714641013959_preal,X3: produc5063651213027957927_preal,Y3: produc5063651213027957927_preal] :
( ( refl_o513130203077362342_preal @ A2 @ R )
=> ( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X3 @ Y3 ) @ R )
=> ( member7052443626638417872_preal @ Y3 @ A2 ) ) ) ).
% refl_onD2
thf(fact_639_refl__onD2,axiom,
! [A2: set_int,R: set_Pr958786334691620121nt_int,X3: int,Y3: int] :
( ( refl_on_int @ A2 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
=> ( member_int @ Y3 @ A2 ) ) ) ).
% refl_onD2
thf(fact_640_refl__onD1,axiom,
! [A2: set_se7472626482866788967_preal,R: set_Pr3180418311277671623_preal,X3: set_Pr3959665804823141767_preal,Y3: set_Pr3959665804823141767_preal] :
( ( refl_o8885767970008106886_preal @ A2 @ R )
=> ( ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ X3 @ Y3 ) @ R )
=> ( member6785577539073195184_preal @ X3 @ A2 ) ) ) ).
% refl_onD1
thf(fact_641_refl__onD1,axiom,
! [A2: set_Pr6738245714641013959_preal,R: set_Pr5164066273998628039_preal,X3: produc7488332866442972391_preal,Y3: produc7488332866442972391_preal] :
( ( refl_o4543496873498767334_preal @ A2 @ R )
=> ( ( member7581708263213832976_preal @ ( produc4416245248500966231_preal @ X3 @ Y3 ) @ R )
=> ( member7273733640249002768_preal @ X3 @ A2 ) ) ) ).
% refl_onD1
thf(fact_642_refl__onD1,axiom,
! [A2: set_Dedekind_preal,R: set_Pr3959665804823141767_preal,X3: dedekind_preal,Y3: dedekind_preal] :
( ( refl_o1982607829261058197_preal @ A2 @ R )
=> ( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ X3 @ Y3 ) @ R )
=> ( member6871284927547481791_preal @ X3 @ A2 ) ) ) ).
% refl_onD1
thf(fact_643_refl__onD1,axiom,
! [A2: set_Pr3959665804823141767_preal,R: set_Pr6738245714641013959_preal,X3: produc5063651213027957927_preal,Y3: produc5063651213027957927_preal] :
( ( refl_o513130203077362342_preal @ A2 @ R )
=> ( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ X3 @ Y3 ) @ R )
=> ( member7052443626638417872_preal @ X3 @ A2 ) ) ) ).
% refl_onD1
thf(fact_644_refl__onD1,axiom,
! [A2: set_int,R: set_Pr958786334691620121nt_int,X3: int,Y3: int] :
( ( refl_on_int @ A2 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
=> ( member_int @ X3 @ A2 ) ) ) ).
% refl_onD1
thf(fact_645_refl__onD,axiom,
! [A2: set_se7472626482866788967_preal,R: set_Pr3180418311277671623_preal,A: set_Pr3959665804823141767_preal] :
( ( refl_o8885767970008106886_preal @ A2 @ R )
=> ( ( member6785577539073195184_preal @ A @ A2 )
=> ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_646_refl__onD,axiom,
! [A2: set_Pr6738245714641013959_preal,R: set_Pr5164066273998628039_preal,A: produc7488332866442972391_preal] :
( ( refl_o4543496873498767334_preal @ A2 @ R )
=> ( ( member7273733640249002768_preal @ A @ A2 )
=> ( member7581708263213832976_preal @ ( produc4416245248500966231_preal @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_647_refl__onD,axiom,
! [A2: set_Dedekind_preal,R: set_Pr3959665804823141767_preal,A: dedekind_preal] :
( ( refl_o1982607829261058197_preal @ A2 @ R )
=> ( ( member6871284927547481791_preal @ A @ A2 )
=> ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_648_refl__onD,axiom,
! [A2: set_Pr3959665804823141767_preal,R: set_Pr6738245714641013959_preal,A: produc5063651213027957927_preal] :
( ( refl_o513130203077362342_preal @ A2 @ R )
=> ( ( member7052443626638417872_preal @ A @ A2 )
=> ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_649_refl__onD,axiom,
! [A2: set_int,R: set_Pr958786334691620121nt_int,A: int] :
( ( refl_on_int @ A2 @ R )
=> ( ( member_int @ A @ A2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_650_refl__on__empty,axiom,
refl_o1982607829261058197_preal @ bot_bo4848840305443107682_preal @ bot_bo5832510953819287539_preal ).
% refl_on_empty
thf(fact_651_refl__on__empty,axiom,
refl_o513130203077362342_preal @ bot_bo5832510953819287539_preal @ bot_bo3249957383623713587_preal ).
% refl_on_empty
thf(fact_652_refl__on__empty,axiom,
refl_on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% refl_on_empty
thf(fact_653_ssubst__Pair__rhs,axiom,
! [R: dedekind_preal,S: dedekind_preal,R2: set_Pr3959665804823141767_preal,S2: dedekind_preal] :
( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_654_ssubst__Pair__rhs,axiom,
! [R: produc5063651213027957927_preal,S: produc5063651213027957927_preal,R2: set_Pr6738245714641013959_preal,S2: produc5063651213027957927_preal] :
( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_655_ssubst__Pair__rhs,axiom,
! [R: int,S: int,R2: set_Pr958786334691620121nt_int,S2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_656_dbl__inc__def,axiom,
( neg_nu2849763295382964967d_real
= ( ^ [X4: dedekind_real] : ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ X4 @ X4 ) @ one_on6069100329679821595d_real ) ) ) ).
% dbl_inc_def
thf(fact_657_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_658_refl__on__domain,axiom,
! [A2: set_se7472626482866788967_preal,R: set_Pr3180418311277671623_preal,A: set_Pr3959665804823141767_preal,B: set_Pr3959665804823141767_preal] :
( ( refl_o8885767970008106886_preal @ A2 @ R )
=> ( ( member4141329389715116816_preal @ ( produc7167390414934094679_preal @ A @ B ) @ R )
=> ( ( member6785577539073195184_preal @ A @ A2 )
& ( member6785577539073195184_preal @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_659_refl__on__domain,axiom,
! [A2: set_Pr6738245714641013959_preal,R: set_Pr5164066273998628039_preal,A: produc7488332866442972391_preal,B: produc7488332866442972391_preal] :
( ( refl_o4543496873498767334_preal @ A2 @ R )
=> ( ( member7581708263213832976_preal @ ( produc4416245248500966231_preal @ A @ B ) @ R )
=> ( ( member7273733640249002768_preal @ A @ A2 )
& ( member7273733640249002768_preal @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_660_refl__on__domain,axiom,
! [A2: set_Dedekind_preal,R: set_Pr3959665804823141767_preal,A: dedekind_preal,B: dedekind_preal] :
( ( refl_o1982607829261058197_preal @ A2 @ R )
=> ( ( member7052443626638417872_preal @ ( produc5992578386459872151_preal @ A @ B ) @ R )
=> ( ( member6871284927547481791_preal @ A @ A2 )
& ( member6871284927547481791_preal @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_661_refl__on__domain,axiom,
! [A2: set_Pr3959665804823141767_preal,R: set_Pr6738245714641013959_preal,A: produc5063651213027957927_preal,B: produc5063651213027957927_preal] :
( ( refl_o513130203077362342_preal @ A2 @ R )
=> ( ( member7273733640249002768_preal @ ( produc2372827984048905047_preal @ A @ B ) @ R )
=> ( ( member7052443626638417872_preal @ A @ A2 )
& ( member7052443626638417872_preal @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_662_refl__on__domain,axiom,
! [A2: set_int,R: set_Pr958786334691620121nt_int,A: int,B: int] :
( ( refl_on_int @ A2 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
=> ( ( member_int @ A @ A2 )
& ( member_int @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_663_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7286324685880884523d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ K ) ) )
= ( uminus7714077491378687647d_real @ ( neg_nu2849763295382964967d_real @ ( numera6440378960330123505d_real @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_664_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_665_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu2849763295382964967d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ K ) ) )
= ( uminus7714077491378687647d_real @ ( neg_nu7286324685880884523d_real @ ( numera6440378960330123505d_real @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_666_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_667_Set_Ois__empty__def,axiom,
( is_emp4133903051233259897_preal
= ( ^ [A6: set_Pr3959665804823141767_preal] : ( A6 = bot_bo5832510953819287539_preal ) ) ) ).
% Set.is_empty_def
thf(fact_668_Set_Ois__empty__def,axiom,
( is_empty_int
= ( ^ [A6: set_int] : ( A6 = bot_bot_set_int ) ) ) ).
% Set.is_empty_def
thf(fact_669_linear__order__on__singleton,axiom,
! [X3: dedekind_preal] : ( order_8671473668144347851_preal @ ( insert4264952892349296358_preal @ X3 @ bot_bo4848840305443107682_preal ) @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ X3 @ X3 ) @ bot_bo5832510953819287539_preal ) ) ).
% linear_order_on_singleton
thf(fact_670_linear__order__on__singleton,axiom,
! [X3: produc5063651213027957927_preal] : ( order_7849749076495584732_preal @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) @ ( insert8416738607890598583_preal @ ( produc2372827984048905047_preal @ X3 @ X3 ) @ bot_bo3249957383623713587_preal ) ) ).
% linear_order_on_singleton
thf(fact_671_linear__order__on__singleton,axiom,
! [X3: int] : ( order_4471489696718655927on_int @ ( insert_int @ X3 @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ bot_bo1796632182523588997nt_int ) ) ).
% linear_order_on_singleton
thf(fact_672_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_673_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_int @ M2 )
= ( numeral_numeral_int @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_674_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_675_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_676_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_677_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_678_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_679_add__numeral__left,axiom,
! [V: num,W: num,Z: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ V ) @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ W ) @ Z ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_680_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_681_add__numeral__left,axiom,
! [V: num,W: num,Z: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_682_neg__numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M2 = N ) ) ).
% neg_numeral_eq_iff
thf(fact_683_neg__numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% neg_numeral_le_iff
thf(fact_684_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_685_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_686_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_687_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_688_numeral__neq__neg__numeral,axiom,
! [M2: num,N: num] :
( ( numeral_numeral_int @ M2 )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_689_neg__numeral__neq__numeral,axiom,
! [M2: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_690_lnear__order__on__empty,axiom,
order_8671473668144347851_preal @ bot_bo4848840305443107682_preal @ bot_bo5832510953819287539_preal ).
% lnear_order_on_empty
thf(fact_691_lnear__order__on__empty,axiom,
order_7849749076495584732_preal @ bot_bo5832510953819287539_preal @ bot_bo3249957383623713587_preal ).
% lnear_order_on_empty
thf(fact_692_lnear__order__on__empty,axiom,
order_4471489696718655927on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% lnear_order_on_empty
thf(fact_693_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_694_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_695_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_696_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_697_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_698_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_699_neg__numeral__le__numeral,axiom,
! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_700_not__numeral__le__neg__numeral,axiom,
! [M2: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_701_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_702_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ X3 ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ X3 ) @ one_on6069100329679821595d_real ) ) ).
% one_plus_numeral_commute
thf(fact_703_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_704_one__plus__numeral__commute,axiom,
! [X3: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_705_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_706_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_707_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_708_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_709_neg__numeral__le__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_710_neg__one__le__numeral,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).
% neg_one_le_numeral
thf(fact_711_neg__numeral__le__neg__one,axiom,
! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_712_not__numeral__le__neg__one,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_713_not__one__le__neg__numeral,axiom,
! [M2: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% not_one_le_neg_numeral
thf(fact_714_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y3: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ V ) ) @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ Y3 ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(167)
thf(fact_715_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% semiring_norm(167)
thf(fact_716_add__neg__numeral__special_I6_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( inc @ M2 ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_717_add__neg__numeral__special_I6_J,axiom,
! [M2: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M2 ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_718_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_719_add__neg__numeral__special_I5_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_720_add__inc,axiom,
! [X3: num,Y3: num] :
( ( plus_plus_num @ X3 @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus_num @ X3 @ Y3 ) ) ) ).
% add_inc
thf(fact_721_numeral__inc,axiom,
! [X3: num] :
( ( numera6440378960330123505d_real @ ( inc @ X3 ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ X3 ) @ one_on6069100329679821595d_real ) ) ).
% numeral_inc
thf(fact_722_numeral__inc,axiom,
! [X3: num] :
( ( numeral_numeral_nat @ ( inc @ X3 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% numeral_inc
thf(fact_723_numeral__inc,axiom,
! [X3: num] :
( ( numeral_numeral_int @ ( inc @ X3 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% numeral_inc
thf(fact_724_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8665065731627423525d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ K ) ) )
= ( uminus7714077491378687647d_real @ ( neg_nu8665065731627423525d_real @ ( numera6440378960330123505d_real @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_725_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_726_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_727_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_728_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_729_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_730_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_731_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_732_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_733_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_734_not__neg__one__le__neg__numeral__iff,axiom,
! [M2: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
= ( M2 != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_735_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( numera6440378960330123505d_real @ N ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_736_diff__numeral__special_I5_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_737_Diff__empty,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( minus_2885960186335841006_preal @ A2 @ bot_bo5832510953819287539_preal )
= A2 ) ).
% Diff_empty
thf(fact_738_Diff__empty,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% Diff_empty
thf(fact_739_empty__Diff,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( minus_2885960186335841006_preal @ bot_bo5832510953819287539_preal @ A2 )
= bot_bo5832510953819287539_preal ) ).
% empty_Diff
thf(fact_740_empty__Diff,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_741_Diff__cancel,axiom,
! [A2: set_Pr3959665804823141767_preal] :
( ( minus_2885960186335841006_preal @ A2 @ A2 )
= bot_bo5832510953819287539_preal ) ).
% Diff_cancel
thf(fact_742_Diff__cancel,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ A2 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_743_Diff__insert0,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ~ ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ( minus_8964681152028453838_preal @ A2 @ ( insert5864466517235936087_preal @ X3 @ B2 ) )
= ( minus_8964681152028453838_preal @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_744_Diff__insert0,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ~ ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ( minus_2831991452770004782_preal @ A2 @ ( insert8416738607890598583_preal @ X3 @ B2 ) )
= ( minus_2831991452770004782_preal @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_745_Diff__insert0,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ~ ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ B2 ) )
= ( minus_2885960186335841006_preal @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_746_Diff__insert0,axiom,
! [X3: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
= ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_747_insert__Diff1,axiom,
! [X3: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ X3 @ B2 )
=> ( ( minus_8964681152028453838_preal @ ( insert5864466517235936087_preal @ X3 @ A2 ) @ B2 )
= ( minus_8964681152028453838_preal @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_748_insert__Diff1,axiom,
! [X3: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ X3 @ B2 )
=> ( ( minus_2831991452770004782_preal @ ( insert8416738607890598583_preal @ X3 @ A2 ) @ B2 )
= ( minus_2831991452770004782_preal @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_749_insert__Diff1,axiom,
! [X3: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ X3 @ B2 )
=> ( ( minus_2885960186335841006_preal @ ( insert2800550009750661751_preal @ X3 @ A2 ) @ B2 )
= ( minus_2885960186335841006_preal @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_750_insert__Diff1,axiom,
! [X3: int,B2: set_int,A2: set_int] :
( ( member_int @ X3 @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_751_diff__self,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ A )
= zero_z580800474297136991d_real ) ).
% diff_self
thf(fact_752_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_753_diff__0__right,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% diff_0_right
thf(fact_754_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_755_diff__zero,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% diff_zero
thf(fact_756_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_757_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ A )
= zero_z580800474297136991d_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_758_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_759_add__diff__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_760_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_761_diff__add__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_762_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_763_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_764_add__diff__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( minus_5539002012860128047d_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_765_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_766_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_767_add__diff__cancel__left_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_768_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_769_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_770_add__diff__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_771_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_772_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_773_add__diff__cancel__right_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_774_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_775_minus__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( minus_5539002012860128047d_real @ A @ B ) )
= ( minus_5539002012860128047d_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_776_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_777_Diff__eq__empty__iff,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( ( minus_2885960186335841006_preal @ A2 @ B2 )
= bot_bo5832510953819287539_preal )
= ( ord_le845437053512250663_preal @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_778_Diff__eq__empty__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( ( minus_minus_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_779_insert__Diff__single,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( insert2800550009750661751_preal @ A @ ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) )
= ( insert2800550009750661751_preal @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_780_insert__Diff__single,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= ( insert_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_781_dbl__simps_I2_J,axiom,
( ( neg_nu8665065731627423525d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% dbl_simps(2)
thf(fact_782_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_783_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_784_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_785_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_786_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_787_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_788_diff__numeral__special_I9_J,axiom,
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real )
= zero_z580800474297136991d_real ) ).
% diff_numeral_special(9)
thf(fact_789_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_790_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_791_verit__minus__simplify_I3_J,axiom,
! [B: dedekind_real] :
( ( minus_5539002012860128047d_real @ zero_z580800474297136991d_real @ B )
= ( uminus7714077491378687647d_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_792_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_793_diff__0,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ zero_z580800474297136991d_real @ A )
= ( uminus7714077491378687647d_real @ A ) ) ).
% diff_0
thf(fact_794_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_795_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_796_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_797_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_798_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_799_uminus__add__conv__diff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( minus_5539002012860128047d_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_800_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_801_diff__minus__eq__add,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ ( uminus7714077491378687647d_real @ B ) )
= ( plus_p4060926892116697567d_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_802_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_803_diff__numeral__special_I12_J,axiom,
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= zero_z580800474297136991d_real ) ).
% diff_numeral_special(12)
thf(fact_804_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_805_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_806_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_807_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_808_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_809_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( numera6440378960330123505d_real @ N ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_810_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_811_diff__numeral__special_I6_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( numera6440378960330123505d_real @ ( inc @ M2 ) ) ) ).
% diff_numeral_special(6)
thf(fact_812_diff__numeral__special_I6_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( inc @ M2 ) ) ) ).
% diff_numeral_special(6)
thf(fact_813_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_814_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_815_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ one_on6069100329679821595d_real )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_816_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_817_diff__eq__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ( minus_5539002012860128047d_real @ A @ B )
= ( minus_5539002012860128047d_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_818_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_819_diff__right__commute,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ C ) @ B )
= ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_820_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_821_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_822_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_823_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_824_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_825_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_826_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: dedekind_real,Z3: dedekind_real] : ( Y5 = Z3 ) )
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( minus_5539002012860128047d_real @ A4 @ B3 )
= zero_z580800474297136991d_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_827_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_828_group__cancel_Osub1,axiom,
! [A2: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( minus_5539002012860128047d_real @ A2 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( minus_5539002012860128047d_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_829_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_830_diff__eq__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( minus_5539002012860128047d_real @ A @ B )
= C )
= ( A
= ( plus_p4060926892116697567d_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_831_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_832_eq__diff__eq,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( A
= ( minus_5539002012860128047d_real @ C @ B ) )
= ( ( plus_p4060926892116697567d_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_833_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_834_add__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_835_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_836_diff__diff__eq2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_837_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_838_diff__add__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C )
= ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_839_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_840_diff__add__eq__diff__diff__swap,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_841_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_842_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_843_add__implies__diff,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ C @ B )
= A )
=> ( C
= ( minus_5539002012860128047d_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_844_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_845_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_846_diff__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C )
= ( minus_5539002012860128047d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_847_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_848_minus__diff__commute,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ B ) @ A )
= ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_849_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_850_insert__Diff__if,axiom,
! [X3: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal,A2: set_se7472626482866788967_preal] :
( ( ( member6785577539073195184_preal @ X3 @ B2 )
=> ( ( minus_8964681152028453838_preal @ ( insert5864466517235936087_preal @ X3 @ A2 ) @ B2 )
= ( minus_8964681152028453838_preal @ A2 @ B2 ) ) )
& ( ~ ( member6785577539073195184_preal @ X3 @ B2 )
=> ( ( minus_8964681152028453838_preal @ ( insert5864466517235936087_preal @ X3 @ A2 ) @ B2 )
= ( insert5864466517235936087_preal @ X3 @ ( minus_8964681152028453838_preal @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_851_insert__Diff__if,axiom,
! [X3: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal,A2: set_Pr6738245714641013959_preal] :
( ( ( member7273733640249002768_preal @ X3 @ B2 )
=> ( ( minus_2831991452770004782_preal @ ( insert8416738607890598583_preal @ X3 @ A2 ) @ B2 )
= ( minus_2831991452770004782_preal @ A2 @ B2 ) ) )
& ( ~ ( member7273733640249002768_preal @ X3 @ B2 )
=> ( ( minus_2831991452770004782_preal @ ( insert8416738607890598583_preal @ X3 @ A2 ) @ B2 )
= ( insert8416738607890598583_preal @ X3 @ ( minus_2831991452770004782_preal @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_852_insert__Diff__if,axiom,
! [X3: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal,A2: set_Pr3959665804823141767_preal] :
( ( ( member7052443626638417872_preal @ X3 @ B2 )
=> ( ( minus_2885960186335841006_preal @ ( insert2800550009750661751_preal @ X3 @ A2 ) @ B2 )
= ( minus_2885960186335841006_preal @ A2 @ B2 ) ) )
& ( ~ ( member7052443626638417872_preal @ X3 @ B2 )
=> ( ( minus_2885960186335841006_preal @ ( insert2800550009750661751_preal @ X3 @ A2 ) @ B2 )
= ( insert2800550009750661751_preal @ X3 @ ( minus_2885960186335841006_preal @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_853_insert__Diff__if,axiom,
! [X3: int,B2: set_int,A2: set_int] :
( ( ( member_int @ X3 @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_int @ A2 @ B2 ) ) )
& ( ~ ( member_int @ X3 @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
= ( insert_int @ X3 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_854_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_855_num__induct,axiom,
! [P: num > $o,X3: num] :
( ( P @ one )
=> ( ! [X: num] :
( ( P @ X )
=> ( P @ ( inc @ X ) ) )
=> ( P @ X3 ) ) ) ).
% num_induct
thf(fact_856_le__num__One__iff,axiom,
! [X3: num] :
( ( ord_less_eq_num @ X3 @ one )
= ( X3 = one ) ) ).
% le_num_One_iff
thf(fact_857_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_858_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_859_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_860_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_861_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_862_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_863_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_864_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_865_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_866_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_867_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_868_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_869_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_870_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_871_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_872_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_873_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_874_diff__shunt__var,axiom,
! [X3: set_Pr3959665804823141767_preal,Y3: set_Pr3959665804823141767_preal] :
( ( ( minus_2885960186335841006_preal @ X3 @ Y3 )
= bot_bo5832510953819287539_preal )
= ( ord_le845437053512250663_preal @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_875_diff__shunt__var,axiom,
! [X3: set_int,Y3: set_int] :
( ( ( minus_minus_set_int @ X3 @ Y3 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_876_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_5539002012860128047d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] : ( plus_p4060926892116697567d_real @ A4 @ ( uminus7714077491378687647d_real @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_877_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_878_diff__conv__add__uminus,axiom,
( minus_5539002012860128047d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] : ( plus_p4060926892116697567d_real @ A4 @ ( uminus7714077491378687647d_real @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_879_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_880_group__cancel_Osub2,axiom,
! [B2: dedekind_real,K: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( B2
= ( plus_p4060926892116697567d_real @ K @ B ) )
=> ( ( minus_5539002012860128047d_real @ A @ B2 )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ K ) @ ( minus_5539002012860128047d_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_881_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_882_Diff__insert,axiom,
! [A2: set_Pr3959665804823141767_preal,A: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal] :
( ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ A @ B2 ) )
= ( minus_2885960186335841006_preal @ ( minus_2885960186335841006_preal @ A2 @ B2 ) @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) ) ).
% Diff_insert
thf(fact_883_Diff__insert,axiom,
! [A2: set_int,A: int,B2: set_int] :
( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% Diff_insert
thf(fact_884_insert__Diff,axiom,
! [A: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ A @ A2 )
=> ( ( insert5864466517235936087_preal @ A @ ( minus_8964681152028453838_preal @ A2 @ ( insert5864466517235936087_preal @ A @ bot_bo6482683055655779539_preal ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_885_insert__Diff,axiom,
! [A: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ A @ A2 )
=> ( ( insert8416738607890598583_preal @ A @ ( minus_2831991452770004782_preal @ A2 @ ( insert8416738607890598583_preal @ A @ bot_bo3249957383623713587_preal ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_886_insert__Diff,axiom,
! [A: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ A @ A2 )
=> ( ( insert2800550009750661751_preal @ A @ ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_887_insert__Diff,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_888_Diff__insert2,axiom,
! [A2: set_Pr3959665804823141767_preal,A: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal] :
( ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ A @ B2 ) )
= ( minus_2885960186335841006_preal @ ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ A @ bot_bo5832510953819287539_preal ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_889_Diff__insert2,axiom,
! [A2: set_int,A: int,B2: set_int] :
( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
= ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_890_Diff__insert__absorb,axiom,
! [X3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ~ ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ( minus_8964681152028453838_preal @ ( insert5864466517235936087_preal @ X3 @ A2 ) @ ( insert5864466517235936087_preal @ X3 @ bot_bo6482683055655779539_preal ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_891_Diff__insert__absorb,axiom,
! [X3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ~ ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ( minus_2831991452770004782_preal @ ( insert8416738607890598583_preal @ X3 @ A2 ) @ ( insert8416738607890598583_preal @ X3 @ bot_bo3249957383623713587_preal ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_892_Diff__insert__absorb,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ~ ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ( minus_2885960186335841006_preal @ ( insert2800550009750661751_preal @ X3 @ A2 ) @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_893_Diff__insert__absorb,axiom,
! [X3: int,A2: set_int] :
( ~ ( member_int @ X3 @ A2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_894_subset__Diff__insert,axiom,
! [A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal,X3: set_Pr3959665804823141767_preal,C2: set_se7472626482866788967_preal] :
( ( ord_le6746582767776264711_preal @ A2 @ ( minus_8964681152028453838_preal @ B2 @ ( insert5864466517235936087_preal @ X3 @ C2 ) ) )
= ( ( ord_le6746582767776264711_preal @ A2 @ ( minus_8964681152028453838_preal @ B2 @ C2 ) )
& ~ ( member6785577539073195184_preal @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_895_subset__Diff__insert,axiom,
! [A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal,X3: produc7488332866442972391_preal,C2: set_Pr6738245714641013959_preal] :
( ( ord_le6462023905233536103_preal @ A2 @ ( minus_2831991452770004782_preal @ B2 @ ( insert8416738607890598583_preal @ X3 @ C2 ) ) )
= ( ( ord_le6462023905233536103_preal @ A2 @ ( minus_2831991452770004782_preal @ B2 @ C2 ) )
& ~ ( member7273733640249002768_preal @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_896_subset__Diff__insert,axiom,
! [A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal,X3: produc5063651213027957927_preal,C2: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ A2 @ ( minus_2885960186335841006_preal @ B2 @ ( insert2800550009750661751_preal @ X3 @ C2 ) ) )
= ( ( ord_le845437053512250663_preal @ A2 @ ( minus_2885960186335841006_preal @ B2 @ C2 ) )
& ~ ( member7052443626638417872_preal @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_897_subset__Diff__insert,axiom,
! [A2: set_int,B2: set_int,X3: int,C2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C2 ) )
& ~ ( member_int @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_898_dbl__def,axiom,
( neg_nu8665065731627423525d_real
= ( ^ [X4: dedekind_real] : ( plus_p4060926892116697567d_real @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_899_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).
% dbl_def
thf(fact_900_numeral__One,axiom,
( ( numera6440378960330123505d_real @ one )
= one_on6069100329679821595d_real ) ).
% numeral_One
thf(fact_901_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_902_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_903_add__One,axiom,
! [X3: num] :
( ( plus_plus_num @ X3 @ one )
= ( inc @ X3 ) ) ).
% add_One
thf(fact_904_subset__insert__iff,axiom,
! [A2: set_se7472626482866788967_preal,X3: set_Pr3959665804823141767_preal,B2: set_se7472626482866788967_preal] :
( ( ord_le6746582767776264711_preal @ A2 @ ( insert5864466517235936087_preal @ X3 @ B2 ) )
= ( ( ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ord_le6746582767776264711_preal @ ( minus_8964681152028453838_preal @ A2 @ ( insert5864466517235936087_preal @ X3 @ bot_bo6482683055655779539_preal ) ) @ B2 ) )
& ( ~ ( member6785577539073195184_preal @ X3 @ A2 )
=> ( ord_le6746582767776264711_preal @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_905_subset__insert__iff,axiom,
! [A2: set_Pr6738245714641013959_preal,X3: produc7488332866442972391_preal,B2: set_Pr6738245714641013959_preal] :
( ( ord_le6462023905233536103_preal @ A2 @ ( insert8416738607890598583_preal @ X3 @ B2 ) )
= ( ( ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ord_le6462023905233536103_preal @ ( minus_2831991452770004782_preal @ A2 @ ( insert8416738607890598583_preal @ X3 @ bot_bo3249957383623713587_preal ) ) @ B2 ) )
& ( ~ ( member7273733640249002768_preal @ X3 @ A2 )
=> ( ord_le6462023905233536103_preal @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_906_subset__insert__iff,axiom,
! [A2: set_Pr3959665804823141767_preal,X3: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ B2 ) )
= ( ( ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ord_le845437053512250663_preal @ ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) ) @ B2 ) )
& ( ~ ( member7052443626638417872_preal @ X3 @ A2 )
=> ( ord_le845437053512250663_preal @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_907_subset__insert__iff,axiom,
! [A2: set_int,X3: int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
= ( ( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
& ( ~ ( member_int @ X3 @ A2 )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_908_Diff__single__insert,axiom,
! [A2: set_Pr3959665804823141767_preal,X3: produc5063651213027957927_preal,B2: set_Pr3959665804823141767_preal] :
( ( ord_le845437053512250663_preal @ ( minus_2885960186335841006_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) ) @ B2 )
=> ( ord_le845437053512250663_preal @ A2 @ ( insert2800550009750661751_preal @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_909_Diff__single__insert,axiom,
! [A2: set_int,X3: int,B2: set_int] :
( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_910_Compl__insert,axiom,
! [X3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( uminus7914843409941547902_preal @ ( insert2800550009750661751_preal @ X3 @ A2 ) )
= ( minus_2885960186335841006_preal @ ( uminus7914843409941547902_preal @ A2 ) @ ( insert2800550009750661751_preal @ X3 @ bot_bo5832510953819287539_preal ) ) ) ).
% Compl_insert
thf(fact_911_Compl__insert,axiom,
! [X3: int,A2: set_int] :
( ( uminus1532241313380277803et_int @ ( insert_int @ X3 @ A2 ) )
= ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% Compl_insert
thf(fact_912_uminus__numeral__One,axiom,
( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% uminus_numeral_One
thf(fact_913_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_914_real__diff__def,axiom,
( minus_5539002012860128047d_real
= ( ^ [R4: dedekind_real,S3: dedekind_real] : ( plus_p4060926892116697567d_real @ R4 @ ( uminus7714077491378687647d_real @ S3 ) ) ) ) ).
% real_diff_def
thf(fact_915_dbl__dec__def,axiom,
( neg_nu7286324685880884523d_real
= ( ^ [X4: dedekind_real] : ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ X4 @ X4 ) @ one_on6069100329679821595d_real ) ) ) ).
% dbl_dec_def
thf(fact_916_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_917_diff__numeral__special_I8_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( neg_nu5286250102780226903d_real @ one @ M2 ) ) ).
% diff_numeral_special(8)
thf(fact_918_diff__numeral__special_I8_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ one @ M2 ) ) ).
% diff_numeral_special(8)
thf(fact_919_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( neg_nu5286250102780226903d_real @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_920_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_921_minus__sub__one__diff__one,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( neg_nu5286250102780226903d_real @ M2 @ one ) ) @ one_on6069100329679821595d_real )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) ) ).
% minus_sub_one_diff_one
thf(fact_922_minus__sub__one__diff__one,axiom,
! [M2: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M2 @ one ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).
% minus_sub_one_diff_one
thf(fact_923_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_924_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_925_add__neg__numeral__special_I3_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( neg_nu5286250102780226903d_real @ M2 @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_926_add__neg__numeral__special_I3_J,axiom,
! [M2: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
= ( neg_numeral_sub_int @ M2 @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_927_add__neg__numeral__special_I2_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ one_on6069100329679821595d_real )
= ( neg_nu5286250102780226903d_real @ one @ M2 ) ) ).
% add_neg_numeral_special(2)
thf(fact_928_add__neg__numeral__special_I2_J,axiom,
! [M2: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
= ( neg_numeral_sub_int @ one @ M2 ) ) ).
% add_neg_numeral_special(2)
thf(fact_929_DiffI,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ A2 )
=> ( ~ ( member6785577539073195184_preal @ C @ B2 )
=> ( member6785577539073195184_preal @ C @ ( minus_8964681152028453838_preal @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_930_DiffI,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ A2 )
=> ( ~ ( member7273733640249002768_preal @ C @ B2 )
=> ( member7273733640249002768_preal @ C @ ( minus_2831991452770004782_preal @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_931_DiffI,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ A2 )
=> ( ~ ( member7052443626638417872_preal @ C @ B2 )
=> ( member7052443626638417872_preal @ C @ ( minus_2885960186335841006_preal @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_932_DiffI,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ A2 )
=> ( ~ ( member_int @ C @ B2 )
=> ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_933_Diff__iff,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ ( minus_8964681152028453838_preal @ A2 @ B2 ) )
= ( ( member6785577539073195184_preal @ C @ A2 )
& ~ ( member6785577539073195184_preal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_934_Diff__iff,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ ( minus_2831991452770004782_preal @ A2 @ B2 ) )
= ( ( member7273733640249002768_preal @ C @ A2 )
& ~ ( member7273733640249002768_preal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_935_Diff__iff,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ ( minus_2885960186335841006_preal @ A2 @ B2 ) )
= ( ( member7052443626638417872_preal @ C @ A2 )
& ~ ( member7052443626638417872_preal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_936_Diff__iff,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
= ( ( member_int @ C @ A2 )
& ~ ( member_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_937_sub__num__simps_I1_J,axiom,
( ( neg_nu5286250102780226903d_real @ one @ one )
= zero_z580800474297136991d_real ) ).
% sub_num_simps(1)
thf(fact_938_sub__num__simps_I1_J,axiom,
( ( neg_numeral_sub_int @ one @ one )
= zero_zero_int ) ).
% sub_num_simps(1)
thf(fact_939_diff__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ M2 @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_940_diff__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ M2 @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_941_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y3: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ V ) @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ Y3 ) )
= ( plus_p4060926892116697567d_real @ ( neg_nu5286250102780226903d_real @ V @ W ) @ Y3 ) ) ).
% semiring_norm(165)
thf(fact_942_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y3: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y3 ) ) ).
% semiring_norm(165)
thf(fact_943_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y3: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ V ) ) @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ W ) @ Y3 ) )
= ( plus_p4060926892116697567d_real @ ( neg_nu5286250102780226903d_real @ W @ V ) @ Y3 ) ) ).
% semiring_norm(166)
thf(fact_944_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y3: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
= ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y3 ) ) ).
% semiring_norm(166)
thf(fact_945_add__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( neg_nu5286250102780226903d_real @ M2 @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_946_add__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ M2 @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_947_add__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ N @ M2 ) ) ).
% add_neg_numeral_simps(2)
thf(fact_948_add__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ N @ M2 ) ) ).
% add_neg_numeral_simps(2)
thf(fact_949_diff__numeral__simps_I4_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( neg_nu5286250102780226903d_real @ N @ M2 ) ) ).
% diff_numeral_simps(4)
thf(fact_950_diff__numeral__simps_I4_J,axiom,
! [M2: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( neg_numeral_sub_int @ N @ M2 ) ) ).
% diff_numeral_simps(4)
thf(fact_951_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_952_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( neg_numeral_sub_int @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_953_diff__numeral__special_I2_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ one_on6069100329679821595d_real )
= ( neg_nu5286250102780226903d_real @ M2 @ one ) ) ).
% diff_numeral_special(2)
thf(fact_954_diff__numeral__special_I2_J,axiom,
! [M2: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int )
= ( neg_numeral_sub_int @ M2 @ one ) ) ).
% diff_numeral_special(2)
thf(fact_955_add__neg__numeral__special_I1_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) )
= ( neg_nu5286250102780226903d_real @ one @ M2 ) ) ).
% add_neg_numeral_special(1)
thf(fact_956_add__neg__numeral__special_I1_J,axiom,
! [M2: num] :
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) )
= ( neg_numeral_sub_int @ one @ M2 ) ) ).
% add_neg_numeral_special(1)
thf(fact_957_DiffE,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ ( minus_8964681152028453838_preal @ A2 @ B2 ) )
=> ~ ( ( member6785577539073195184_preal @ C @ A2 )
=> ( member6785577539073195184_preal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_958_DiffE,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ ( minus_2831991452770004782_preal @ A2 @ B2 ) )
=> ~ ( ( member7273733640249002768_preal @ C @ A2 )
=> ( member7273733640249002768_preal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_959_DiffE,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ ( minus_2885960186335841006_preal @ A2 @ B2 ) )
=> ~ ( ( member7052443626638417872_preal @ C @ A2 )
=> ( member7052443626638417872_preal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_960_DiffE,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ~ ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% DiffE
thf(fact_961_DiffD1,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ ( minus_8964681152028453838_preal @ A2 @ B2 ) )
=> ( member6785577539073195184_preal @ C @ A2 ) ) ).
% DiffD1
thf(fact_962_DiffD1,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ ( minus_2831991452770004782_preal @ A2 @ B2 ) )
=> ( member7273733640249002768_preal @ C @ A2 ) ) ).
% DiffD1
thf(fact_963_DiffD1,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ ( minus_2885960186335841006_preal @ A2 @ B2 ) )
=> ( member7052443626638417872_preal @ C @ A2 ) ) ).
% DiffD1
thf(fact_964_DiffD1,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ( member_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_965_DiffD2,axiom,
! [C: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal,B2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ C @ ( minus_8964681152028453838_preal @ A2 @ B2 ) )
=> ~ ( member6785577539073195184_preal @ C @ B2 ) ) ).
% DiffD2
thf(fact_966_DiffD2,axiom,
! [C: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal,B2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ C @ ( minus_2831991452770004782_preal @ A2 @ B2 ) )
=> ~ ( member7273733640249002768_preal @ C @ B2 ) ) ).
% DiffD2
thf(fact_967_DiffD2,axiom,
! [C: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal,B2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ C @ ( minus_2885960186335841006_preal @ A2 @ B2 ) )
=> ~ ( member7052443626638417872_preal @ C @ B2 ) ) ).
% DiffD2
thf(fact_968_DiffD2,axiom,
! [C: int,A2: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
=> ~ ( member_int @ C @ B2 ) ) ).
% DiffD2
thf(fact_969_neg__numeral__class_Osub__def,axiom,
( neg_nu5286250102780226903d_real
= ( ^ [K2: num,L2: num] : ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ K2 ) @ ( numera6440378960330123505d_real @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_970_neg__numeral__class_Osub__def,axiom,
( neg_numeral_sub_int
= ( ^ [K2: num,L2: num] : ( minus_minus_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_971_sub__inc__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( inc @ N ) @ one )
= ( numeral_numeral_int @ N ) ) ).
% sub_inc_One_eq
thf(fact_972_sub__non__positive,axiom,
! [N: num,M2: num] :
( ( ord_less_eq_int @ ( neg_numeral_sub_int @ N @ M2 ) @ zero_zero_int )
= ( ord_less_eq_num @ N @ M2 ) ) ).
% sub_non_positive
thf(fact_973_sub__non__negative,axiom,
! [N: num,M2: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( neg_numeral_sub_int @ N @ M2 ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% sub_non_negative
thf(fact_974_eq__numeral__iff__iszero_I8_J,axiom,
! [Y3: num] :
( ( one_on6069100329679821595d_real
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y3 ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_975_eq__numeral__iff__iszero_I8_J,axiom,
! [Y3: num] :
( ( one_one_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_976_eq__numeral__iff__iszero_I7_J,axiom,
! [X3: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X3 ) )
= one_on6069100329679821595d_real )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ X3 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_977_eq__numeral__iff__iszero_I7_J,axiom,
! [X3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) )
= one_one_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X3 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_978_dbl__simps_I4_J,axiom,
( ( neg_nu8665065731627423525d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_979_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_980_minus__diff__minus,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( uminus7714077491378687647d_real @ ( minus_5539002012860128047d_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_981_minus__diff__minus,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_982_add__diff__add,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real,D: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D ) )
= ( plus_p4060926892116697567d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ ( minus_5539002012860128047d_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_983_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_984_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_985_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1453304368769599722d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_986_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_987_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_988_one__add__one,axiom,
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_989_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_990_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_991_dbl__simps_I3_J,axiom,
( ( neg_nu8665065731627423525d_real @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_992_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_993_add__neg__numeral__special_I9_J,axiom,
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_994_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_995_diff__numeral__special_I10_J,axiom,
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ one_on6069100329679821595d_real )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_996_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_997_diff__numeral__special_I11_J,axiom,
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_998_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_999_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_1000_iszero__def,axiom,
( ring_1453304368769599722d_real
= ( ^ [Z2: dedekind_real] : ( Z2 = zero_z580800474297136991d_real ) ) ) ).
% iszero_def
thf(fact_1001_iszero__def,axiom,
( ring_1_iszero_int
= ( ^ [Z2: int] : ( Z2 = zero_zero_int ) ) ) ).
% iszero_def
thf(fact_1002_iszero__0,axiom,
ring_1453304368769599722d_real @ zero_z580800474297136991d_real ).
% iszero_0
thf(fact_1003_iszero__0,axiom,
ring_1_iszero_int @ zero_zero_int ).
% iszero_0
thf(fact_1004_not__iszero__numeral,axiom,
! [W: num] :
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ W ) ) ).
% not_iszero_numeral
thf(fact_1005_not__iszero__1,axiom,
~ ( ring_1453304368769599722d_real @ one_on6069100329679821595d_real ) ).
% not_iszero_1
thf(fact_1006_not__iszero__1,axiom,
~ ( ring_1_iszero_int @ one_one_int ) ).
% not_iszero_1
thf(fact_1007_eq__iff__iszero__diff,axiom,
( ( ^ [Y5: dedekind_real,Z3: dedekind_real] : ( Y5 = Z3 ) )
= ( ^ [X4: dedekind_real,Y4: dedekind_real] : ( ring_1453304368769599722d_real @ ( minus_5539002012860128047d_real @ X4 @ Y4 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_1008_eq__iff__iszero__diff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X4: int,Y4: int] : ( ring_1_iszero_int @ ( minus_minus_int @ X4 @ Y4 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_1009_numeral__Bit0,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ ( bit0 @ N ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ ( numera6440378960330123505d_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_1010_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_1011_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_1012_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1013_eq__numeral__iff__iszero_I9_J,axiom,
! [X3: num] :
( ( ( numera6440378960330123505d_real @ X3 )
= zero_z580800474297136991d_real )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ X3 ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_1014_eq__numeral__iff__iszero_I9_J,axiom,
! [X3: num] :
( ( ( numeral_numeral_int @ X3 )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X3 ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_1015_eq__numeral__iff__iszero_I10_J,axiom,
! [Y3: num] :
( ( zero_z580800474297136991d_real
= ( numera6440378960330123505d_real @ Y3 ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_1016_eq__numeral__iff__iszero_I10_J,axiom,
! [Y3: num] :
( ( zero_zero_int
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_1017_not__iszero__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( numeral_numeral_int @ one ) ) ).
% not_iszero_Numeral1
thf(fact_1018_not__iszero__neg__1,axiom,
~ ( ring_1453304368769599722d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% not_iszero_neg_1
thf(fact_1019_not__iszero__neg__1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_iszero_neg_1
thf(fact_1020_eq__numeral__iff__iszero_I1_J,axiom,
! [X3: num,Y3: num] :
( ( ( numeral_numeral_int @ X3 )
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X3 @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_1021_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1022_eq__numeral__iff__iszero_I11_J,axiom,
! [X3: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X3 ) )
= zero_z580800474297136991d_real )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ X3 ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1023_eq__numeral__iff__iszero_I11_J,axiom,
! [X3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) )
= zero_zero_int )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ X3 ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1024_eq__numeral__iff__iszero_I12_J,axiom,
! [Y3: num] :
( ( zero_z580800474297136991d_real
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y3 ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1025_eq__numeral__iff__iszero_I12_J,axiom,
! [Y3: num] :
( ( zero_zero_int
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1026_not__iszero__neg__Numeral1,axiom,
~ ( ring_1453304368769599722d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_1027_not__iszero__neg__Numeral1,axiom,
~ ( ring_1_iszero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_1028_eq__numeral__iff__iszero_I2_J,axiom,
! [X3: num,Y3: num] :
( ( ( numera6440378960330123505d_real @ X3 )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y3 ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ X3 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_1029_eq__numeral__iff__iszero_I2_J,axiom,
! [X3: num,Y3: num] :
( ( ( numeral_numeral_int @ X3 )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X3 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_1030_eq__numeral__iff__iszero_I3_J,axiom,
! [X3: num,Y3: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X3 ) )
= ( numera6440378960330123505d_real @ Y3 ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ X3 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_1031_eq__numeral__iff__iszero_I3_J,axiom,
! [X3: num,Y3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) )
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X3 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_1032_eq__numeral__iff__iszero_I4_J,axiom,
! [X3: num,Y3: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X3 ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y3 ) ) )
= ( ring_1453304368769599722d_real @ ( neg_nu5286250102780226903d_real @ Y3 @ X3 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_1033_eq__numeral__iff__iszero_I4_J,axiom,
! [X3: num,Y3: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ Y3 ) ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ Y3 @ X3 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_1034_eq__numeral__iff__iszero_I6_J,axiom,
! [Y3: num] :
( ( one_on6069100329679821595d_real
= ( numera6440378960330123505d_real @ Y3 ) )
= ( ring_1453304368769599722d_real @ ( neg_nu5286250102780226903d_real @ one @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_1035_eq__numeral__iff__iszero_I6_J,axiom,
! [Y3: num] :
( ( one_one_int
= ( numeral_numeral_int @ Y3 ) )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y3 ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_1036_eq__numeral__iff__iszero_I5_J,axiom,
! [X3: num] :
( ( ( numera6440378960330123505d_real @ X3 )
= one_on6069100329679821595d_real )
= ( ring_1453304368769599722d_real @ ( neg_nu5286250102780226903d_real @ X3 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_1037_eq__numeral__iff__iszero_I5_J,axiom,
! [X3: num] :
( ( ( numeral_numeral_int @ X3 )
= one_one_int )
= ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X3 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_1038_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_nu5286250102780226903d_real @ one @ ( bit0 @ L ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_1039_sub__num__simps_I2_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit0 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bitM @ L ) ) ) ) ).
% sub_num_simps(2)
thf(fact_1040_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7286324685880884523d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_1041_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_1042_remove__def,axiom,
( remove5625372653177682828_preal
= ( ^ [X4: produc5063651213027957927_preal,A6: set_Pr3959665804823141767_preal] : ( minus_2885960186335841006_preal @ A6 @ ( insert2800550009750661751_preal @ X4 @ bot_bo5832510953819287539_preal ) ) ) ) ).
% remove_def
thf(fact_1043_remove__def,axiom,
( remove_int
= ( ^ [X4: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).
% remove_def
thf(fact_1044_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1045_member__remove,axiom,
! [X3: set_Pr3959665804823141767_preal,Y3: set_Pr3959665804823141767_preal,A2: set_se7472626482866788967_preal] :
( ( member6785577539073195184_preal @ X3 @ ( remove3864610947230394476_preal @ Y3 @ A2 ) )
= ( ( member6785577539073195184_preal @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1046_member__remove,axiom,
! [X3: produc7488332866442972391_preal,Y3: produc7488332866442972391_preal,A2: set_Pr6738245714641013959_preal] :
( ( member7273733640249002768_preal @ X3 @ ( remove616724958711913164_preal @ Y3 @ A2 ) )
= ( ( member7273733640249002768_preal @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1047_member__remove,axiom,
! [X3: produc5063651213027957927_preal,Y3: produc5063651213027957927_preal,A2: set_Pr3959665804823141767_preal] :
( ( member7052443626638417872_preal @ X3 @ ( remove5625372653177682828_preal @ Y3 @ A2 ) )
= ( ( member7052443626638417872_preal @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1048_member__remove,axiom,
! [X3: int,Y3: int,A2: set_int] :
( ( member_int @ X3 @ ( remove_int @ Y3 @ A2 ) )
= ( ( member_int @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_1049_div__by__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% div_by_0
thf(fact_1050_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1051_div__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% div_0
thf(fact_1052_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1053_div__by__1,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% div_by_1
thf(fact_1054_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1055_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1056_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1057_div__self,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ).
% div_self
thf(fact_1058_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1059_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_1060_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_1061_sub__num__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit1 @ K ) @ one )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% sub_num_simps(5)
thf(fact_1062_sub__num__simps_I4_J,axiom,
! [K: num] :
( ( neg_numeral_sub_int @ ( bit0 @ K ) @ one )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% sub_num_simps(4)
thf(fact_1063_dbl__inc__simps_I3_J,axiom,
( ( neg_nu2849763295382964967d_real @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_1064_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_1065_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_nu5286250102780226903d_real @ one @ ( bit1 @ L ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_1066_sub__num__simps_I3_J,axiom,
! [L: num] :
( ( neg_numeral_sub_int @ one @ ( bit1 @ L ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ L ) ) ) ) ).
% sub_num_simps(3)
thf(fact_1067_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1068_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_1069_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1070_num_Oexhaust,axiom,
! [Y3: num] :
( ( Y3 != one )
=> ( ! [X22: num] :
( Y3
!= ( bit0 @ X22 ) )
=> ~ ! [X33: num] :
( Y3
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1071_inc_Osimps_I3_J,axiom,
! [X3: num] :
( ( inc @ ( bit1 @ X3 ) )
= ( bit0 @ ( inc @ X3 ) ) ) ).
% inc.simps(3)
thf(fact_1072_inc_Osimps_I2_J,axiom,
! [X3: num] :
( ( inc @ ( bit0 @ X3 ) )
= ( bit1 @ X3 ) ) ).
% inc.simps(2)
thf(fact_1073_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_1074_numeral__Bit1,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ ( bit1 @ N ) )
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ ( numera6440378960330123505d_real @ N ) ) @ one_on6069100329679821595d_real ) ) ).
% numeral_Bit1
thf(fact_1075_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_1076_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_1077_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_1078_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_1079_numeral__BitM,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ ( bitM @ N ) )
= ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ ( bit0 @ N ) ) @ one_on6069100329679821595d_real ) ) ).
% numeral_BitM
thf(fact_1080_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_1081_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_1082_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_1083_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_1084_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_1085_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_1086_divide__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1087_divide__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ C @ A )
= ( divide9119111104558704680d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1088_divide__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ C )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1089_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1090_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1091_division__ring__divide__zero,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% division_ring_divide_zero
thf(fact_1092_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1093_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_1094_divide__self__if,axiom,
! [A: dedekind_real] :
( ( ( A = zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= zero_z580800474297136991d_real ) )
& ( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ) ).
% divide_self_if
thf(fact_1095_divide__self,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ).
% divide_self
thf(fact_1096_one__eq__divide__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( one_on6069100329679821595d_real
= ( divide9119111104558704680d_real @ A @ B ) )
= ( ( B != zero_z580800474297136991d_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1097_divide__eq__1__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= one_on6069100329679821595d_real )
= ( ( B != zero_z580800474297136991d_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1098_divide__minus1,axiom,
! [X3: dedekind_real] :
( ( divide9119111104558704680d_real @ X3 @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ X3 ) ) ).
% divide_minus1
thf(fact_1099_add__divide__distrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ C ) @ ( divide9119111104558704680d_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_1100_minus__divide__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_1101_minus__divide__divide,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_1102_minus__divide__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ A @ ( uminus7714077491378687647d_real @ B ) ) ) ).
% minus_divide_right
thf(fact_1103_right__inverse__eq,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ A @ B )
= one_on6069100329679821595d_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_1104_nonzero__minus__divide__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ A @ ( uminus7714077491378687647d_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1105_nonzero__minus__divide__divide,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1106_divide__eq__minus__1__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( ( B != zero_z580800474297136991d_real )
& ( A
= ( uminus7714077491378687647d_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1107_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_1108_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_1109_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1110_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1111_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1112_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1113_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1114_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1115_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1116_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1117_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_1118_divides__aux__eq,axiom,
! [Q: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_1119_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_1120_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5403075989570733136od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_1121_divmod__algorithm__code_I2_J,axiom,
! [N: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1122_divmod__algorithm__code_I2_J,axiom,
! [N: num] :
( ( unique5403075989570733136od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1123_divmod__algorithm__code_I1_J,axiom,
! [M2: num] :
( ( unique5405566460079783412od_nat @ M2 @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(1)
thf(fact_1124_divmod__algorithm__code_I1_J,axiom,
! [M2: num] :
( ( unique5403075989570733136od_int @ M2 @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(1)
thf(fact_1125_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( unique5405566460079783412od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_1126_divmod__algorithm__code_I6_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5403075989570733136od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M2 @ N )
=> ( ( unique5403075989570733136od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique4034149617335721244ep_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( unique5403075989570733136od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_1127_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 ) )
=> ( ! [K3: int,L3: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L3 ) )
=> ( ( ~ ( ( member_int @ K3 @ ( 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 @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K3 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_1128_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1129_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1130_dvd__0__left__iff,axiom,
! [A: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ zero_z580800474297136991d_real @ A )
= ( A = zero_z580800474297136991d_real ) ) ).
% dvd_0_left_iff
thf(fact_1131_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_1132_dvd__0__right,axiom,
! [A: dedekind_real] : ( dvd_dv2022643831595171964d_real @ A @ zero_z580800474297136991d_real ) ).
% dvd_0_right
thf(fact_1133_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_1134_dvd__add__triv__left__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ A @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( dvd_dv2022643831595171964d_real @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1135_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1136_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1137_dvd__add__triv__right__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ A @ ( plus_p4060926892116697567d_real @ B @ A ) )
= ( dvd_dv2022643831595171964d_real @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1138_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1139_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1140_minus__dvd__iff,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ ( uminus7714077491378687647d_real @ X3 ) @ Y3 )
= ( dvd_dv2022643831595171964d_real @ X3 @ Y3 ) ) ).
% minus_dvd_iff
thf(fact_1141_minus__dvd__iff,axiom,
! [X3: int,Y3: int] :
( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
= ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% minus_dvd_iff
thf(fact_1142_dvd__minus__iff,axiom,
! [X3: dedekind_real,Y3: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ X3 @ ( uminus7714077491378687647d_real @ Y3 ) )
= ( dvd_dv2022643831595171964d_real @ X3 @ Y3 ) ) ).
% dvd_minus_iff
thf(fact_1143_dvd__minus__iff,axiom,
! [X3: int,Y3: int] :
( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
= ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% dvd_minus_iff
thf(fact_1144_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1145_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_1146_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_1147_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_1148_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_1149_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_1150_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_1151_div__add,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_1152_div__add,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_1153_div__diff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_1154_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_1155_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_1156_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_1157_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_1158_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_1159_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_1160_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_1161_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1162_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1163_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_1164_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_1165_dvd__add__right__iff,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ A @ B )
=> ( ( dvd_dv2022643831595171964d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( dvd_dv2022643831595171964d_real @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1166_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1167_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_1168_dvd__add__left__iff,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ A @ C )
=> ( ( dvd_dv2022643831595171964d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( dvd_dv2022643831595171964d_real @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1169_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1170_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_1171_dvd__add,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ A @ B )
=> ( ( dvd_dv2022643831595171964d_real @ A @ C )
=> ( dvd_dv2022643831595171964d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1172_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1173_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_1174_zdvd__zdiffD,axiom,
! [K: int,M2: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M2 @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M2 ) ) ) ).
% zdvd_zdiffD
thf(fact_1175_dvd__diff,axiom,
! [X3: dedekind_real,Y3: dedekind_real,Z: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ X3 @ Y3 )
=> ( ( dvd_dv2022643831595171964d_real @ X3 @ Z )
=> ( dvd_dv2022643831595171964d_real @ X3 @ ( minus_5539002012860128047d_real @ Y3 @ Z ) ) ) ) ).
% dvd_diff
thf(fact_1176_dvd__diff,axiom,
! [X3: int,Y3: int,Z: int] :
( ( dvd_dvd_int @ X3 @ Y3 )
=> ( ( dvd_dvd_int @ X3 @ Z )
=> ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% dvd_diff
thf(fact_1177_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_1178_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_1179_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_1180_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1181_dvd__div__eq__iff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1182_dvd__0__left,axiom,
! [A: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ zero_z580800474297136991d_real @ A )
=> ( A = zero_z580800474297136991d_real ) ) ).
% dvd_0_left
thf(fact_1183_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_1184_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1185_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1186_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_1187_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_1188_one__dvd,axiom,
! [A: dedekind_preal] : ( dvd_dv5067560330434688496_preal @ one_on9143529541772854033_preal @ A ) ).
% one_dvd
thf(fact_1189_one__dvd,axiom,
! [A: dedekind_real] : ( dvd_dv2022643831595171964d_real @ one_on6069100329679821595d_real @ A ) ).
% one_dvd
thf(fact_1190_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_1191_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_1192_dvd__field__iff,axiom,
( dvd_dv2022643831595171964d_real
= ( ^ [A4: dedekind_real,B3: dedekind_real] :
( ( A4 = zero_z580800474297136991d_real )
=> ( B3 = zero_z580800474297136991d_real ) ) ) ) ).
% dvd_field_iff
thf(fact_1193_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1194_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1195_dvd__div__eq__0__iff,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ B @ A )
=> ( ( ( divide9119111104558704680d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1196_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1197_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1198_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1199_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1200_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1201_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1202_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1203_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1204_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1205_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1206_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1207_dvd__neg__div,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ B @ A )
=> ( ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_1208_dvd__neg__div,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_1209_dvd__div__neg,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( dvd_dv2022643831595171964d_real @ B @ A )
=> ( ( divide9119111104558704680d_real @ A @ ( uminus7714077491378687647d_real @ B ) )
= ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_1210_dvd__div__neg,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_1211_zdvd__antisym__nonneg,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M2 @ N )
=> ( ( dvd_dvd_int @ N @ M2 )
=> ( M2 = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1212_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1213_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1214_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_1215_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_1216_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_1217_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_1218_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_1219_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_1220_even__minus,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_1221_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( unique5405566460079783412od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_1222_divmod__algorithm__code_I7_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_num @ M2 @ N )
=> ( ( unique5403075989570733136od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less_num @ M2 @ N )
=> ( ( unique5403075989570733136od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique4034149617335721244ep_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( unique5403075989570733136od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_1223_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_1224_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_1225_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_1226_even__mask__div__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% even_mask_div_iff
thf(fact_1227_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1228_mult__zero__left,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% mult_zero_left
thf(fact_1229_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1230_mult__zero__right,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% mult_zero_right
thf(fact_1231_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1232_mult__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1233_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1234_mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1235_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1236_mult__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1237_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1238_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1239_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1240_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1241_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ W ) @ Z ) )
= ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1242_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1243_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1244_numeral__times__numeral,axiom,
! [M2: num,N: num] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( times_times_num @ M2 @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1245_numeral__times__numeral,axiom,
! [M2: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1246_numeral__times__numeral,axiom,
! [M2: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1247_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1248_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1249_mult_Oright__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.right_neutral
thf(fact_1250_mult_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.right_neutral
thf(fact_1251_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1252_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1253_mult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% mult_1
thf(fact_1254_mult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% mult_1
thf(fact_1255_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_1256_mult__minus__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( uminus7714077491378687647d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1257_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1258_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1259_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1260_zdvd__reduce,axiom,
! [K: int,N: int,M2: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M2 ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1261_zdvd__period,axiom,
! [A: int,D: int,X3: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_1262_zdvd__mult__cancel,axiom,
! [K: int,M2: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M2 ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1263_preal__mult__1,axiom,
! [Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z )
= Z ) ).
% preal_mult_1
thf(fact_1264_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1265_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1266_real__mult__1,axiom,
! [Z: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z )
= Z ) ).
% real_mult_1
thf(fact_1267_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_1268_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_1269_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_1270_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_1271_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y2: dedekind_preal,X2: dedekind_preal,Y1: dedekind_preal,X3: dedekind_preal,Y3: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y2 )
= ( plus_p3173629198307831117_preal @ X2 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ X1 ) @ ( times_3000655703912201937_preal @ Y3 @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ Y2 ) @ ( times_3000655703912201937_preal @ Y3 @ X2 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ X2 ) @ ( times_3000655703912201937_preal @ Y3 @ Y2 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ Y1 ) @ ( times_3000655703912201937_preal @ Y3 @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1272_preal__add__mult__distrib2,axiom,
! [W: dedekind_preal,X3: dedekind_preal,Y3: dedekind_preal] :
( ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X3 ) @ ( times_3000655703912201937_preal @ W @ Y3 ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1273_preal__add__mult__distrib,axiom,
! [X3: dedekind_preal,Y3: dedekind_preal,W: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X3 @ Y3 ) @ W )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X3 @ W ) @ ( times_3000655703912201937_preal @ Y3 @ W ) ) ) ).
% preal_add_mult_distrib
thf(fact_1274_real__add__mult__distrib,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,W: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ Z1 @ Z22 ) @ W )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ Z1 @ W ) @ ( times_2157731159493324635d_real @ Z22 @ W ) ) ) ).
% real_add_mult_distrib
% Conjectures (1)
thf(conj_0,conjecture,
( ( ord_le2716243287969276982d_real @ ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ x1 @ y1 ) @ bot_bo5832510953819287539_preal ) ) ) @ ( dedekind_Abs_Real @ ( image_7293650632215208851_preal @ dedekind_realrel @ ( insert2800550009750661751_preal @ ( produc5992578386459872151_preal @ x2 @ y2 ) @ bot_bo5832510953819287539_preal ) ) ) )
= ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ x1 @ y2 ) @ ( plus_p3173629198307831117_preal @ x2 @ y1 ) ) ) ).
%------------------------------------------------------------------------------